Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Mullins, in a series of papers, developed a surface dynamics for phase interfaces whose evolution is controlled by mass diffusion within the interface. It is our purpose here to embed Mullin's theory within a general framework based on balance laws for mass and capillary forces in conjunction with a version of the second law, appropriate to a purely mechanical theory, which asserts that the rate at which the free energy increases cannot be greater than the energy inflow plus the power supplied. We develop an appropriate constitutive theory, and deduce general and approximate equations for the evolution of the interface.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   We have found that when suprathreshold cathodal stimuli were applied to the epicardium of canine ventricle, impulse propagation originated at a "virtual cathode" with dimensions greater than those of the physical cathode. We report the two- dimensional geometry of the virtual cathode as a function of stimulus strength; the results are compared with the predictions of an anisotropic, bidomain model of cardiac conduction recently developed in our laboratories. Data were collected in six pentobarbital-anesthetized dogs by using a small plaque electrode sewn to the left ventricular epicardium. Arrival times at closely spaced bipolar electrodes oriented radially around a central cathode were obtained as a function of stimulus strength and fiber orientation. The dimensions of the virtual cathode were determined by linear back- extrapolation of arrival times to the time of stimulation. The directional dependence of the conduction velocity was consistent with previous reports: at 1 mA, longitudinal (0- degrees) and transverse (90-degrees) velocities were 0.60 +/- 0.03 and 0.29 +/- 0.02 m/sec, respectively. At 7 mA, the longitudinal velocity was 0.75 +/- 0.05 m/sec, whereas there was no significant change in the transverse velocity. In contrast to conduction velocity, the virtual cathode was smallest in the longitudinal orientation and largest between 45-degrees and 60-degrees. Virtual cathode size was dependent on both orientation and stimulus strength: at 0-degrees, the virtual cathode was small (approximately 1 mm) and relatively constant over the range of 1-7 mA; at oblique orientations (45- degrees-90-degrees), it displayed a roughly logarithmic dependence on stimulus strength, approximately 1 mm at 1 mA and approximately 3 mm at 7 mA. The bidomain, anisotropic model reproduced both the stimulus strength and the fiber-orientation dependence of the virtual cathode geometry when the intracellular and extracellular anisotropies were 10:1 and 4:1, respectively, but not when the two anisotropies were equal. We suggest that the virtual cathode provides a direct measure of the determinants of cardiac activation; its complex geometry appears to reflect the bidomain, anisotropic nature of cardiac muscle.

Abstract:   Simple inequalities are presented for the viscosity solution of a Hamilton-Jacobi equation in N space dimension when neither the initial data nor the Hamiltonian need be convex (or concave). The initial data are uniformly Lipschitz and can be written as the sum of a convex function in a group of variables and a concave function in the remaining variables, therefore including the nonconvex Riemann problem. The inequalities become equalities wherever a "maxmin" equals a "minmax" and thus a representation formula for this problem is then obtained, generalizing the classical Hopf's formulas.

Abstract:   Limit analysis of prismatic torsion bars was the earliest attempt to apply plasticity theory to a continuum. The simplicity of the problem made it feasible to use the two- dimensional Prandtl stress function, defined for the elastic torsion problems, for the plastic stress distributions as well. The gradient of the stress functions for plastic torsion has a constant magnitude, and hence a function of this type assumes the profile of a sand hill. This sand hill analogy of Nadai (1950, The Theory of Flow and Fracture of Solids, McGraw-Hill, U.K.) gave a visual sense of possible non-smoothness of such stress functions and thus discontinuous stress fields. Many stress functions of plastic torsion for relatively simple cross-sections have been constructed graphically. However, collapse modes in terms of warping functions were much less reported. In this paper, we shall establish a duality theorem which relates the correct stress function to the correct warping function, thus providing the means to obtain complete static and kinematic solutions. This dual variational principle leads naturally to a general numerical algorithm which guarantees convergence and accuracy. In this paper, we shall only present three exact solutions to verify the theorem, to demonstrate the possible non-smooth feature of the solutions and to reiterate this effective dual variational approach to limit analysis in general.

Abstract:   We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set GAMMA-0 a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

Abstract:   We construct a unique weak solution of the nonlinear PDE which asserts each level set evolves in time according to its mean curvature. This weak solution allows us then to define for any compact set GAMMA-0 a unique generalized motion by mean curvature, existing for all time. We investigate the various geometric properties and pathologies of this evolution.

Abstract:   Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite-difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first- arrival-time field. The upwind finite-difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first-order upwind finite-difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

Abstract:   Seismic traveltimes can be computed efficiently on a regular grid by an upwind finite-difference method. The method solves a conservation law that describes changes in the gradient components of the traveltime field. The traveltime field itself is easily obtained from the solution of the conservation law by numerical integration. The conservation law derives from the eikonal equation, and its solution depicts the first- arrival-time field. The upwind finite-difference scheme can be implemented in fully vectorized form, in contrast to a similar scheme proposed recently by Vidale. The resulting traveltime field is useful both in Kirchhoff migration and modeling and in seismic tomography. Many reliable methods exist for the numerical solution of conservation laws, which appear in fluid mechanics as statements of the conservation of mass, momentum, etc. A first-order upwind finite-difference scheme proves accurate enough for seismic applications. Upwind schemes are stable because they mimic the behavior of fluid flow by using only information taken from upstream in the fluid. Other common difference schemes are unstable, or overly dissipative, at shocks (discontinuities in flow variables), which are time gradient discontinuities in our approach to solving the eikonal equation.

Abstract:   We derive an "eikonal-curvature" equation to describe the propagation of action potential wavefronts in myocardium. This equation is used to study the effects of fiber orientation on propagation in the myocardial wall. There are significant computational advantages to the use of an eikonal-curvature equation over a full ionic model of action potential spread. With this model, it is shown that the experimentally observed misalignment of spreading action potential "ellipses" from fiber orientation in level myocardial surfaces is adequately explained by the rotation of fiber orientation through the myocardial wall. Additionally, it is shown that apparently high propagation velocities on the epicardial and endocardial surfaces are the result of propagation into the midwall region and acceleration along midwall fibers before reemergence at an outer surface at a time preceding what could be accomplished with propagation along the surface alone.

Abstract:   Hamilton-Jacobi (H-J) equations are frequently encountered in applications, e.g., in control theory and differential games. H-J equations are closely related to hyperbolic conservation laws-in one space dimension the former is simply the integrated version of the latter. Similarity also exists for the multidimensional case, and this is helpful in the design of difference approximations. In this paper high-order essentially nonoscillatory (ENO) schemes for H-J equations are investigated, which yield uniform high-order accuracy in smooth regions and sharply resolve discontinuities in the derivatives. The ENO scheme construction procedure is adapted from that for hyperbolic conservation laws. The schemes are numerically tested on a variety of one-dimensional and two-dimensional problems, including a problem related to control optimization, and high-order accuracy in smooth regions, good resolution of discontinuities in the derivatives, and convergence to viscosity solutions are observed.

Abstract:   We study shock reflection for the two 2D Burgers equation. This model equation is an asymptotic limit of the Euler equations, and retains many of the features of the full equations. A von Neumann type analysis shows that the 2D Burgers equation has detachment, sonic, and Crocco points in complete analogy with gas dynamics. Numerical solutions support the detachment/sonic criterion for transition from regular to Mach reflection. There is also strong numerical evidence that the reflected shock in the 2D Burgers Mach reflection forms a smooth wave near the Mach stem, as proposed by Colella and Henderson in their study of the Euler equations.

Abstract:   This paper is concerned with the asymptotic behavior, as epsilon arrow pointing down and to the right 0, of the solution (u(epsilon), upsilon(epsilon)) Of the second initial-boundary value problem of the reaction-diffusion system: [GRAPHICS] where gamma > 0 is a constant. When upsilon is-an-element-of (- 2 square-root 3/9, 2 square-root 3/9), f is bistable in the sense that the ordinary differential equation u(t) = f(u, upsilon) has two stable solutions u = h-(upsilon) and u = h+(upsilon) and one unstable solution u = h0(upsilon), where h- (upsilon) , h0(upsilon) , and h+(upsilon) are the three solutions of the algebraic equation f(u, upsilon) = 0 . We show that, when the initial data of upsilon is in the interval (-2 square-root 3/9, 2 square-root 3/9) , the solution (u(epsilon), upsilon(epsilon)) of the system tends to a limit (u, upsilon) which is a solution of a free boundary problem, as long as the free boundary problem has a unique classical solution. The function u is a ''phase'' function in the sense that it coincides with h+(upsilon) in one region OMEGA+ and with h- (upsilon) in another region OMEGA- . The common boundary (free boundary or interface) of the two regions OMEGA- and OMEGA+ moves with a normal velocity equal to V(upsilon), where V(.) is a function that can be calculated. The local (in time) existence of a unique classical solution to the free boundary problem is also established. Further we show that if initially u(., 0) - h0(upsilon(.,0)) takes both positive and negative values, then an interface will develop in a short time O(epsilon\ln epsilon\) near the hypersurface where u(x, 0) - h0(upsilon(x, 0)) = 0.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   Because the stress resulting from compositional inhomogeneities are long range, the local stress, diffusional flux and equilibrium conditions at a point depend on the entire composition distribution in a specimen. For a thin plate with a one-dimensional composition profile, this dependence is simple; the local stress depends on the local composition and on both the average composition and the first moment of the composition profile, neither of which are local. A theory of diffusion and equilibrium in a thin plate is developed, based on a free energy that depends on composition, its gradients and strain, and has a term for chemical effects at the plate boundary. Under certain assumptions, a standard diffusion equation is derived, with all of the non-local stress effects in the boundary conditions. Solutions are altered by these new conditions. Spontaneous bending is often a natural result of diffusion.

Abstract:   This paper discusses a generalized Stefan problem which allows for supercooling and superheating and for capillarity in the interface between phases. Simple solutions are obtained indicating the chief differences between this problem and the classical Stefan problem. A weak formulation of the general problem is given.

Abstract:   A stable algorithm is proposed for image restoration based on the "mean curvature motion" equation. Existence and uniqueness of the "viscosity" solution of the equation are proved, a L(infinity) stable algorithm is given, experimental results are shown, and the subjacent vision model is compared with those introduced recently by several vision researchers. The algorithm presented appears to be the sharpest possible among the multiscale image smoothing methods preserving uniqueness and stability.

Abstract:   Recent theoretical advances in the mathematical treatment of geometric interface motion make more tractable the theory of a wide variety of materials science problems where the interface velocity is not controlled by long-range-diffusion. Among the interface motion problems that can be modelled as geometric are certain types of phase changes, crystal growth, domain growth, grain growth. ion beam and chemical etching, and coherency stress driven interface migration. We provide an introduction to nine mathematical methods for solving such problems, give the limits of applicability of the methods, and discuss the relations among them theoretically and their uses in computation. Comparisons of some of them are made by displaying how the same physical problems are treated in the various applicable methods.

Abstract:   The notion of viscosity solutions of scalar fully nonlinear partial differential equations of second order provides a framework in which startling comparison and uniqueness theorems, existence theorems, and theorems about continuous dependence may now be proved by very efficient and striking arguments. The range of important applications of these results is enormous. This article is a self-contained exposition of the basic theory of viscosity solutions.

Abstract:   A fully nonlinear evolution equation governing the propagation of a premixed flame through a large-scale spatially periodic shear flow is derived, and steady-state solutions are obtained numerically. The gas density is assumed to be constant across the flame, but the local normal burning speed is allowed to vary with the local strain and curvature along the flame front in order to investigate the influence of the length scale of the external flow on the average propagation speed of the wrinkled flame. At fixed values of the amplitude of the flow- field variations an increase in the length scale (relative to the flame thickness) is found to result in an increase in the average flame propagation speed, in accordance with the predictions of earlier theoretical investigations and with experimental observations for the regime of large-scale turbulence. The propagation speed of the wrinkled flame is calculated to exhibit the experimentally observed bending effect, the tendency of the rate of change of the burning velocity to decrease with increasing turbulence intensity at low fixed turbulence Reynolds numbers. It is shown also how the average flame speed depends on the ratio of the transverse to longitudinal length scale associated with the periodic flow.

Abstract:   A fully nonlinear evolution equation governing the propagation of a premixed flame through a large-scale spatially periodic shear flow is derived, and steady-state solutions are obtained numerically. The gas density is assumed to be constant across the flame, but the local normal burning speed is allowed to vary with the local strain and curvature along the flame front in order to investigate the influence of the length scale of the external flow on the average propagation speed of the wrinkled flame. At fixed values of the amplitude of the flow- field variations an increase in the length scale (relative to the flame thickness) is found to result in an increase in the average flame propagation speed, in accordance with the predictions of earlier theoretical investigations and with experimental observations for the regime of large-scale turbulence. The propagation speed of the wrinkled flame is calculated to exhibit the experimentally observed bending effect, the tendency of the rate of change of the burning velocity to decrease with increasing turbulence intensity at low fixed turbulence Reynolds numbers. It is shown also how the average flame speed depends on the ratio of the transverse to longitudinal length scale associated with the periodic flow.

Abstract:   This paper reports the progress made in multiquadrics (MQ) as a spatial approximation scheme for systems of governing equations of engineering and physics by minimizing the spatial truncation errors without excessive refinement. Although MQ is defined over the general n-dimensional real space, this paper is limited to two spatial dimensions defined over a general non- convex irregular region containing inhomogeneously scattered nodes. We have developed a strictly conservative interpolation scheme over such irregular regions from which the partial derivative estimates are obtained. In addition, we developed a non-iterative scheme to be used with domain decomposition to ensure derivative continuity over contiguous regions. Jump discontinuities for shock and material interfaces are likewise treated by appropriate modification of the algorithm. We have compared the relative errors of the derivative estimates defined over an irregular region consisting of inhomogeneously scattered nodes obtained by the MQ and Voronoi mesh schemes. The MQ relative errors of the derivative estimates are three orders of magnitude better than those obtained from the Voronoi mesh method. (In our previous papers, we have shown that MQ is superior in its derivative estimates over regular gridded regions.) We have also used MQ to estimate derivatives within a very narrow "shock" region with similar excellent results. While comparing spatial approximation schemes for PDE's, we found the MQ results to be superior in accuracy and were calculated by far fewer operations than standard finite difference schemes. Other authors have likewise used MQ successfully to solve integral equations.

Abstract:   For realistic interfacial energies, the equations of anisotropic motion-by-curvature exhibit backward-parabolic behavior over portions of their domain, thereby inducing phenomena such as the formation of facets and wrinkles. In this paper, a physically consistent regularized equation that may be used to analyze such phenomena is derived.

Abstract:   We describe all multiscale causal, local, stable and shape preserving filterings. This classification contains the classical "morphological" operators, and some new ones.

Abstract:   Stationary premixed flames in dual-source flow are considered. The significant features of the dual-source system are that the sources are of finite strength, and that a stagnation point is located between the sources. A new mathematical model for front propagation and advection is introduced that tracks the front along streamlines. The equations for the stationary fronts of the dual-source system are solved numerically. The assumption of constant-density potential flow is made to simplify the problem and to illustrate the effects of the geometry alone. It is shown that for sufficiently slow burning velocity (or equivalently, small source separation), three stationary states exist for closed, free flames, but one of them is unstable. In addition, several types of burner-attached flames are observed. Quasi-stationary evolution of a closed, free flame exhibits a change of topology and hysteresis. Nonclosed flames are predicted if local extinction due to flow strain is allowed.

Abstract:   Stationary premixed flames in dual-source flow are considered. The significant features of the dual-source system are that the sources are of finite strength, and that a stagnation point is located between the sources. A new mathematical model for front propagation and advection is introduced that tracks the front along streamlines. The equations for the stationary fronts of the dual-source system are solved numerically. The assumption of constant-density potential flow is made to simplify the problem and to illustrate the effects of the geometry alone. It is shown that for sufficiently slow burning velocity (or equivalently, small source separation), three stationary states exist for closed, free flames, but one of them is unstable. In addition, several types of burner-attached flames are observed. Quasi-stationary evolution of a closed, free flame exhibits a change of topology and hysteresis. Nonclosed flames are predicted if local extinction due to flow strain is allowed.

Abstract:   We study the limiting behavior of solutions to appropriately rescaled versions of the Allen-Cahn equation, a simplified model for dynamic phase transitions. We rigorously establish the existence in the limit of a phase-antiphase interface evolving according to mean curvature motion. This assertion is valid for all positive time, the motion interpreted in the generalized sense of Evans-Spruck and Chen-Giga-Goto after the onset of geometric singularities.

Abstract:   This paper describes a method for tracking contact discontinuities and material interfaces that arise in the solution of hyperbolic systems of conservation laws. Numerical results arc presented to show that the fronts are resolved to within a mesh interval and smooth portions of the solution are computed to within the accuracy of the underlying numerical scheme.

Abstract:   A numerical study was conducted to understand how a vortex, when convected at moderate speeds across a premixed flame, can induce velocities that pull the flame along with the vortex, causing flame elongation and unsteady flame stretch. If the vortex-induced velocity that opposes flame motion is sufficiently large, the flame cannot propagate over the vortex and thus temporarily remains attached to the moving vortex. A flame attachment criterion is discussed; when the criterion is met the vortex forms cusps and pockets in the flame structure similar to those observed experimentally. The net result of increasing the vortex convection velocity is to reduce the residence time of the vortex in the flame, which reduces the degree of flame wrinkling. Vortex pairs that exert an extensive strain on the flame were found to have significantly longer residence times of interaction than vortices that exert compressive strain; this difference in residence time helps to explain why extensive strain on a flame is more probable in turbulent flames than compressive strain. The calculated images of the laminar flame shape show encouraging agreement with experiment, which is another indication that flame-interface simulations are a promising way to represent very wrinkled turbulent premixed flames in a numerically efficient manner.

Abstract:   The level-set flow of Evans-Spruck and Chen-Giga-Goto is generalized to a Riemannian manifold, using recent techniques of Crandall-Ishii for viscosity solutions. Generally speaking, the motion is not unique for noncompact closed sets, but the definition can be modified to make the motion unique. We give examples to show: (1) a smooth set can develop an interior that originates from infinity (2) in the case of a Grayson neckpinch, the evolving function u(x,t) need not remain C2.

Abstract:   We consider a macroscopic model of the excitation process in the anisotropic myocardium involving the transmembrane, extracellular, and extracardiac potentials upsilon, u(e), and u0. The model is described by a reaction-diffusion (R-D) system, and the component upsilon exhibits a front-like behavior reflecting the features of the excitation process. In numerical simulations, the presence of a moving excitation layer imposes severe constraints on the time and space steps to achieve stability and accuracy; consequently, application of the model is very costly in terms of computer time. An approximate model has been derived from the R-D system by means of a singular perturbation technique, and it is described by an eikonal equation, nonlinear and elliptic, in the activation time psi(x). Larger space steps are possible with this equation. From psi(x), we can derive, for a given instant t, the transmembrane potential upsilon and subsequently, by solving an elliptic problem, we can compute the corresponding extracellular and extracardiac potentials u(e) and u0. The results of the R-D and the eikonal models applied to a portion of the ventricular wall are in excellent agreement; moreover, the eikonal model requires only a small fraction of the computer time needed by the R-D system. Therefore, for large- scale simulations of the excitation process, only the eikonal model has been used, and we investigate its ability to cope with complex situations such as front-front collisions and related potential patterns.

Abstract:   We describe all multiscale movie filtering which are causal, local, shape preserving and galilean invariant.

Abstract:   A constrained optimization type of numerical algorithm for removing noise from images is presented. The total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using the gradient-projection method. This amounts to solving a time dependent partial differential equation on a manifold determined by the constraints. As t --> infinity the solution converges to a steady state which is the denoised image. The numerical algorithm is simple and relatively fast. The results appear to be state-of-the-art for very noisy images. The method is noninvasive, yielding sharp edges in the image. The technique could be interpreted as a first step of moving each level set of the image normal to itself with velocity equal to the curvature of the level set divided by the magnitude of the gradient of the image, and a second step which projects the image back onto the constraint set.

Abstract:   In this paper we shall describe a numerical method for the solution of curve flow problems in which the normal velocity of the curve depends locally on the position, normal and curvature of the curve. The method involves approximating the curve by a number of line elements (segments) which are only allowed to move in a direction normal to the element. Hence the normal of each line element remains constant throughout the evolution. In regions of high curvature elements naturally tend to accumulate. The method easily deals with the formation of cusps as found in flame propagation problems and is computationally comparable to a naive marker particle method. As a test of the method we present a number of numerical experiments related to mean curvature flow and flows associated with flame propagation and bushfires.

Abstract:   This paper deals with the mathematical theory of invariant curve evolution. We present a high-level procedure for the formulation of geometric heat flows which are invariant with respect to a given Lie group. This approach is based on the classical theory of differential invariants. The affine group is then analyzed in detail. Indeed, we give a rather complete description of the properties of the affine geometric heat equation. We moreover extend the results of [38] from the convex to the nonconvex case. The paper concludes with a summary of recent applications of curve evolution theory to image analysis.

Abstract:   In this work we investigate, by means of numerical simulations, the performance of two mathematical models describing the spread of excitation in a three-dimensional block representing anisotropic cardiac tissue. The first model is characterized by a reaction-diffusion system in the transmembrane and extracellular potentials v and u. The second model is derived from the first by means of a perturbation technique. It is characterized by an eikonal equation, nonlinear and elliptic in the activation time psi(x). The level surfaces psi(x) = t represent the wave-front positions. The numerical procedures based on the two models were applied to test functions and to excitation processes elicited by local stimulations in a relatively small block. The results are in excellent agreement, and for the same problem the computation time required by the eikonal equation is a small fraction of that needed for the reaction-diffusion system. Thus we have strong evidence that the eikonal equation provides a reliable and numerically efficient model of the excitation process. Moreover, numerical simulations have been performed to validate an approximate model for the extracellular potential based on knowledge of the excitation sequence. The features of the extracellular potential distribution affected by the anisotropic conductivity of the medium were investigated.

Abstract:   We consider an evolution which starts as a flow of smooth surfaces in nonparametric form propagating in space with normal speed equal to the mean curvature of the current surface The boundaries of the surfaces are assumed to remain fixed. G. Huisken has shown that if the boundary of the domain over which this flow is considered satisfies the ''mean curvature'' condition of H. Jenkins and J. Serrin (that is, the boundary of the domain is convex ''in the mean'') then the corresponding initial boundary value problem with Dirichlet boundary data the smooth initial data admits a smooth SolUtion for all time. In this paper we consider the case of arbitrary domains with smooth boundaries not necessarily satisfying the condition of Jenkins-Serrin. In this case, even if the flow starts with smooth initial data and homogeneous Dirichlet boundary data, singularities may develop in finite time at the boundary of the domain and the solution will not satisfy the boundary condition. We prove, however. existence of solutions that are smooth inside the domain for all time and become smooth up to the boundary after elapsing of a sufficiently long period of time. From that moment on such solutions assume the boundary values in the classical sense. We also give sufficient conditions that guarantee the existence of classical solutions for all time t greater-than-or-equal-to 0. In addition. we establish estimates of the rate at which solutions tend to zero as t --> infinity.

Abstract:   We consider the problem of coexistence of two competing species mediated by the presence of a predator. We employ a reaction- diffusion model equation with Lotka-Volterra interaction, and speculate that the possibility of coexistence is enhanced by differences in the diffusion rates of the prey and their predator. In the limit where the diffusion rate of the prey tends to zero, a new equation is derived and the dynamics of spatial segregation is discussed by means of the interfacial dynamics approach. Also, we show that spatial segregation permits periodic and chaotic dynamics for certain parameter ranges.

Abstract:   The connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction-diffusion equations is rigorously established. Two different scalings are studied. In the first, the limiting geometric equation is a first-order equation; in the second, it is a generalization of the mean curvature equation. Intrinsic definitions for the geometric equations are obtained, and uniqueness under a geometric condition on the initial surface is proved. In particular, in the case of the mean curvature equation, this condition is satisfied by surfaces that are strictly starshaped, that have positive mean curvature, or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions.

Abstract:   The connection between the weak theories for a class of geometric equations and the asymptotics of appropriately rescaled reaction-diffusion equations is rigorously established. Two different scalings are studied. In the first, the limiting geometric equation is a first-order equation; in the second, it is a generalization of the mean curvature equation. Intrinsic definitions for the geometric equations are obtained, and uniqueness under a geometric condition on the initial surface is proved. In particular, in the case of the mean curvature equation, this condition is satisfied by surfaces that are strictly starshaped, that have positive mean curvature, or that satisfy a condition that interpolates between the positive mean curvature and the starshape conditions.

Abstract:   An algorithm for shape offsetting is presented that is based on level-set propagation. This algorithm avoids the topological problems encountered in traditional offsetting algorithms, and it deals with curvature singularities by including an 'entropy condition' in its numerical implementation.

Abstract:   A simple phase field model for one component melt growth is presented. which includes anisotropy in a certain form. The formation of various dendritic patterns can be shown by a series of numerical simulations of this model. Qualitative relations between the shapes of crystals and some physical parameters are discussed. Also it is shown that noises give a crucial influence on the side branch structure of dendrites in some situations.

Abstract:   We consider applications of Hamilton-Jacobi equations for which the initial data is only assumed to be in L(infinity). Such problems arise for example when one attempts to describe several characteristic singularities of the compressible Euler equations such as contact and acoustic surfaces, propagating from the same discontinuous initial front. These surfaces represent the level sets of solutions to a Hamilton-Jacobi equation which belongs to a special class. For such Hamilton- Jacobi equations we prove the existence and regularity of solutions for any positive time and convergence to initial data along rays of geometrical optics at any point where the gradient of the initial data exists. Finally, we present numerical algorithms for efficiently capturing singular fronts with complicated topologies such as corners and cusps. The approach of using Hamilton-Jacobi equations for capturing fronts has been used in [14] for fronts propagating with curvature-dependent speed.

Abstract:   The propagation of a two-dimensional wave front in an excitable medium is dependent on the curvature of the front; current theories of excitable reaction-diffusion models predict that, when reaction is much faster than diffusion, the normal wave speed (N) is approximately related to the curvature of the wave front (kappa), the plane wave speed (c), and the diffusion coefficient of the propagator variable (D), by the ''eikonal'' equation, N = c - Dkappa. We show that a simple model for intracellular calcium (Ca2+) wave propagation does not obey the eikonal equation, and postulate an alternative curvature equation that is dependent on the parameter values used in the model. This new curvature relation is confirmed by numerical simulations. We raise the possibility that different models for Ca2+ wave propagation will have qualitatively different spiral wave patterns, providing a new way of distinguishing between proposed models. The theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of Ca2+.

Abstract:   The propagation of a two-dimensional wave front in an excitable medium is dependent on the curvature of the front; current theories of excitable reaction-diffusion models predict that, when reaction is much faster than diffusion, the normal wave speed (N) is approximately related to the curvature of the wave front (kappa), the plane wave speed (c), and the diffusion coefficient of the propagator variable (D), by the ''eikonal'' equation, N = c - Dkappa. We show that a simple model for intracellular calcium (Ca2+) wave propagation does not obey the eikonal equation, and postulate an alternative curvature equation that is dependent on the parameter values used in the model. This new curvature relation is confirmed by numerical simulations. We raise the possibility that different models for Ca2+ wave propagation will have qualitatively different spiral wave patterns, providing a new way of distinguishing between proposed models. The theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of Ca2+.

Abstract:   The propagation of a two-dimensional wave front in an excitable medium is dependent on the curvature of the front; current theories of excitable reaction-diffusion models predict that, when reaction is much faster than diffusion, the normal wave speed (N) is approximately related to the curvature of the wave front (kappa), the plane wave speed (c), and the diffusion coefficient of the propagator variable (D), by the ''eikonal'' equation, N = c - Dkappa. We show that a simple model for intracellular calcium (Ca2+) wave propagation does not obey the eikonal equation, and postulate an alternative curvature equation that is dependent on the parameter values used in the model. This new curvature relation is confirmed by numerical simulations. We raise the possibility that different models for Ca2+ wave propagation will have qualitatively different spiral wave patterns, providing a new way of distinguishing between proposed models. The theory developed here also necessitates a reconsideration of methods previously used to measure the intracellular diffusion coefficient of Ca2+.

Abstract:   Computation of three-dimensional (3-D) Rayleigh-Taylor instability in compressible fluids is performed on a MIMD computer. A second-order TVD scheme is applied with a fully parallelized algorithm to the 3-D Euler equations. The computational program is implemented for a 3-D study of bubble evolution in the Rayleigh-Taylor instability with varying bubble aspect ratio and for large-scale simulation of a 3-D random fluid interface. The numerical solution is compared with the experimental results by Taylor.

Abstract:   A level set formulation for the solution of the Hamilton-Jacobi equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is prescribed on a set of closed bounded noncharacteristic curves. A time dependent Hamilton-Jacobi equation is derived such that the zero level set at various time t of this solution is precisely the set of points (x, y) for which u(x, y) = t. This gives a fast and simple numerical method for generating the viscosity solution to F = 0. The level set capturing idea was first introduced by Osher and Sethian [J. Comput. Phys., 79 (1988), pp. 12-49], and the observation that this is useful for an important computer vision problem of this type was then made by Kimmel and Bruckstein in [Technion (Israel) Computer Science Report, CIS #9209, 1992] following Bruckstein [Comput. Vision Graphics Image Process, 44 (1988), pp. 139-154]. Finally, it is noted that an extension to many space dimensions is immediate.

Abstract:   A level set formulation for the solution of the Hamilton-Jacobi equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is prescribed on a set of closed bounded noncharacteristic curves. A time dependent Hamilton-Jacobi equation is derived such that the zero level set at various time t of this solution is precisely the set of points (x, y) for which u(x, y) = t. This gives a fast and simple numerical method for generating the viscosity solution to F = 0. The level set capturing idea was first introduced by Osher and Sethian [J. Comput. Phys., 79 (1988), pp. 12-49], and the observation that this is useful for an important computer vision problem of this type was then made by Kimmel and Bruckstein in [Technion (Israel) Computer Science Report, CIS #9209, 1992] following Bruckstein [Comput. Vision Graphics Image Process, 44 (1988), pp. 139-154]. Finally, it is noted that an extension to many space dimensions is immediate.

Abstract:   A new affine invariant scale-space for planar curves is presented in this work. The scale-space is obtained from the solution of a novel nonlinear curve evolution equation which admits affine invariant solutions. This flow was proved to be the affine analogue of the well known Euclidean shortening flow. The evolution also satisfies properties such as causality, which makes it useful in defining a scale-space. Using an efficient numerical algorithm for curve evolution, this continuous affine flow is implemented, and examples are presented. The affine-invariant progressive smoothing property of die evolution equation is demonstrated as well.

Abstract:   Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: ''architectural requirements'' like locality, recursivity and causality in the scale space, ''stability requirements'' like the comparison principle and ''morphological requirements'', which correspond to shape- preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.

Abstract:   A formalism for a flamelet's evolution of its spatial distribution is derived from a field equation which is slightly more general than Williams' field equation. Unlike Williams' field equation, the field equation used here, though non- linear, has the property that an arbitrary linear combination of interface solutions (Heavyside type of functions) is also a solution. We therefore can describe the location of the flamelet with two interfaces rather than one, both moving relative to the flow in the same direction. The volume between these two interfaces is on average conserved; this makes it possible to define a probability density for the spatial distribution of the flamelet, and thereby derive equations describing the evolution of the spatial distribution of folds and wrinkles of the flame front. Three main conclusions are reached in this paper using this formalism, through the exact analytical study of a flamelet in an arbitrary 1-d velocity field, and through the numerical study of a flamelet in a simulated 2-d turbulent velocity field. (1) The rate of advancement u(M) of the average location of the flame front can be smaller than the turbulent flame speed u(T) at short times, and sometimes even smaller than the laminar flame speed u(L) (at short times). It is shown, in the case of an arbitrary 1-d velocity field, that u(M) = u(T) only after cusps have formed on the flamelet, and u(M) < u(L) < u(T) before. (2) If the turbulence is too weak or too strong compared with the laminar flame speed, the dispersion of the flame is, at short times, increased by the turbulence and reduced by the laminar flame speed. (3) The dispersion of the flame is skewed towards the direction of the flame's propagation at all times, even before cusp formation.

Abstract:   Bence, Merriman and Osher [BMO] have proposed a new numerical algorithm for computing mean curvature flow, in terms of solutions of the usual heat equation, continually reinitialized after short time steps. This paper employs nonlinear semigroup theory to reconcile their algorithm with the ''level-set'' approach to mean curvature flow of Osher-Sethian [OS], Evans- Spruck [ES], and Chen-Giga-Goto [CGG].

Abstract:   A new algorithm that determines the evolution of a surface eroding under reactive-ion etching is presented. The surface motion is governed by both the Hamilton-Jacobi equation and the entropy condition for a given etch rate. The trajectories of ''shocks'' and ''rarefaction waves'' are then directly tracked, and thus this method may be regarded as a generalization of the method of characteristics. This allows slope discontinuities to be accurately calculated without artificial diffusion. The algorithm is compared with ''geometric'' surface evolution methods, such as the line-segment method.

Abstract:   We propose a new model for active contours based on a geometric partial differential equation. Our model is intrinsec, stable (satisfies the maximum principle) and permits a rigorous mathematical analysis. It enables us to extract smooth shapes (we cannot retrieve angles) and it can be adapted to find several contours simultaneously. Moreover, as a consequence of the stability, we can design robust algorithms which can be engineed with no parameters in applications. Numerical experiments are presented.

Abstract:   A new approach to digital implementation of continuous-scale mathematical morphology is presented. The approach is based on discretization of evolution equations associated with continuous multiscale morphological operations. Those equations, and their corresponding numerical implementation, can be derived either directly from mathematical morphology definitions or from curve evolution theory. The advantages of the proposed approach over the classical discrete morphology are demonstrated.

Abstract:   Numerical modeling of propagating fronts in non-uniform two- dimensional flow fields is performed in order to simulate the effect of such flows on premixed flame fronts. In particular, the influence of the flow disturbance intensity (u') on the mean front propagation rate (S-T) is examined. A second-order numerical technique is employed that combines the level set (G- equation) formulation to describe the self-propagation of the front and a multidimensional upwind technique to describe the convection of the front by the flow field. In this way the effect of the non-dimensional disturbance intensity (u'/S-L) on the non-dimensional propagation rate (S-T/S-L) at values of u'/S-L >> 1 is computed. The dependence of the laminar propagation speed (S-L) on the flame stretch (including both the front curvature and the velocity strain effects) is incorporated in this formulation. We focus on front propagation in simulated Taylor-Couette flows in the ''Taylor vortex'' regime and the results are found to compare favorably with recent experiments on the propagation of isothermal chemical fronts in this flow. The formation of ''islands'' of reactants is observed and its relation to front propagation rates is discussed.

Abstract:   The time necessary to cover a path on a grey-scale image is the sum of the grey-level values along the path. The geodesic time between two points in a grey-scale image is defined as the smallest amount of time allowing to link these points. The geodesic time allows the definition of generalized geodesic distances, erosions, dilations, and skeletons by influence zones. An application to minimal path extraction on grey-scale images is presented.

Abstract:   A mathematical model is developed for melting of a multilayered medium while a heat source traverses one boundary. The finite- element method uses moving meshes, front-tracking using spines, an automatic time-step algorithm, and an efficient solution of the linearized equations. A novel solution method allows the fixed-mesh code to work unchanged but allows a moving mesh in other problems. The finite-element method is applied when the heater mesh moves with respect to the multilayered medium mesh. The same technique allows parallel processing for finite- element codes. The model is applied to several test problems and then to the title problem.

Abstract:   A mathematical model is developed for melting of a multilayered medium while a heat source traverses one boundary. The finite- element method uses moving meshes, front-tracking using spines, an automatic time-step algorithm, and an efficient solution of the linearized equations. A novel solution method allows the fixed-mesh code to work unchanged but allows a moving mesh in other problems. The finite-element method is applied when the heater mesh moves with respect to the multilayered medium mesh. The same technique allows parallel processing for finite- element codes. The model is applied to several test problems and then to the title problem.

Abstract:   The authors define a new class of filters for noise elimination and edge enhancement by using shock filters and anisotropic diffusion. Some nonlinear partial differential equations used as models for these filters are studied. The authors develop recursive and unconditional stable schemes which drastically reduce the computational effort of the algorithms. A new fast recursive approach to linear Gaussian filters is also shown by using the heat equation.

Abstract:   The dynamics of inviscid multicomponent fluids may be modelled by the Euler equations, augmented by one (or more) additional species equation(s). Attempts to compute solutions for extended Euler models in conservation form, show strong oscillations and other computational inaccuracies near material interfaces. These are due to erroneous pressure fluctuations generated by the conservative wave model. This problem does not occur in single component computations and arises only in the presence of several species. A nonconservative (primitive) Euler formulation is proposed, which results in complete elimination of the oscillations. The numerical algorithm uses small viscous perturbations to remove leading order conservation errors and is conservative to the order of numerical approximation. Numerical experiments show clean monotonic solution profiles, with acceptably small conservation error for shocks of weak to moderate strengths. (C) 1994 Academic Press, Inc.

Abstract:   A coupled level set method for the motion of multiple junctions is proposed. The new method extends the ''Hamilton-Jacobi'' level set formulation of Osher and Sethian. It retains the feature of tracking fronts by following level sets and allows the specification of arbitrary velocities on each front, The diffusion equation is shown to generate curvature dependent motion and this is used to develop an algorithm to move multiple junctions with curvature-dependent speed. Systems of reaction-diffusion equations are shown to possess inherent properties which prohibit efficient numerical solutions when applied to curvature-dependent motion. (C) 1994 Academic Press, Inc.

Abstract:   The premixed flame situated in a spatially periodic flow field is examined using the passive propagation model with the local flame speed affected by stretch and nonequidiffusion. Numerical solution shows that the average flame speed increases with either increasing fluctuation amplitude or increasing wavelength of the imposed flow field, and that the flame surface can locally extinguish for sufficiently large fluctuation amplitude of the imposed flow. Perturbation solutions in the weakly wrinkled flame and the thin flame limits are presented. The formation of comers on the flame surface in the thin flame limit is illustrated, and the structure of the comer is further found to resemble that of the Bunsen flame. The premixed flame situated in a two-dimensional periodic flow field is also analyzed in the Huygens limit, leading to the observation that flame surface discontinuities exist in the form of cones.

Abstract:   An accurate finite difference scheme for the flow by curvature in R2 is presented, and its convergence theorem is stated. The numerical scheme has a correction term which is effective in locating points uniformly and the effect prevents the computation from breaking down.

Abstract:   The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Abstract:   The nonlinear interfacial instability of a liquid jet in a coflowing compressible airstream is studied numerically. A high-resolution scheme which has second-order accuracy in space and time is coupled with a Lagrangian marker particle algorithm to visualize the large-scale motion of the interfaces in compressible flow. A numerical algorithm based on an approximate equation of state of a compressible liquid is developed to allow this two-fluid system to be governed by the nonlinear unsteady Euler equations in conservative form. The initial growth of small disturbances given by the simulations agrees well with linear theory. The process of jet disruption in compressible flow is demonstrated to consist of the formation of liquid spikes, interweaving of the gas and liquid and stretching and detachment of the liquid main center core.

Abstract:   A level set approach for computing solutions to incompressible two-phase flow is presented. The interface between the two fluids is considered to be sharp and is described as the zero level set of a smooth function. We use a second-order projection method which implements a second-order upwinded procedure for differencing the convection terms. A new treatment of the level set method allows us to include large density and viscosity ratios as well as surface tension. We consider the motion of air bubbles in water and falling water drops in air. (C) 1994 Academic Press, Inc.

Abstract:   A certain class of surface motions, including those of a relativistic membrane minimizing the three-dimensional volume swept out in Minkowski space, is shown to be equivalent to three-dimensional steady-state irrotational inviscid isentropic gas dynamics. The SU(infinity) Nahm equations turn out to correspond to motions where the time t at which the surface moves through the point r is a harmonic function of the three space coordinates. The solution also implies the linearisation of a non-trivial-looking scalar field theory.

Abstract:   I. We study Brakke's motion of varifolds by mean curvature in the special case that the initial surface is an integral cycle, giving a new existence proof by mean of elliptic regularization. Under a uniqueness hypothesis, we obtain a weakly continuous family of currents solving Brakke's motion. II. These currents remain within the corresponding level-set motion by mean curvature, as defined by Evans-Spruck and Chen- Giga-Goto. Now let T0 be the reduced boundary of a bounded set of finite perimeter in R(n). If the level-set motion of the support of T0 does not develop positive Lebesgue measure, then there corresponds a unique integral n-current T, partial derivative = T0, whose time-slices form a unit density Brakke motion. Using Brakke's Regularity Theorem, spt T is smooth H(n)-almost everywhere. In consequence, almost every level-set of the level-set flow is smooth H(n)-almost everywhere in space-time.

Abstract:   The authors treat some applications of Hamilton-Jacobi equations to the study of a flame front propagation model and the rendez-vous problem. The solution of both problems requires the determination of the level sets of the viscosity solution for the corresponding equation. In the flame front propagation model described here, it is assumed that the evolution is driven by a vector field satisfying a transversality condition at time t = 0. The evolution in the normal direction with variable velocity c(x) greater than or equal to 0 is considered as a special case. This approach is constructive, permitting the numerical solution of such problems.

Abstract:   We study the limiting behavior (the macroscopic limit) of an appropriately scaled spin system with Glauber-Kawasaki dynamics. We rigorously establish the existence in the limit of an interface evolving according to motion by mean curvature. This limit is valid for all positive times, past possible geometric singularities of the motion, which is interpreted in the viscosity sense.

Abstract:   We study the limiting behavior (the macroscopic limit) of an appropriately scaled spin system with Glauber-Kawasaki dynamics. We rigorously establish the existence in the limit of an interface evolving according to motion by mean curvature. This limit is valid for all positive times, past possible geometric singularities of the motion, which is interpreted in the viscosity sense.

Abstract:   We describe a numerical technique to generate logically rectangular body-fitted interior and exterior grids. The technique is based on solving a Hamilton-Jacobi-type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Coordinate grid lines are kept smooth through curvature terms which regularize the equation of motion, and upwind difference schemes which satisfy the correct entropy conditions of front propagation. The resulting algorithm can be used to generate two- and three- dimensional interior and exterior grids around reasonably complex bodies which may contain sharp corners and significant variations in curvature. The technique may also be easily extended to problems of boundary-fitted moving grids. (C) 1994 Academic Press, Inc.

Abstract:   We analyze the problem of stationary self-propagating fronts in potential flow. The issues of local existence and uniqueness for solutions of the ODE describing stationary fronts, multiplicity of solutions and linearized stability of a stationary front as a solution of the (hyperbolic) evolution equation are addressed. The results are illustrated in the case of the dual-source system, which is a simple model of a combustion system in which local extinction may arise. Model extensions for combustion applications are presented.

Abstract:   In this paper, we study generalized ''viscosity'' solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Abstract:   Evolutions of closed planar polygons are studied in this work. In the first part of the paper, the general theory of linear polygon evolutions is presented, and two specific problems are analyzed. The first one is a polygonal analog of a novel affine-invariant differential curve evolution, for which the convergence of planar curves to ellipses was proved. In the polygon case, convergence to polygonal approximation of ellipses, polygonal ellipses, is proven. The second one is related to cyclic pursuit problems, and convergence, either to polygonal ellipses or to polygonal circles, is proven. In the second part, two possible polygonal analogues of the well-known Euclidean curve shortening flow are presented. The models follow from geometric considerations. Experimental results show that an arbitrary initial polygon converges to either regular or irregular polygonal approximations of circles when evolving according to the proposed Euclidean flows.

Abstract:   In this paper, area preserving multi-scale representations of planar curves are described. This allows smoothing without shrinkage at the same time preserving all the scale-space properties. The representations are obtained deforming the curve via geometric heat flows while simultaneously magnifying the plane by a homethety which keeps the enclosed area constant. When the Euclidean geometric heat now is used, the resulting representation is Euclidean invariant, and similarly it is affine invariant when the affine one is used. The flows are geometrically intrinsic to the curve, and exactly satisfy all the basic requirements of scale-space representations. In the case of the Euclidean heat flow, it is completely local as well. The same approach is used to define length preserving geometric flows. A similarity (scale) invariant geometric heat flow is studied as well in this work.

Abstract:   Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods and overcomes some of their limitations. Our techniques can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of our model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian to model propagating solid/liquid interfaces with curvature dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature, It is moved by solving a ''Hamilton-Jacobi'' type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity-of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. We present a variety of ways of computing evolving front, including narrow bands, reinitializations, and different stopping criteria. The efficacy of the scheme is demonstrated with numerical experiments on some synthesized images and some low contrast medical images.

Abstract:   This paper considers the questions of convergence of: (i) MUSCL type (i.e. second-order, TVD) finite-difference approximations towards the entropic weak solution of scalar, one-dimensional conservation laws with strictly convex flux and (ii) higher- order schemes (filtered to ''preserve'' an upper-bound on some weak second-order finite differences) towards the viscosity solution of scalar, multi-dimensional Hamilton-Jacobi equations with convex Hamiltonians.

Abstract:   Level-set approach to the motion of surfaces is presented. Applications to geometrical models in condensed matter physics are given. The finger solution and its generalizations, which were reported very recently, are derived in a simple way.

Abstract:   In this paper, a curve evolution approach for the computation of geodesic curves on 3D surfaces is presented. The algorithm is based on deforming, via the curve shortening flow, an arbitrary initial curve ending at two given surface points. The 3D curve shortening flow is first transformed into an equivalent 2D one. This 2D flow is implemented, using an efficient numerical algorithm for curve evolution with fixed end points.

Abstract:   We prove the convergence of an approximation scheme recently proposed by Bence, Merriman, and Osher for computing motions of hypersurfaces by mean curvature. Our proof is based on viscosity solutions methods.

Abstract:   We present some difference approximation schemes which converge to the entropy solution of a scalar conservation law having a convex flux. The numerical methods described here take their origin from approximation schemes for Hamilton-Jacobi-Bellman equations related to optimal control problems and exhibit several interesting features: the convergence result still holds for quite arbitrary time steps, the main assumption for convergence can be interpreted as a discrete analogue of Oleinik's entropy condition, numerical diffusion around the shocks is very limited. Some tests are included in order to compare the performances of these methods with other classical methods (Godunov, TVD).

Abstract:   We continue Our investigation of the ''level-set'' technique for describing the generalized evolution of hypersurfaces moving according to their mean curvature. The principal assertion of this paper is a kind of reconciliation with the geometric measure theoretic approach pioneered by K. Brakke: we prove that almost every level set of the solution to the mean curvature evolution PDE is in fact a unit-density varifold moving according to its mean curvature. In particular, a.e. level set is endowed with a kind of ''geometric structure.'' The proof utilizes compensated compactness methods to pass to limits in various geometric expressions.

Abstract:   We discuss three different affine invariant evolution processes for smoothing planar curves. The first one is derived from a geometric heat-type flow, both the initial and the smoothed curves being differentiable. The second smoothing process is obtained from a discretization of this affine heat equation. In this case, the curves are represented by planar polygons. The third process is based on B-spline approximations. For this process, the initial curve is a planar polygon, and the smoothed curves are differentiable and even analytic. We show that, in the limit, all three affine invariant smoothing processes collapse any initial curve into an elliptic point.

Abstract:   A method is introduced to decrease the computational labor of the standard level set method for propagating interfaces. The fast approach uses only points close to the curve at every time step. We describe this new algorithm and compare its efficiency and accuracy with the standard level set approach. (c) 1995 Academic Press, Inc.

Abstract:   We present a nerv algorithm for determining minimal length paths between two regions on a three dimensional surface, The numerical implementation is based on finding equal geodesic distance contours from a given area, These contours are calculated as zero sets of a bivariate function designed to evolve so as to track the equal distance curves on the given surface, The algorithm produces all paths of minimal length between the source and destination areas on the surface given as height values on a rectangular grid.

Abstract:   A new algorithm for recovering depth to a Lambertian C-1 smooth object given its gray-level image under uniform illumination from the viewing direction is presented. To recover depth, an almost arbitrarily initialized surface is numerically propagated on a rectangular grid, so that a level set of this surface tracks the height contours of the depth function. The image shading controls the propagation of the surface. When the light direction is tilted with respect to the viewing direction the problem is solved by tracking the projection of equal- height contours defined with respect to the light source direction. This projection approach provides a solution that overcomes ambiguity problems encountered in previous work, while the level set approach of implementing the contour propagation overcomes numerical problems and some of the topology problems of the evolving contours. (C) 1995 Academic Press, Inc.

Abstract:   A new algorithm for recovering depth to a Lambertian C-1 smooth object given its gray-level image under uniform illumination from the viewing direction is presented. To recover depth, an almost arbitrarily initialized surface is numerically propagated on a rectangular grid, so that a level set of this surface tracks the height contours of the depth function. The image shading controls the propagation of the surface. When the light direction is tilted with respect to the viewing direction the problem is solved by tracking the projection of equal- height contours defined with respect to the light source direction. This projection approach provides a solution that overcomes ambiguity problems encountered in previous work, while the level set approach of implementing the contour propagation overcomes numerical problems and some of the topology problems of the evolving contours. (C) 1995 Academic Press, Inc.

Abstract:   We present a controlled image smoothing and enhancement method based on a curvature flow interpretation of the geometric heat equation. Compared to existing techniques, the model has several distinct advantages. (i) It contains just one enhancement parameter. (ii) The scheme naturally inherits a stopping criterion from the image; continued application of the scheme produces no further change. (iii) The method is one of the fastest possible schemes based on a curvature-controlled approach.

Abstract:   We study a sharp-interface model for phase transitions which incorporates the interaction of tile phase boundaries with the walls of a container Omega. In this model, the interfaces move by their mean curvature and are normal to partial derivative Omega. We first establish local-in-time existence and uniqueness of smooth solutions for the mean curvature equation with a normal contact angle condition. We then discuss global solutions by interpreting the equation and the boundary condition in a weak (viscosity) sense. Finally, we investigate the relation of the aforementioned model with a transition- layer model. We prove that if Omega is convex, the transition- layer solutions converge to the sharp-interface solutions as the thickness of the layer tends to zero. We conclude with a discussion of the difficulties that arise in establishing this result in nonconvex domains.

Abstract:   We apply a level set formulation to the problem of surface advancement in a two-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner a nd cusp development, a nd accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. The development of this model and algorithm naturally extends to three dimensions in a straightforward manner and is described in part II of this paper (in press). (C) 1995 Academic Press, Inc.

Abstract:   We undertake to develop a general theory of two-dimensional shape by elucidating several principles which any such theory should meet. The principles are organized around two basic intuitions: first, if a boundary were changed only slightly, then, in general, its shape would change only slightly. This leads us to propose an operational theory of shape based on incremental contour deformations. The second intuition is that not all contours are shapes, but rather only those that can enclose ''physical'' material. A theory of contour deformation is derived from these principles, based on abstract conservation principles and Hamilton-Jacobi theory. These principles are based on the work of Sethian (1985a, c), the Osher-Sethian (1988), level set formulation the classical shock theory of Lax (1971; 1973), as well as curve evolution theory for a curve evolving as a function of the curvature and the relation to geometric smoothing of Gage-Hamilton-Grayson (1986; 1989). The result is a characterization of the computational elements of shape: deformations, parts, bends, and seeds, which show where to place the components of a shape. The theory unifies many of the diverse aspects of shapes, and leads to a space of shapes (the reaction/diffusion space), which places shapes within a neighborhood of ''similar'' ones. Such similarity relationships underlie descriptions suitable for recognition.

Abstract:   This paper presents two-dimensional direct numerical simulations of a passive flame surface passing through homogeneous isotropic turbulence. The flame was represented by a field variable, G(x, t), whose isocontours constitute flame surfaces. One well known complication in analyzing premixed combustion in a homogeneous environment is decoupling the effect of the decaying turbulent velocity field from the dynamics of the flame surface. To overcome this, the velocity field was made stationary by introducing a random forcing term into the Navier Stokes equations. Forcing was done over two different ranges of wavenumbers (k(f) = 10-14, and k(f) = 80- 84) thus creating turbulence with different length scales and inertial range power laws. By comparing the response of the flame to the two types of turbulence it was possible to determine the effect the spectral distribution energy has on the surface topology and mean rate of propagation. Indeed, the flames were found to be remarkably sensitive to the spectral distribution of the turbulent energy, and not just its magnitude. Furthermore, a k(-5/3) inertial range was shown to produce a flame surface that was preferentially wrinkled at intermediate to small scales for purely geometric reasons. By defining a surface area spectrum it was possible to rationalize this result by recognizing that flame surface area is closely related to the dissipation spectrum of the scalar field. Collectively the results suggest that knowledge of the energy spectrum al a minimum is required to predict a turbulent flame speed under general circumstances.

Abstract:   The formation of drops resulting from the breakup of an axisymmetric Newtonian liquid jet injected vertically into another immiscible Newtonian liquid at various Reynolds numbers is investigated here. The full transient from startup to breakup into drops was simulated numerically by solving the time-dependent axisymmetric equations of motion and continuity using a combination of the volume-of-fluid (VOF) and continuous-surface-force (CSF) methods. The numerical simulation results compare well with previous experimental data and are significantly more accurate than previous simplified analyses based on drop formation before and after jetting over a wide range of conditions. (C) 1995 American Institute of Physics.

Abstract:   We present a new implementation of an algorithm aimed at recovering a 3D shape from its 2D gray-level picture. In order to reconstruct the shape of the object, an almost arbitrarily initialized 3D function is propagated on a rectangular grid, so that a level set of this function tracks the height contours of the shape. The method imports techniques from differential geometry, fluid dynamics, and numerical analysis and provides an accurate shape from shading algorithm. The method solves some topological problems and gracefully handles cases of non- smooth surfaces that give rise to shocks in the propagating contours. Real and synthetic images of 3D profiles were submitted to the algorithm and the reconstructed surfaces are presented, demonstrating the effectiveness of the proposed method.

Abstract:   The level-set approach of Osher & Sethian to tracking interfaces is successfully adapted to the simulation of a premixed turbulent open V-flame including the effects of exothermicity and baroclinicity. In accord with experimental observations this algorithm, along with a flame anchoring scheme, predicts flame cusping for a case in which a strong vortex pair interacts with the flame front. The computed velocity and scalar statistics obtained for the turbulent V- flame compare reasonably well with experimental results by Cheng & Shepherd, and demonstrate the importance of flame- generated vorticity in the determination of flame dynamics and product velocity characteristics.

Abstract:   The medial axis transform (MAT) of a shape, better known as its skeleton, is frequently used in shape analysis and related areas. In this paper a new approach for determining the skeleton of an object is presented. The boundary is segmented at points of maximal positive curvature and a distance map from each of the segments is calculated. The skeleton is then located by applying simple rules to the zero sets of distance map differences. A framework is proposed for numerical approximation of distance maps that is consistent with the continuous case and hence does nor suffer from digitization bias due to metrication errors of the implementation on the grid. Subpixel accuracy in distance map calculation is obtained by using gray-level information along the boundary of the shape in the numerical scheme. The accuracy of the resulting efficient skeletonization algorithm is demonstrated by several examples. (C) 1995 Academic Press, Inc.

Abstract:   A new approach for the reconstruction of a smooth three- dimensional object from its two-dimensional gray-level image is presented. An algorithm based on topological properties of simple smooth surfaces is provided to solve the problem of global reconstruction. Classifying singular points in the shading image as maxima, minima, and two kinds of saddle points serves as the key to the solution of the problem. The global reconstruction procedure, being deterministic and using topological properties of the surface, performs better than other approaches proposed so far that are based on classification of singular points according to the behavior of characteristics in their neighborhood. The proposed algorithm is simple and easy to implement and lends itself to a parallel implementation. (C) 1995 Academic Press, Inc.

Abstract:   We show how optimization of the Nordstrom and Mumford-Shah functionals can be used to develop a type of curve-evolution that is able to preserve salient features of closed curves while simultaneously suppressing noise and irrelevant details. The idea is to characterize a curve by means of its angle- function and apply the appropriate dynamics to this representation. Upon convergence, the resulting form of the contour is reconstructed from the representation.

Abstract:   This paper introduces a discrete scheme for mean curvature motion using a morphological image processing approach. An axiomatic approach of image processing and the mean curvature partial differential equation (PDE) are briefly presented, then the properties of the proposed scheme are studied. In particular, consistency and convergence are proved. The applications of mean curvature motion in image denoising and form evolution are developed and experiences are presented.

Abstract:   We apply a level set formulation to the problem of surface advancement in three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi-type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. The equations of motion of a unified model, including the effects of isotropic and unidirectional deposition and etching, visibility, surface diffusion, reflection, and material dependent etch/deposition rates are presented and adapted to a level set formulation. In Part I of this paper, the basic equations and algorithms for two- dimensional simulations were developed. In this paper, the extension to three dimensions is presented. We show a large collection of simulations, including three-dimensional etching and deposition into cavities under the effects of visibility, directional and source flux functions, evolution of lithographic profiles, discontinuous etch rates through multiple materials, and non-convex sputter yield flux functions. In Part III of this paper, effects of reflection and re-emission and surface diffusion Will be presented. (C) 1995 Academic Press, Inc.

Abstract:   In this paper, we study generalized ''viscosity'' solutions of the mean curvature evolution which were introduced by Chen, Giga, and Goto and by Evans and Spruck. We devote much of our attention to solutions whose initial value is a compact, smooth, rotationally symmetric hypersurface given by rotating a graph around an axis. Our main result is the regularity of the solution except at isolated points in spacetime and estimates on the number of such points.

Abstract:   A new geometric approach to the affine geometry of curves in the plane and to affine-invariant curve shortening is presented. We describe methods of approximating the affine curvature with discrete finite difference approximations, based on a general theory of approximating differential invariants of Lie group actions by joint invariants. Applications to computer vision are indicated. (C) 1996 Academic Press, Inc.

Abstract:   The kinematic effects of a space-time forced velocity held upon a thin premixed flame, stabilized above a circular cross- section burner, are studied in order to point out the non- linearities due to a sufficiently high velocity perturbation level whose RMS amplitudes remain nonetheless inferior to the normal burning velocity. The present calculation proposes to seek a solution using the characteristics method, without any linearized calculation, to express these effects. A front evolution equation is interpreted as the differentiated form of a conservation equation of the radial distance between two points of the front. These modelling results are used to interpret experiments of a vibrating flame subjected to a space-time sinusoidal velocity held. In this last case, the limit of cusps formation is represented as a similarity law expressing the nondimensional perturbation amplitude versus a Strouhal number of the aero-acoustic reactive medium.

Abstract:   We present a fully conservative, high resolution approach to front tracking for nonlinear systems of conservation laws in two space dimensions. An underlying uniform Cartesian grid is used, with some cells cut by the front into two subcells. The front is moved by solving a Riemann problem normal to each segment of the front and using the motion of the strongest wave to give an approximate location of the front at the end of the time step. A high resolution finite volume method is then applied on the resulting slightly irregular grid to update all cell values. A ''large time step'' wave propagation algorithm is used that remains stable in the small cut cells with a time step that is chosen with respect to the uniform grid cells. Numerical results on a radially symmetric problem show that pointwise convergence with order between 1 and 2 is obtained in both the cell values and location of the front. Other computations are also presented. (C) 1996 Academic Press, Inc.

Abstract:   A new finite element method is discussed for approximating evolving interfaces in R(n) whose normal velocity equals mean curvature plus a forcing function. The method is insensitive to singularity formation and retains the local structure of the limit problem and, thus, exhibits a computational complexity typical of R(n-1) without having the drawbacks of front- tracking strategies. A graded dynamic mesh around the propagating front is the sole partition present at any time step and is significantly smaller than a full mesh. Time stepping is explicit, but stability constraints force small time steps only when singularities develop, whereas relatively large time steps are allowed before or past singularities, when the evolution is smooth. The explicit marching scheme also guarantees that at most one layer of elements has to be added or deleted per time step, thereby making mesh updating simple and, thus, practical. Performance and potentials are fully documented via a number of numerical simulations in 2D, 3D, 4D, and 8D, with axial symmetries. They include tori and cones for the mean curvature flow, minimal and prescribed mean curvature surfaces with given boundary, fattening for smooth driving force, and volume constraint. (C) 1996 Academic Press, Inc.

Abstract:   For the computation of nonlinear solutions of Hamilton-Jacobi scalar equations in two space dimensions, we develop high order accurate numerical schemes that can be applied to complicated geometries. Previously, the recently developed essentially nonoscillatory (ENO) technology has been applied in simple domains like squares or rectangles using dimension-by-dimension algorithms. On arbitrary two dimensional closed or multiply connected domains, first order monotone methods were used. In this paper, we propose two different techniques to construct high order accurate methods using the ENO philosophy. Namely, any arbitrary domain is triangulated by finite elements into which two dimensional ENO polynomials are constructed. These polynomials are then differentiated to compute a high order accurate numerical solution. These new techniques are shown to be very useful in the computation of numerical solutions of various applications without significantly increasing CPU running times as compared to dimension-by-dimension algorithms. Furthermore, these methods are stable and no spurious oscillations are detected near singular points or curves. (C) 1996 Academic Press, Inc.

Abstract:   Exact, non-planar travelling solutions of the eikonal equation on an infinite plane are presented for the first time. These solutions are matched to produce corrugated wave fronts and patterns such as 'spot' solutions as well as extended parabolic type wave fronts. The stability of these solutions is also analysed. The variational equation which belongs to a generalised Wangerin class of differential equations is solved, first with the aid of the Liouville-Green approximation for the estimated eigenvalues characterising stability and then by a more elaborate shooting-matching method. All of the three types of travelling solutions are found to be geometrically stable. It is suggested that some of these predictions are experimentally testable.

Abstract:   A fast marching level set method is presented for monotonically advancing fronts, which leads to an extremely fast scheme for solving the Eikonal equation. Level set methods are numerical techniques for computing the position of propagating fronts. They rely on an initial value partial differential equation for a propagating level set function and use techniques borrowed from hyperbolic conservation laws. Topological changes, corner and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. This paper describes a particular case of such methods for interfaces whose speed depends only on local position. The technique works by coupling work on entropy conditions for interface motion, the theory of viscosity solutions for Hamilton-Jacobi equations, and fast adaptive narrow band level set methods. The technique is applicable to a variety of problems, including shape-from-shading problems, lithographic development calculations in microchip manufacturing, and arrival time problems in control theory.

Abstract:   Despite continuing research and steady progress in the field of digital halftones, there is a feeling that a wide gap in quality still remains between the best available and the best achievable results. A glance at man-made halftones readily confirms this feeling. To bridge this gap, we propose the use of computer-generated, yet gridless halftones. This involves solving the halftoning problem on the continuous 2D plane rather than on the usual discrete grid of pixels. In this article we outline this new approach, describe its expected advantages over existing techniques and demonstrate some of them via a prototype system, DigiDurer, developed for this purpose. (C) 1996 Academic Press, Inc.

Abstract:   We present an asymptotic theory for the dynamics of detonation when the radius of curvature of the detonation shock is large compared to the one-dimensional, steady, Chapman-Jouguet (CJ) detonation reaction-zone thickness. The analysis considers additional time-dependence in the slowly varying reaction zone to that considered in previous works. The detonation is assumed to have a sonic point in the reaction-zone structure behind the shock, and is referred to as an eigenvalue detonation. A new, iterative method is used to calculate the eigenvalue relation, which ultimately is expressed as an intrinsic, partial differential equation (PDE) for the motion of the shock surface. Two cases are considered for an ideal equation of state. The first corresponds to a model of a condensed-phase explosive, with modest reaction rate sensitivity, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D-n, the first normal time derivative of the normal shock velocity, D-n, and the shock curvature, kappa. The second case corresponds to a gaseous explosive mixture, with the large reaction rate sensitivity of Arrhenius kinetics, and the intrinsic shock surface PDE is a relation between the normal detonation shock velocity, D-n, its first and second normal time derivatives of the normal shock velocity, D-n, D-n, and the shock curvature, kappa, and its first normal time derivative of the curvature, kappa. For the second case, one obtains a one-dimensional theory of pulsations of plane CJ detonation and a theory that predicts the evolution of self-sustained cellular detonation. Versions of the theory include the limits of near-CJ detonation, and when the normal detonation velocity is significantly below its CJ value. The curvature of the detonation can also be of either sign, corresponding to both diverging and converging geometries.

Abstract:   Flame propagation through an array of vortices was studied with a model which incorporated the variation of the local burning velocity with stretch. Assuming incompressible how, the mean burning velocities were calculated and compared to those of the Huygens limit. It was found that stretch causes a decrease in the mean burning velocity, and a mechanism which explains this trend was identified. The study also demonstrated that, as expected, stretch only has a significant effect on the mean burning velocity for vortices whose size is of the same order as that of the flame thickness.

Abstract:   A level set formulation is derived for incompressible, immiscible Navier-Stokes equations separated by a free surface. The interface is identified as the zero level set of a smooth function. Eulerian finite difference methods based on this level set formulation are proposed. These methods are robust and efficient and are capable of computing interface singularities such as merging and reconnection. Numerical experiments are presented to demonstrate the effectiveness of the methods. (C) 1996 Academic Press, Inc.

Abstract:   An implicit, pseudo-solid domain mapping technique is described that facilitates finite element analysis of free and moving boundary problems. The technique is based on an implicit, full- Newton strategy,free of restrictions on mesh structure; this leads to many advantages over existing domain mapping techniques. The fully coupled approach using Newton's method is particularly effective for problems with strong coupling between the internal bulk physics and the governing physics at unknown free boundary locations. It is also useful when the distinguishing conditions which constrain the free boundary shape provide only an implicit dependence on the boundary location. Unstructured meshes allow for efficient resolution of internal and boundary layers and other regions of strong local variations in the solution and they also reduce the amount of user interaction required to define a problem since the meshes may be generated automatically. The technique is readily applied to steady or transient problems in complex geometries of two and three dimensions. Examples are shown that include free and moving boundary problems from solidification and capillary hydrodynamics. (C) 1996 Academic Press, Inc.

Abstract:   We present a class of PDE-based algorithms suitable for image denoising and enhancement. The techniques are applicable to both salt-and-pepper gray-scale noise and full-image continuous noise present in black and white images, gray-scale images, texture images, and color images. At the core, the techniques rely on two fundamental ideas. First, a level set formulation is used for evolving curves; use of this technique to flow isointensity contours under curvature is known to remove noise and enhance images. Second, the particular form of the curvature how is governed by a minimax switch which selects a range of denoising dependent on the size of switching window. Our approach has several virtues. First, it contains only one enhancement parameter, which in most cases is automatically chosen. Second, the scheme automatically stops smoothing at a point which depends on the switching window size; continued application of the scheme produces no further change. Third, the method is one of the fastest possible schemes based on a curvature-controlled approach. (C) 1996 Academic Press, Inc.

Abstract:   A novel approach for the computation of optical how based on an L(1) type minimization is presented, It is shown that the approach has inherent advantages since it does not smooth the flow-velocity across the edges and hence preserves edge information, A numerical approach based on computation of evolving curves is proposed for computing the optical flow field. Computations are carried out on a number of real image sequences in order to illustrate the theory as well as the numerical approach.

Abstract:   Image processing via mathematical morphology has traditionally used geometry to intuitively understand morphological signal operators and set or lattice algebra to analyze them in the space domain, In this paper, we provide a unified view and analytic tools for a recently growing part of morphological image processing that is based on ideas from differential calculus and dynamical systems, This part includes both recent and some earlier ideas on using partial differential or difference equations (PDEs) to model distance propagation or nonlinear multiscale processes in images. We briefly review some nonlinear difference equations that implement discrete distance transforms and relate them to numerical solutions of the eikonal equation of optics. We also review some nonlinear PDEs that model the evolution of multiscale morphological operators and use morphological derivatives. Among the new ideas presented, we develop some general 2-D max/min-sum difference equations that model the space dynamics of 2-D morphological systems (including the distance computations) and some nonlinear signal transforms, called slope transforms, that can analyze these systems in a transform domain in ways conceptually similar to the application of Fourier transforms to linear systems. Thus, distance transforms are shown to be bandpass slope filters, We view the analysis of the multiscale morphological PDEs and of the eikonal PDE solved via weighted distance tranforms as a unified area in nonlinear image processing, which we call differential morphology, and briefly discuss its potential applications to image processing and computer vision.

Abstract:   The propagation of a premixed flame through a large-scale vortical how field is studied numerically by solving a front propagation equation governing the evolution of a scalar field whose zero-level surface defines the location of a self- propagating interface. The flame front is considered to propagate normal to itself with a constant speed, and the density variation across the flame is considered to be zero. Average burning rates are calculated for large velocity fluctuation intensities, as the formulation allows naturally for the formation of pockets of unburned gas downstream from the reaction front. The propagation rate of the corrugated flame front is found to vary periodically with time, with an average that varies linearly with velocity fluctuation intensity at large intensities. A mechanism is identified for the decreasing sensitivity of the average burning rate to increases in the fluctuation intensity, over an intermediate range of intensities, in the zero-viscosity limit.

Abstract:   We utilize the level-set method to interpret geometrically what it means to solve the heat equation with multivalued initial data. We prove that in one space dimension, the limits of ''geometrically natural'' approximations instantly unfold multivalued initial data, according to an equal-area rule. In. higher dimensions, the limits of certain ''analytically natural'' approximations display similar effects.

Abstract:   We give an extension of the level set formulation of Osher and Sethian, which describes the dynamics of surfaces that propagate under the influence of their own curvature. We consider an extension of their original algorithms for finite domains that includes boundary conditions. We discuss this extension in the context of a specific application that comes from the theory of detonation shock dynamics (DSD). We give an outline of the theory of DSD which includes the formulation of the boundary conditions that comprise the engineering model. We give the formulation of the level set method, as applied to our application with finite boundary conditions. We develop a numerical method to implement arbitrarily complex 2D boundary conditions and give a few representative calculations. We also discuss the dynamics of level curve motion and point out restrictions that arise when applying boundary conditions. (C) 1996 Academic Press, Inc.

Abstract:   The Rayleigh-Taylor instability is a gravity driven instability of a contact surface between fluids of different densities, The growth of this instability is sensitive to numerical or physical mass diffusion. For this reason, high resolution of the contact discontinuity is particularly important, In this paper, we address this problem using a second-order TVD finite difference scheme with artificial compression. We describe our numerical simulations of the 3D Rayleigh-Taylor instability using this scheme. The numerical solutions are compared to (a) the exact 2D solution in the linear regime and (b) numerical solutions using the TVD scheme and the front tracking method. The computational program is used to study the evolution of a single bubble and 3D bubble merger, i.e., the nonlinear evolution of a single mode and the process of nonlinear mode- mode interaction. (C) 1996 Academic Press, Inc.

Abstract:   We study the unique affine invariant morphological scale space in three dimensions. We discuss its properties and show that it improves the dynamic shape model. We explain the algorithms and display the first numerical experiments.

Abstract:   The flow of a closed surface of codimension 1 in R(R) driven by curvature is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon > 0. Conforming piecewise linear finite elements, with mass lumping, over a quasi-uniform and weakly acute mesh of size h are further used for space discretization, and combined with forward differences for time discretization with uniform time- step tau. The resulting explicit schemes are the basis for an efficient algorithm, the so-called dynamic mesh algorithm, and exhibit finite speed of propagation and discrete nondegeneracy. No iteration is required, not even to handle the obstacle constraints. The zero level set of the fully discrete solution is shown to converge past singularities to the true interface, provided tau, h(2) approximate to 0(epsilon(4)) and no fattening occurs. If the more stringent relations tau, h(2) approximate to 0(epsilon(6)) are enforced, then an interface rate of convergence O(epsilon) is derived in the vicinity of regular points, along with a companion O(epsilon(1/2)) for type I singularities. For smooth flows, an interface rate of convergence of O(epsilon(2)) is proven, provided tau, h(2) approximate to O(epsilon(5)) and exact integration is used for the potential term. The analysis is based on constructing fully discrete barriers via an explicit parabolic projection with quadrature, which bears some intrinsic interest, Lipschitz properties of viscosity solutions of the level set approach, and discrete nondegeneracy. These basic ingredients are also discussed.

Abstract:   We are interested in the study of a fast algorithm introduced by Brenier computing the discrete Legendre-Fenchel transform of a real function. We present convergence results and show how the order of convergence grows with the regularity of the Function to be transformed. applications to Hamilton-Jacobi equations for front propagation problems and conservation laws are presented.

Abstract:   A coupled level set method for the motion of multiple junctions (of, e.g., solid, liquid, and grain boundaries), which follows the gradient flow for an energy functional consisting of surface tension (proportional to length) and bulk energies (proportional to area), is developed. The approach combines the level set method of S. Osher and J. A. Sethian with a theoretical variational formulation of the motion by F. Reitich and H. M. Soner. The resulting method uses as many level set functions as there are regions and the energy functional is evaluated entirely in terms of level set functions. The gradient projection method leads to a coupled system of perturbed (by curvature terms) Hamilton-Jacobi equations. The coupling is enforced using a single Lagrange multiplier associated with a constraint which essentially prevents (a) regions from overlapping and (b) the development of a vacuum. The numerical implementation is relatively simple and the results agree with (and go beyond) the theory as given in [12]. Other applications of this methodology, including the decomposition of a domain into subregions with minimal interface length, are discussed. Finally, some new techniques and results in level set methodology are presented. (C) 1996 Academic Press, Inc.

Abstract:   We present an Eulerian, fixed grid, approach to solve the motion of an incompressible fluid, in two and three dimensions, in which the vorticity is concentrated on a lower dimensional set. Our approach uses a decomposition of the vorticity of the form xi = P(phi) eta, in which both phi (the level set function) and eta (the vorticity strength vector) are smooth. We derive coupled equations for phi and eta which give a regularization of the problem. The regularization is topological and is automatically accomplished through the use of numerical schemes whose viscosity shrinks to zero with grid size. There is no need for explicit filtering, even when singularities appear in the front, The method also has the advantage of automatically allowing topological changes such as merging of surfaces. Numerical examples, including two and three dimensional vortex sheets, two-dimensional vortex dipole sheets, and point vortices, are given. To our knowledge, this is the first three-dimensional vortex sheet calculation in which the sheet evolution feeds back to the calculation of the fluid velocity. Vortex in cell calculations for three- dimensional vortex sheets were done earlier by Trygvasson et al. (C) 1996 Academic Press, Inc.

Abstract:   Shape modeling is an important constituent of computer vision as well as computer graphics research. Shape models aid the tasks of object representation and recognition. This paper presents a new approach to shape modeling which retains some of the attractive features of existing methods, and overcomes some of their limitations. Our technique can be applied to model arbitrarily complex shapes, which include shapes with significant protrusions, and to situations where no a priori assumption about the object's topology is made. A single instance of our model, when presented with an image having more than one object of interest, has the ability to split freely to represent each object. This method is based on the ideas developed by Osher and Sethian to model propagating solid/liquid interfaces with curvature-dependent speeds. The interface (front) is a closed, nonintersecting, hypersurface flowing along its gradient field with constant speed or a speed that depends on the curvature. It is moved by solving a ''Hamilton-Jacobi'' type equation written for a function in which the interface is a particular level set. A speed term synthesized from the image is used to stop the interface in the vicinity of object boundaries. The resulting equation of motion is solved by employing entropy-satisfying upwind finite difference schemes. We also introduce a new algorithm for rapid advancement of the front using what we call a narrow-band update scheme. The efficacy of the scheme is demonstrated with numerical experiments on low contrast medical images.

Abstract:   A new framework for computing the Euclidean distance and weighted distance from the boundary of a given digitized shape is presented. The distance is calculated with sub-pixel accuracy. The algorithm is based on an equal distance contour evolution process. The moving contour is embedded as a level set in a time varying function of higher dimension. This representation of the evolving contour makes possible the use of an accurate and stable numerical scheme, due to Osher and Sethian [22]. The relation between the classical shape from shading problem and the weighted distance transform is presented, as well as an algorithm that calculates the geodesic distance transform on surfaces.

Abstract:   In this paper, we study the asymptotics of Fitzhugh-Nagumo-type systems of reaction-diffusion equations with bistable nonlinearity. In the limit, we obtain an interface moving with normal velocity determined by the dynamics and the scaling.

Abstract:   We present a shape-recovery technique in two dimensions and three dimensions with specific applications in modeling anatomical shapes from medical images. This algorithm models extremely corrugated structures like the brain, is topologically adaptable, and runs in O(N log N) time, where N is the total number of points in the domain. Our technique is based on a level set shape-recovery scheme recently introduced by the authors and the fast marching method for computing solutions to static Hamilton-Jacobi equations.

Abstract:   In this paper we describe some numerical simulations in the context of mean curvature flow. We recover a few different approaches in modeling the evolution of an interface Sigma which evolves according to the law: V = k + g where V is the velocity in the inward normal direction, k is the sum of the principal curvatures and g is a given forcing term. We will discuss about the phenomenon of fattening or nonuniqueness of the solution, recalling what is known about this subject. Finally we show some interesting numerical simulations that suggest evidence of fattening starting from different initial interfaces. Of particular interest is the result obtained for a torus in R(4) which would be a first example of a regular and compact surface showing evidence of fattening in the case of pure motion by mean curvature (no forcing term).

Abstract:   The passive propagation of wrinkled, non-folding, premixed flames in quiescent and spatially periodic how fields is investigated by employing the scalar held, G-equation formulation. Rather than solving the G-equation directly, we transform it into a g-equation, which is a differential equation governing the evolution of the slope of the flame shape in two-dimensional flows. For the Landau limit of flame propagation with constant flame speed, the resulting g-equation degenerates to a quasi-linear wave equation in a quiescent flow. For the stretch-affected propagation mode in which the flame propagation speed is curvature-dependent, the resulting g-equation is in the general form of the Burgers' equation. Analytical solutions were obtained for several flame and flow types, revealing some interesting characteristics of the geometry and propagation of the flame, including the formation of cusps and their inner structure, and the augmentation of the average burning velocity through flame wrinkling.

Abstract:   In this paper, we discuss some uses of curve evolution theory for problems in computer vision. We concentrate on three problem areas: shape theory, active contours, and geometric invariant scale spaces. The solutions to these key problems will all be based on flows which are obtained in a completely natural manner from geometric and physical principles.

Abstract:   In this paper, we analyze geometric active contour models from a curve evolution point of view and propose some modifications based on gradient flows relative to certain new feature-based Riemannian metrics. This leads to a novel edge-detection paradigm in which the feature of interest may be considered to lie at the bottom of a potential well. Thus an edge-seeking curve is attracted very naturally and efficiently to the desired feature. Comparison with the Allen-Cahn model clarifies some of the choices made in these models, and suggests inhomogeneous models which may in return be useful in phase transitions. We also consider some 3-dimensional active surface models based on these ideas. The justification of this model rests on the careful study of the viscosity solutions of evolution equations derived from a level-set approach.

Abstract:   We develop a level set theory for the mean curvature evolution of surfaces with arbitrary co-dimension, thus generalizing the previous work [8, 15] on hypersurfaces. The main idea is to surround the evolving surface of codimension-k in R(d) by a family of hypersurfaces (the level sets of a function) evolving with normal velocity equal to the sum of the (d - k) smallest principal curvatures. The existence and the uniqueness of a weak (level-set) solution is easily established by using mainly the results of [8] and the theory of viscosity solutions for second order nonlinear parabolic equations. The level set solutions coincide with the classical solutions whenever the latter exist. The proof of this connection uses a careful analysis of the squared distance from the surfaces. It is also shown that varifold solutions constructed by Brakke [7] are included in the level-set solutions. The idea of surrounding the evolving surface by a family of hypersurfaces with a certain property is related to the barriers of De Giorgi. An introduction to the theory of barriers and its connection to the level set solutions is also provided.

Abstract:   We present a unified approach to noise removal, image enhancement, and shape recovery in images. The underlying approach relies on the level set formulation of curve and surface motion, which leads to a class of PDE-based algorithms. Beginning with an image, the first stage of this approach removes noise and enhances the image by evolving the image under flow controlled by min/max curvature and by the mean curvature. This stage is applicable to both salt-and-pepper grey-scale noise and full-image continuous noise present in black and white images, grey-scale images, texture images, and color images. The noise removal/enhancement schemes applied in this stage contain only one enhancement parameter, which in most cases is automatically chosen. The other key advantage of our approach is that a stopping criteria is automatically picked from the image; continued application of the scheme produces no further change. The second stage of our approach is the shape recovery of a desired object; we again exploit the level set approach to evolve an initial curve/surface toward the desired boundary, driven by an image-dependent speed function that automatically stops at the desired boundary.

Abstract:   In this paper, we analyze the behavior of the anisotropic diffusion model of Perona and Malik. The main idea is to express the anisotropic diffusion equation as coming from a certain optimization problem, so its behavior can be analyzed based on the shape of the corresponding energy surface. We show that anisotropic diffusion is the steepest descent method for solving an energy minimization problem. It is demonstrated that an anisotropic diffusion is well posed when there exists a unique global minimum for the energy functional and that the ill posedness of a certain anisotropic diffusion is caused by the fact that its energy functional has an infinite number of global minima that are dense in the image space. We give a sufficient condition for an anisotropic diffusion to be well posed and a sufficient and necessary condition for it to be ill posed due to the dense global minima. The mechanism of smoothing and edge enhancement of anisotropic diffusion is illustrated through a particular orthogonal decomposition of the diffusion operator into two parts: one that diffuses tangentially to the edges and therefore acts as an anisotropic smoothing operator, and the other that flows normally to the edges and thus acts as an enhancement operator.

Abstract:   We discuss the parallel implemention of some algorithms for Hamilton-Jacobi equations based on dynamic programming. Our model problem is the first order partial differential equation related to the isotropic motion of fronts in the normal direction with velocity c(x). We present a local version of the serial algorithm which is well suited for the parallelisation via a domain decomposition strategy and we discuss the performance obtained using PVM and Linda on a cluster of UNIX workstations.

Abstract:   A front dynamics approach is developed to study the evolution of planar curves whose normal speed depends on curvature. The formulation is similar to Whitham's shock dynamics theory for the propagation of shock Nave in gases but assumes a different propagation rule. Equations that describe the motion of the front are obtained, and these are evolution equations for the normal direction and local are length of the front. The solution of these equations leads to the front positions using an appropriate integration along rays. A similarity solution of the equations is found for the evolution of an initial corner. Free-boundary problems for the motion of a junction connecting front segments are discussed. A numerical method is presented to calculate the evolution of any number of front segments. The segments can be closed or open, connected to Nail boundaries or not, or connected to other segments at 3-segment junctions. Several sample problems are considered to illustrate the method. An extension of the method for curvature-dependent motion under a constant area constraint is also discussed.

Abstract:   Visual tasks often require a hierarchical representation of shapes and images in scales ranging from coarse to fine. A variety of linear and nonlinear smoothing techniques, such as Gaussian smoothing, anisotropic diffusion, regularization, etc., have been proposed, leading to scalespace representations. We propose a geometric smoothing method based on local curvature for shapes and images. The deformation by curvature, or the geometric heat equation, is a special case of the reaction-diffusion framework proposed in [41]. For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by are-length at each infinitesimal step. For images, the smoothing is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge location is left intact. Curvature deformation smoothing for shape has a number of desirable properties: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semigroup property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature. The reassembly of these smoothed level sets into a smoothed image follows a number of mathematical properties; it is shown that the extension from smoothing shapes to smoothing images is mathematically sound due to a number of recent results [21]. A generalization of these results [14] justifies the extension of the entire entropy scale space for shapes [42] to one for images, where each iso-intensity level curve is deformed by a combination of constant and curvature deformation. The scheme has been implemented and is illustrated for several medical, aerial, and range images. (C) 1996 Academic Press, Inc.

Abstract:   Accurately segmenting and quantifying structures is a key issue in biomedical image analysis. The two conventional methods of image segmentation, region-based segmentation, and boundary finding, often suffer from a variety of limitations. Here we propose a method which endeavors to integrate the two approaches in an effort to form a unified approach that is robust to noise and poor initialization. Our approach uses Green's theorem to derive the boundary of a homogeneous region- classified area in the image and integrates this with a gray level gradient-based boundary finder. This combines the perceptual notions of edge/shape information with gray level homogeneity. A number of experiments were performed both on synthetic and real medical images of the brain and heart to evaluate the new approach, and it is shown that the integrated method typically performs better when compared to conventional gradient-based deformable boundary finding. Further, this method yields these improvements with little increase in computational overhead, an advantage derived from the application of the Green's theorem.

Abstract:   Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of ''motion by mean curvature'' is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well.

Abstract:   In this paper, we propose a geometric partial differential equation (PDE) for tracking one or several moving objects from a sequence of images, which is based on a geometric model for active contours. The active contour approach permits us to simultaneously handle both aspects: finding the boundaries and tracking them. We also describe a numerical scheme to solve the geometric equation and we present some numerical experiments.

Abstract:   We propose a finite element algorithm for computing the motion of a surface moving by mean curvature. The algorithm uses the level set formulation so that changes in topology of the surface can be accommodated. Stability is deduced by showing that the discrete solutions satisfy both L(infinity) and W-1,W- 1 bounds. Existence of discrete solutions and connections with Brakke flows are established. Some numerical examples and application to related problems, such as the phase field equations, are also presented.

Abstract:   Finding the shortest path between points on a surface is a challenging global optimization problem. It is difficult to devise an algorithm that is computationally efficient, locally accurate and guarantees to converge to the globally shortest path. In this paper a two stage coarse-to-fine approach for finding the shortest paths is suggested. In the first stage the algorithm of Ref. 10 that combines a 3D length estimator with graph search is used to rapidly obtain an approximation to the globally shortest path. In the second stage the approximation is refined to become a shorter geodesic curve, i.e., a locally optimal path. This is achieved by using an algorithm that deforms an arbitrary initial curve ending at two given surface points via geodesic curvature shortening flow. The 3D curve shortening how is transformed into an equivalent 2D one that is implemented using an efficient numerical algorithm for curve evolution with fixed end points, introduced in Ref. 9.

Abstract:   In the present paper we consider the Allen-Cahn equation in a class of domains consisting of a rectangular part with two attachments on its sides. We establish the existence of stationary solutions with nearly flat interfaces intersecting orthogonally the boundary of the domain at its rectangular part. We also show that the stability of these equilibria depends on the geometry of the domain. Finally we obtain some results regarding the dynamics of the Allen-Cahn equation, namely we construct an approximation of the invariant manifold associated with the equilibria (quasi-invariant manifold). Analysis of the vector field near this manifold suggests that the normal velocity of the flat interfaces is exponentially small in epsilon.

Abstract:   A novel interactive segmentation method has been developed which uses estimated boundaries, generated from cores, to initialize a scale-space boundary evolution process in greyscale medical images. Presented is an important addition to core extraction methodology that improves core generation for objects that are in the presence of interfering objects. The boundary at the scale of the core (BASOC) and its associated width information, both derived from the core, are used to initialize the second stage of the segmentation process. In this automatic refinement stage, the BASOC is allowed to evolve in a spline-snake-like manner that makes use of object-relevant width information to make robust measurements of local edge positions.

Abstract:   The range of surface evolution problems in etching, deposition, and lithography development offers significant challenge for numerical methods in front tracking. Level set methods for evolving interfaces are specifically designed for profiles which can develop sharp corners, change topology, and undergo orders of magnitude changes in speed, They are based on solving a Hamilton-Jacobi type equation for a level set function, using techniques borrowed from hyperbolic conservation laws. Over the past few years, a body of level set methods have been developed with application to microfabrication problems, In this paper, we give an overview of these techniques, describe the implementation in etching, deposition, and lithography simulations, and present a collection of fast level set methods, each aimed at a particular application, In the case of photoresist development and isotropic etching/deposition, the fast marching level set method, introduced by Sethian in [39], [40], can track the three-dimensional photoresist process through a 200x200x 200 rate function grid in under 55 s on a Sparc10. In the case of more complex etching and deposition, the narrow band level set method, introduced in Adalsteinsson and Sethian in [2], can be used to handle problems in which the speed of the interface delicately depends on the orientation of the interface versus an incoming beam, the effects of visibility, surface tension, reflection and re-emission, and complex three-dimensional effects, Our applications include photoresist development, etching/deposition problems under the effects of masking, visibility, complex flux integrations over sources, nonconvex sputter deposition problems, and simultaneous deposition and etch phenomena.

Abstract:   Multiscale representations and progressive smoothing constitute an important topic in different fields as computer vision, CAGD, and image processing. In this work, a multiscale representation of planar shapes is first described. The approach is based on computing classical B-splines of increasing orders, and therefore is automatically affine invariant. The resulting representation satisfies basic scale- space properties at least in a qualitative form, and is simple to implement. The representation obtained in this way is discrete in scale, since classical B-splines are functions in C-k-2, where k is an integer bigger or equal than two. We present a subdivision scheme for the computation of B-splines of finite support at continuous scales. With this scheme, B- splines representations in C-r are obtained for any real r in [0, infinity), and the multiscale representation is extended to continuous scale. The proposed progressive smoothing receives a discrete set of points as initial shape, while the smoothed curves are represented by continuous (analytical) functions, allowing a straightforward computation of geometric characteristics of the shape.

Abstract:   The usual explanation of the gradient-weighted moving finite element (GWMFE) method has been in terms of its variational interpretation. This paper presents a more intuitive geometrical-mechanical interpretation of GWMFE as a balance of forces on the nodes, forces concentrated onto the nodes by the laws of leverage. It also presents significant simplifications in the ''internodal viscosity'' terms for regularization of the nodal movements, plus some simple ''linear internodal tensions'' for regularization of the long-term nodal positioning. These simplifications of the regularizations are especially important in two and three space dimensions. One of the generalizations which follows from the geometrical- mechanical interpretation is a promising but still untested second GWMFE formulation for systems of PDEs. The original MFE method is seen to be the small-slope limit of GWMFE under ''vertical rescaling.'' Reporting on the design and extensive numerical trials of robust and versatile GWMFE systems codes in one and two dimensions is deferred to two forthcoming papers by Carlson and the author [Design and application of a gradient- weighted moving finite element code, Part I, in 1-D, SIAM J. Sci. Comput., to appear] and [Design and application of a gradient-weighted moving finite element code, Part II, in 2-D, SIAM J. Sci. Comput., to appear]. Here only a few illustrative examples are presented involving motion of surfaces by mean curvature, i.e., by surface tension.

Abstract:   We present a numerical method for computing solutions of the incompressible Euler or Navier-Stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties. The method is based on a second-order projection method for variable density flows using an ''approximate projection'' formulation. The boundary between the fluids is tracked with a second-order, volume-of-fluid interface tracking algorithm. We present results for viscious Rayleigh-Taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm. We also present computational results for the Rayleigh-Taylor instability in air-helium and for bubbles and drops in an air-water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts. (C) 1997 Academic Press.

Abstract:   We present a numerical method for computing solutions of the incompressible Euler or Navier-Stokes equations when a principal feature of the flow is the presence of an interface between two fluids with different fluid properties. The method is based on a second-order projection method for variable density flows using an ''approximate projection'' formulation. The boundary between the fluids is tracked with a second-order, volume-of-fluid interface tracking algorithm. We present results for viscious Rayleigh-Taylor problems at early time with equal and unequal viscosities to demonstrate the convergence of the algorithm. We also present computational results for the Rayleigh-Taylor instability in air-helium and for bubbles and drops in an air-water system without surface tension to demonstrate the behavior of the algorithm on problems with large density and viscosity contrasts. (C) 1997 Academic Press.

Abstract:   This paper presents a general framework to generate multi-scale representations of image data. The process is considered as an initial value problem with an acquired image as initial condition and a geometrical invariant as ''driving force'' of an evolutionary process. The geometrical invariants are extracted using the family of Gaussian derivative operators. These operators naturally deal with scale as a free parameter and solve the ill-posedness problem of differentiation. Stability requirements for numerical approximation of evolution schemes using Gaussian derivative operators are derived and establish an intuitive connection between the allowed time-step and scale. This approach has been used to generalize and implement a variety of nonlinear diffusion schemes. Results on test images and medical images are shown.

Abstract:   This paper presents a general framework to generate multi-scale representations of image data. The process is considered as an initial value problem with an acquired image as initial condition and a geometrical invariant as ''driving force'' of an evolutionary process. The geometrical invariants are extracted using the family of Gaussian derivative operators. These operators naturally deal with scale as a free parameter and solve the ill-posedness problem of differentiation. Stability requirements for numerical approximation of evolution schemes using Gaussian derivative operators are derived and establish an intuitive connection between the allowed time-step and scale. This approach has been used to generalize and implement a variety of nonlinear diffusion schemes. Results on test images and medical images are shown.

Abstract:   We present the results of numerical simulations of the three- dimensional thermocapillary motion of deformable viscous drops under the influence of a constant temperature gradient within a second liquid medium. In particular, we examine the effects of shape deformations and convective transport of momentum and energy on the migration velocity of the drop. A numerical method based on a continuum model for the fluid-fluid interface is used to account for finite drop deformations. An oct-tree adaptive grid refinement scheme is integrated into the numerical method in order to track the interface without the need for interface reconstruction. Interface deformations arising from the convection of energy at small Reynolds numbers are found to be negligible. On the other hand, deformations of the drop shape due to inertial effects? though small in magnitude, are found to retard the motion of the drop. The steady drop shapes are found to resemble oblate or prolate spheroids without fore and aft symmetry, with the direction of elongation of the drop depending on the value of the density ratio between the two phases. As in the case of a gas bubble, convection of energy is shown to retard the thermocapillary motion of a viscous drop, as the isotherms get wrapped around the front surface of the drop and effectively reduce the surface temperature gradient which drives the motion. The effect of inertia on the mobility of viscous drops is found to be weaker than that in the case of gas bubbles. (C) 1997 American Institute of Physics.

Abstract:   The segmentation of structure from images is an inherently difficult problem in computer vision and a bottleneck to its widespread application, e.g., in medical imaging, This paper presents an approach for integrating local evidence such as regional homogeneity and edge response to form global structure for figure-ground segmentation. This approach is motivated by a shock-based morphogenetic language, where the growth of four types of shocks results in a complete description of shape, Specifically, objects are randomly hypothesized in the form of fourth-order shocks (seeds) which then grow, merge, split, shrink, and, in general, deform under physically motivated ''forces,'' but slow down and come to a halt near differential structures. Two major issues arise in the segmentation of 3D images using this approach. First, it is shown that the segmentation of 3D images by 3D bubbles is superior to a slice- by-slice segmentation by 2D bubbles or by ''21/2D bubbles'' which are inherently 2D but use 3D information for their deformation. Specifically, the advantages lie in an intrinsic treatment of the underlying geometry and accuracy of reconstruction. Second, gaps and weak edges, which frequently present a significant problem for 2D and 3D segmentation, are regularized by curvature-dependent curve and surface deformations which constitute diffusion processes, The 3D bubbles evolving in the 3D reaction-diffusion space are a powerful tool in the segmentation of medical and other images, as illustrated for several realistic examples. (C) 1997 Academic Press.

Abstract:   We present a novel geometric approach for solving the stereo problem for an arbitrary number of images (greater than or equal to 2). It is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images. The Euler-Lagrange equations which are deduced from the variational principle provide a set of PDE's which are used to deform an initial set of surfaces which then move towards the objects to be detected. The level set implementation of these PDE's potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformation, i.e. to deal with multiple objects. Results of a two dimensional implementation of our theory are presented on synthetic and real images.

Abstract:   In this note, we employ the new geometric active contour models formulated in [25] and [26] for edge detection and segmentation of magnetic resonance imaging (MRI), computed tomography (CT), and ultrasound medical imagery, Our method is based on defining feature-based metrics on a given image which in turn leads to a novel snake paradigm in which the feature of interest mag be considered to lie at the bottom of a potential well, Thus, the snake is attracted very quickly and efficiently to the desired feature.

Abstract:   A novel scheme for the detection of object boundaries is presented. The technique is based on active contours evolving in time according to intrinsic geometric measures of the image. The evolving contours naturally split and merge, allowing the simultaneous detection of several objects and both interior and exterior boundaries. The proposed approach is based on the relation between active contours and the computation of geodesics or minimal distance curves. The minimal distance curve lays in a Riemannian space whose metric is defined by the image content. This geodesic approach for object segmentation allows to connect classical ''snakes'' based on energy minimization and geometric active contours based on the theory of curve evolution. Previous models of geometric active contours are improved, allowing stable boundary detection when their gradients suffer from large variations, including gaps. Formal results concerning existence, uniqueness, stability, and correctness of the evolution are presented as well. The scheme was implemented using an efficient algorithm for curve evolution. Experimental results of applying the scheme to real images including objects with holes and medical data imagery demonstrate its power. The results may be extended to 3D object segmentation as well.

Abstract:   Consider a closed surface in R(n) of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term. This geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon > 0. Conforming piecewise linear finite elements over a quasi- uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time-step tau. It is shown that the zero level set of the fully discrete solution converges past singularities to the true interface, provided tau, h(2) approximate to o(epsilon(3)) and no fattening occurs. If the more stringent relations tau, h(2) approximate to O(epsilon(4)) are enforced, then a linear rate of convergence O(epsilon) for interfaces is derived in the vicinity of regular points, namely those for which the underlying viscosity solution is nondegenerate. Singularities and their smearing effect are also studied. The analysis is based on constructing discrete barriers via a parabolic projection, Lipschitz dependence of viscosity solutions with respect to perturbations of data, and discrete nondegeneracy. These issues are proven, along with quasi optimality in two dimensions of the parabolic projection in L(infinity) with respect to both order and regularity requirements for functions in W-p(2,1).

Abstract:   Consider a closed surface in R(n) of codimension 1 which propagates in the normal direction with velocity proportional to its mean curvature plus a forcing term. This geometric problem is first approximated by a singularly perturbed parabolic double obstacle problem with small parameter epsilon > 0. Conforming piecewise linear finite elements over a quasi- uniform and strongly acute mesh of size h are further used for space discretization and combined with backward differences for time discretization with uniform time-step tau. It is shown that the zero level set of the fully discrete solution converges past singularities to the true interface, provided tau, h(2) approximate to o(epsilon(3)) and no fattening occurs. If the more stringent relations tau, h(2) approximate to O(epsilon(4)) are enforced, then a linear rate of convergence O(epsilon) for interfaces is derived in the vicinity of regular points, namely those for which the underlying viscosity solution is nondegenerate. Singularities and their smearing effect are also studied. The analysis is based on constructing discrete barriers via a parabolic projection, Lipschitz dependence of viscosity solutions with respect to perturbations of data, and discrete nondegeneracy. These issues are proven, along with quasi optimality in two dimensions of the parabolic projection in L(infinity) with respect to both order and regularity requirements for functions in W-p(2,1).

Abstract:   The explicit use of partial differential equations (PDEs) in image processing became a major research topic in the past years. In this work we present a framework for histogram (pixel-value distribution) modification via ordinary and partial differential equations. In this way, the image contrast is improved. We show that the histogram can be modified to achieve any given distribution as the steady state solution of an image now. The contrast modification can be performed while simultaneously reducing noise in a unique PDE, avoiding noise sharpening effects of classical algorithms. The approach is extended to local contrast enhancement as well. A variational interpretation of the flow is presented and theoretical results on the existence of solutions are given. (C) 1997 Academic Press.

Abstract:   The time-dependent motion for a two-layer Couette flow consisting of fluids of different viscosities is simulated numerically by using an algorithm based on the Volume of Fluid (VOF) method. Interfacial tension is included via a continuous surface force (CSF) algorithm. The algorithm is fine-tuned to handle the motion which is driven by a shear-induced interfacial instability due to the viscosity stratification. The code is validated against linear theory. Two prototypical situations are presented, one at a moderately high Reynolds number and the other at a lower Reynolds number. The initial condition is seeded with the eigenmode of largest growth rate, with amplitudes that are varied from those that capture the linear regime to larger values for nonlinear regimes. Issues of free surface advection and viscosity interpolation are discussed. The onset of nonlinearity occurs at the interface and is quadratic, followed by wave steepening. (C) 1997 Academic Press.

Abstract:   A geometric approach for 3D object segmentation and representation is presented. The segmentation is obtained by deformable surfaces moving towards the objects to be detected in the 3D image. The model is based on curvature motion and the computation of surfaces with minimal areas, better known as minimal surfaces. The space where the surfaces are computed is induced from the 3D image (volumetric data) in which the objects are to be detected. The model links between classical deformable surfaces obtained via energy minimization, and intrinsic ones derived from curvature based flows. The new approach is stable, robust, and automatically handles changes in the surface topology during the deformation.

Abstract:   Optimization processes based on ''active models'' play central roles in many areas of computational vision as well as computational geometry. Unfortunately, current models usually require highly complex and sophisticated mathematical machinery and at the same time they suffer from a number of limitations which impose restrictions on their applicability. In this paper a simple class of discrete active models, called migration processes (MPs), is presented. The processes are based on iterated averaging over neighborhoods defined by constant geodesic distance. It is demonstrated that the MP model-a system of self-organizing active particles-has a number of advantages over previous models, both parametric active models (''snakes'') and implicit (contour evolution) models. Due to the generality of the MP model, the process can be applied to derive natural solutions to a variety of optimization problems,including defining (minimal) surface patches given their boundary curves; finding shortest paths joining sets of points; and decomposing objects into ''primitive'' parts. (C) 1997 Pattern Recognition Society.

Abstract:   A second-order accurate interface tracking method for the solution of incompressible Stokes flow problems with moving interfaces on a uniform Cartesian grid is presented. The interface may consist of an elastic boundary immersed in the fluid or san interface between two different fluids. The interface is represented by a cubic spline along which the singularly supported elastic or surface tension force can be computed. The Stokes equations are then discretized using the second-order accurate finite difference methods for elliptic equations with singular sources developed in our previous paper [SIAM J. Numer. Anal., 31(1994), pp. 1019-1044]. The resulting velocities are interpolated to the interface to determine the motion of the interface. An implicit quasi-Newton method is developed that allows reasonable time steps to be used.

Abstract:   We call "natural" image any photograph of an outdoor or indoor scene taken by a standard camera. In such images, most observed objects undergo occlusions and the illumination condition and contrast response of the camera are unknown. Actual Scale Space theories do not incorporate obvious restrictions imposed by the physics of image generation. The heat equation (linear scale space) is not contrast invariant and destroys T-junctions. The same is true for the recently proposed curvature equations (mean curvature motion and affine shortening): They break the symmetry of junctions. To apply directly these models to natural world images, With occlusions, is irrevelant. Returning to the edge detection problem, in which scale space theory originates, we show how level lines can be found in an image without smoothing. As an alternative to edge detection/scale space, we propose to define the line structure in a natural image by its topographic map (set of all level lines). We also show that a modification of morphological scale space can help to the visualization of the topographic map.

Abstract:   A 1D lagrangian formulation, equivalent to the 2D G equation of 2D propagating fronts in the geometrical optics approximation, is introduced. Fractal flames are obtained numerically by this method for flow fields containing a large number of scales.

Abstract:   The study of geometric flows for smoothing, multiscale representation, and analysis of two- and three-dimensional objects has received much attention in the past few years. In this paper, we first survey the geometric smoothing of curves and surfaces via geometric heat-type flows, which are invariant under the groups of Euclidean and affine motions. Second, using the general theory of differential invariants, we determine the general formula for a geometric hypersurface evolution which is invariant under a prescribed symmetry group. As an application, we present the simplest affine invariant flow for (convex) surfaces in three-dimensional space, which, like the affine- invariant curve shortening flow, will be of fundamental importance in the processing of three-dimensional images.

Abstract:   In order to segment an image the use of information at multiple scales is invaluable. The hyperstack, a linking-model-based segmentation technique, uses intensity to link points in adjacent levels of a scale space stack. This approach has been successfully applied to linear multiscale representations. Multiscale representions which satisfy two scale space properties, viz. a causality criterion and a semigroup property in differential form, are valid inputs as well. In this paper we consider linear scale space, gradient-dependent diffusion, and the Euclidean shortening flow. Since no global scale parameter is available in the latter two approaches we compare scale levels based on evolution time, information theoretic measures, and by counting the number of objects. The multiscale representations are compared with respect to their performance in image segmentation tasks on test and MR images. The hyperstack proves to be rather insensitive to the underlying multiscale representation although the nonlinear representations reduced the number of post processing steps. (C) 1997 Academic Press.

Abstract:   This paper presents a method for reconstructing the 3D shape of an object from its endoscope image based on image shading. The primary problem is that the endoscope has a light source near the object surface. Most of the conventional shape from shading methods assumed that the light source was distant from the object surface and simplified the analysis. To deal with the near light source, we use the configuration of the endoscope that the light source of the endoscope is well approximated by an imaginary point source at the projection center. In addition, we introduce a notion of equal distance contours of the object surface; by propagating the contours using the image shading, we reconstruct the object shape. This is an extension of the Kimmel-Bruckstein algorithm of shape from shading to the endoscope images. Experimental results for real medical endoscope images of the stomach wall show the feasibility of this method and also show its promising availability for morphological analyses of tumors on human inner organs. (C) 1997 Academic Press.

Abstract:   We present results of numerical experimentation with a 2-D version of an equation of surface dynamics that has been derived earlier in the context of flame fronts [Fankel & Sivashinsky, 1987, 1988] and solid-liquid interfaces [Frankel, 1988]. Our observations confirm qualitative predictions of Frankel & Sivashinsky [1987, 1988]: the curves develop chaotic cellular pattern and accelerate while imbedding is sustained. However, if we allow self-intersections, in a different range of parameters the equation gives birth to remarkably complex and beautiful fractal-like structures either entirely chaotic or preserving any symmetry if inherited from the initial configuration. This accumulation of complexity is also manifested in exponential growth of the length while diameter of the set increases linearly which results in increasingly dense covering of the plane. Based on these observations we introduce concepts of self-fractalizing family and asymptotic fractal dimension, which turns out to be equal to two.

Abstract:   A scale space for images painted on surfaces is introduced. Based on the geodesic curvature pow of the iso-gray level contours of an image painted on the given surface, the image is evolved and forms the natural geometric scale space. Its geometrical properties are discussed as well as the intrinsic nature of the proposed flow. I.e. the flow is invariant to the bending of the surface.

Abstract:   This paper concerns the extension of the gas-kinetic BGK-type scheme to multicomponent flow calculations. In this new scheme, each component satisfies its individual gas-kinetic BGK equation and the equilibrium states for each component are coupled in space and time to have common temperature and velocity. The particle diffusion in gas mixtures is included naturally in the gas-kinetic model. The current scheme can handle strong shocks and be oscillation-free through the material interface. The scheme guarantees the exact mass conservation for each component and the exact conservation of total momentum and energy in the whole particle system. As a special application, the current scheme is applied to gas vacuum interaction case, where the mass densities for other components are set to zero in the whole domain. The extension of the current approach to three dimensions is straightforward. With the definition of phi = rho((1)) - rho((2)) in the two- component gas flow, similar to the level set method we can follow explicitly the time evolution of the material interface (phi = 0). The numerical results confirm the accuracy and robustness of the BGK-type scheme. (C) 1997 Academic Press.

Abstract:   The present work is concerned with an application of the theory of characteristics to conservation laws with source terms in one space dimension, such as the Euler equations for reacting flows. Space-time paths are introduced on which the flow/chemistry equations decouple to a characteristic set of ODE's for the corresponding homogeneous laws, thus allowing the introduction of functions analogous to the Riemann invariants in classical theory. The geometry of these paths depends on the spatial gradients of the solution. This particular decomposition can be used in the design of efficient unsplit algorithms for the numerical integration of the equations. As a first step, these ideas are implemented for the case of a scalar conservation law with a nonlinear source term. The resulting algorithm belongs to the class of MUSCL-type, shock- capturing schemes. Its accuracy and robustness are checked through a series of tests. The stiffness of the source term is also studied. Then, the algorithm is generalized for a system of hyperbolic equations, namely the Euler equations for reacting flows. A numerical study of unstable detonations is performed. (C) 1997 Academic Press.

Abstract:   Based on artificial compressibility and dual time-stepping, an implicit scheme is developed for solving the steady and unsteady incompressible Navier-Stokes equations for one and two-phase flows. The scheme is centered but, due to its internal dissipation, it needs no staggered grid or upwinding to be stable. Its stability with respect to pseudo and physical times and convergence to a steady state are analyzed for a scalar model equation and partially for the full 2-D Navier- Stokes equations. The scheme is applied to the calculation of flow over a flat plate and steady or unsteady flows in lid- driven cavities. Thanks to a level-set interface tracking method, the method is also applied to model the impingement of a liquid drop on a solid wail. Accurate solutions are obtained compared to analytical, numerical and experimental published results. (C) 1997 Elsevier Science Ltd.

Abstract:   We study Ising models with general spin-flip dynamics obeying the detailed balance law. After passing to suitable macroscopic limits, we obtain interfaces moving with normal velocity depending anisotropically on their principal curvatures and direction. In addition we deduce ( direction-dependent) Kubo- Green-type formulas for the mobility and the Hessian of the surface tension, thus obtaining an explicit description of anisotropy in terms of microscopic quantities. The choice of dynamics affects only the mobility, a scalar function of the direction.

Abstract:   An efficient difference algorithm for computing directly deformation less solid objects suspended in stratified flow in 2D has been developed. The objects are represented by colour functions (or density functions) and predicted by a sharpness preserving scheme that is able to prevent the numerical diffusion across the sharp interface, Pressure distribution is then calculated by a unified solver and the solid object is treated as a mass of material of high sound speed, The motion of the solid object is decomposed into translation and rotation, and the force as well as the torque that cause change in the motion of the solid body are evaluated by an averaging calculation over the region occupied by the solid body. Calculations are conducted on a fixed grid system, Operations for reconstructing moving interfaces or dealing with inner boundary conditions are not necessary.

Abstract:   A second order difference method is developed for the nonlinear moving interface problem of the form u(t) + lambda uu(x) = (beta u(x))(x) - f(x, t), x is an element of [0,alpha) boolean OR (alpha, 1], d alpha/dt = w(t, alpha; u, u(x)), where alpha(t) is the moving interface. The coefficient beta(x, t) and the source term f(x, t) can be discontinuous across alpha(t) and moreover, f(x, t) may have a delta or/and delta- prime function singularity there. As a result, although the equation is parabolic, the solution u and its derivatives may be discontinuous across alpha(t). Two typical interface conditions are considered. One condition occurs in Stefan-like problems in which the solution is known on the interface. A new stable interpolation strategy is proposed. The other type occurs in a one-dimensional model of Peskin's immersed boundary method in which only jump conditions are given across the interface. The Crank-Nicolson difference scheme with modifications near the interface is used to solve for the solution u(x, t) and the interface alpha(t) simultaneously. Several numerical examples, including models of ice-melting and glaciation, are presented. Second order accuracy on uniform grids is confirmed both for the solution and the position of the interface.

Abstract:   This article describes a method for reducing the shape distortions due to scale space smoothing that arise in the computation of 3-D shape cues using operators (derivatives) defined from scale-space representation. More precisely, we are concerned with a general class of methods for deriving 3-D shape cues from a 2-D image data based on the estimation of locally linearized deformations of brightness patterns. This class constitutes a common framework for describing several problems in computer vision (such as shape-from-texture, shape- from disparity-gradients, and motion estimation) and for expressing different algorithms in terms of similar types of visual front-end-operations. It is explained how surface orientation estimates will be biased due to the use of rotationally symmetric smoothing in the image domain. These effects can be reduced by extending the linear scale-space concept into an affine Gaussian scale-space representation and by performing affine shape adaptation of the smoothing kernels. This improves the accuracy of the surface orientation estimates, since the image descriptors, on which the methods are based, will be relative invariant under affine transformations, and the error thus confined to the higher- order terms in the locally linearized perspective transformation. A straightforward algorithm is presented for performing shape adaptation in practice. Experiments on real and synthetic images with known orientation demonstrate that in the presence of moderately high noise levels the accuracy is improved by typically one order of magnitude.

Abstract:   In this paper, a hybrid approach which combines the immersed interface method with the level set approach is presented. The fast version of the immersed interface method is used to solve the differential equations whose solutions and their derivatives may be discontinuous across the interfaces due to the discontinuity of the coefficients or/and singular sources along the interfaces. The moving interfaces then are updated using the newly developed fast level set formulation which involves computation only inside some small tubes containing the interfaces. This method combines the advantage of the two approaches and gives a second-order Eulerian discretization for interface problems. Several key steps in the implementation are addressed in detail. This new approach is then applied to Hele- Shaw flow, an unstable flow involving two fluids with very different viscosity. (C) 1997 Academic Press.

Abstract:   We present a number of three-dimensional axisymmetric free boundary problems for two immiscible fluids, such as air and water. A level set method is used where the interface is the zero level set of a continuous function while the two fluids are solutions of the incompressible Navier-Stokes equation. We examine the rise and distortion of an initially spherical bubble into cap bubbles and toroidal bubbles. Steady solutions for gas bubbles rising in a liquid are computed, with favourable comparisons to experimental data. We also study the inviscid limit and compare our results with a boundary integral method. The problems of an air bubble bursting at a free surface and a liquid drop hitting a free surface are also computed.

Abstract:   A simple level set method for solving Stefan problems is presented. This method can be applied to problems involving dendritic solidification. Our method consists of an implicit finite difference scheme for solving the heat equation and a level set approach for capturing the front between solid and liquid phases of a pure substance. Our method is accurate with respect to some exact solutions of the Stefan problem. Results indicate that this method can handle topology changes and complicated interfacial shapes and that it can numerically simulate many of the physical features of dendritic solidification. (C) 1997 Academic Press.

Abstract:   We introduce a relaxation model for front propagation problems. Our proposed relaxation approximation is a semilinear hyperbolic system without singularities. It yields a direction- dependent normal velocity at the leading term and captures, in the Chapman-Enskog expansion, the higher order curvature dependent corrections, including possible anisotropies. (C) 1997 Academic Press.

Abstract:   We introduce a relaxation model for front propagation problems. Our proposed relaxation approximation is a semilinear hyperbolic system without singularities. It yields a direction- dependent normal velocity at the leading term and captures, in the Chapman-Enskog expansion, the higher order curvature dependent corrections, including possible anisotropies. (C) 1997 Academic Press.

Abstract:   The increasing complexity of VLSI device interconnect features and fabrication technologies encountered by semiconductor etching and deposition simulation necessitates improvements in the robustness, numerical stability, and physical accuracy of the boundary movement method, The volume-mesh-based level set method, integrated with the physical models in SPEEDIE, demonstrates accuracy and robustness for simulations on a wide range of etching/deposition processes The surface profile is reconstructed from the well-behaved level set function without rule-based algorithms, Adaptive gridding is used to accelerate the computation, Our algorithm can be easily extended from two- dimensional (2-D) to three-dimensional (3-D), and applied to model microstructures consisting of multiple materials, Efficiency benchmarks show that this boundary movement method is practical in 2-D, and competitive for larger scale or 3-D modeling applications.

Abstract:   The Hamilton-Jacobi equation describes the dynamics of a hypersurface in R-n. This equation is a nonlinear conservation law and thus has discontinuous solutions. The dependent variable is a surface gradient and the discontinuity is a surface cusp. Here we investigate the intersection of cusp hypersurfaces. These intersections define (n-1)-dimensional Riemann problems for the Hamilton-Jacobi equation. We propose the class of Hamilton-Jacobi equations as a natural higher- dimensional generalization of scalar equations which allow a satisfactory theory of higher-dimensional Riemann problems, The first main result of this paper is a general framework for the study of higher-dimensional Riemann problems for Hamilton- Jacobi equations. The purpose of the framework is to understand the structure of Hamilton-Jacobi wave interactions in an explicit and constructive manner. Specialized to two- dimensional Riemann problems (i.e., the intersection of cusp curves on surfaces embedded in R-3), this framework provides explicit solutions to a number of cases of interest. We are specifically interested in models of deposition and etching, important processes for the manufacture of semiconductor chips. We also define elementary waves as Riemann solutions which possess a common group velocity, Our second main result, for elementary waves, is a complete characterization in terms of algebraic constraints on the data. When satisfied, these constraints allow a consistently defined closed form expression for the solution. We also give a computable characterization for the admissibility of an elementary wave which is inductive in the codimension of the wave, and which generalizes the classical Oleinik condition for scalar conservation laws in one dimension.

Abstract:   A new boundary detection approach for shape modeling is presented. It detects the global minimum of an active contour model's energy between two end points. Initialization is made easier and the curve is not trapped at a local minimum by spurious edges. We modify the ''snake'' energy by including the internal regularization term in the external potential term. Our method is based on finding a path of minimal length in a Riemannian metric. We then make use of a new efficient numerical method to find this shortest path. It is shown that the proposed energy, though based only on a potential integrated along the curve, imposes a regularization effect like snakes. We explore the relation between the maximum curvature along the resulting contour and the potential generated from the image. The method is capable to close contours, given only one point on the objects' boundary by using a topology-based saddle search routine. We show examples of our method applied to real aerial and medical images.

Abstract:   Numerical analysis of conservation laws plays an important role in the implementation of curve evolution equations. This paper reviews the relevant concepts in numerical analysis and the relation between curve evolution, Hamilton-Jacobi partial differential equations, and differential conservation laws. This close relation enables us to introduce finite difference approximations, based on the theory of conservation laws, into curve evolution. It is shown how curve evolution serves as a powerful tool for image analysis, and how these mathematical relations enable us to construct efficient and accurate numerical schemes. Some examples demonstrate the importance of the CFL condition as a necessary condition for the stability of the numerical schemes.

Abstract:   We discuss some properties of barriers and minimal barriers in the sense of De Giorgi for geometric evolutions of subsets of R-n We point out the role played by the regularizations M*(E,f(H)), M*(E, f(H)) in the comparison with others generalized geometric flows, and in the fattening phenomenon.

Abstract:   A shock driven inter-facial instability, known as the Richtmyer-Meshkov instability, is studied numerically in two and three dimensions and in the nonlinear regime. The numerical solution is tested for convergence under computational mesh refinement and is compared with the predictions of a recently developed nonlinear theory based on the Pade approximation and asymptotic matching, Good agreement has been found between numerical solutions and predictions of the nonlinear theory in both two and three dimensions and for both the reflected shock and the reflected rarefaction wave cases. The numerical study is extended to the re-shock experiment in which the fluid interface interacts initially with the incident shock. Later, as the transmitted shock bounces back from the wall, the fluid interface is re-shocked. (C) 1997 American Institute of Physics.

Abstract:   A novel geometric approach for three dimensional object segmentation is presented. The scheme is based on geometric deformable surfaces moving towards the objects to be detected, We show that this model is related to the computation of surfaces of minimal area (local minimal surfaces). The space where these surfaces are computed is induced from the three dimensional image in which the objects are to be detected. The general approach also shows the relation between classical deformable surfaces obtained via energy minimization and geometric ones derived from curvature flows in the surface evolution framework. The scheme is stable, robust, and automatically handles changes in the surface topology during the deformation. Results related to existence, uniqueness, stability, and correctness of the solution to this geometric deformable model are presented as well. Based on an efficient numerical algorithm for surface evolution, we present a number of examples of object detection in real and synthetic images.

Abstract:   Moving-boundary problems arise in numerous important physical phenomena, and often form complex shapes during their evolution. The ability to track the interface in such cases in two dimensions is well established. However, modifying the grid representing the interface as it evolves in three-dimensional space introduces additional issues. In the current work, three- dimensional interfaces are represented by adaptive unstructured grids. The grids are restructured and refined based on the shape and size of the triangular elements in the grid that forms the interfaces. As the interface deforms, points are automatically added to ensure that the accuracy of interface representation remains consistent. Results are presented to show how complex interface features, including surface curvatures and normals, can be captured by modifying an existing method that uses an approximation to the Dupin indicatrix.

Abstract:   The eddy damped quasi-normal Markovian (EDQNM) turbulence theory was applied to a modified Kuramoto-Sivashinsky field equation to develop a spectral model for investigating the single and two-point scalar statistics associated with a flame front (treated as a passive scalar interface) propagating through isotropic turbulence. As a result of the presence of a uniform mean gradient in the scalar field, all correlations involving the scalar were found to be functions of both the wave number, k, and mu, the cosine of the angle between the ik vector and the mean gradient vector. An infinite Legendre expansion separated out the wave number and angle dependencies, where the first term in each series accounted for the isotropic contribution to the correlations and the higher order terms accounted for the anisotropy introduced as a result of the mean gradient. It was found that while strong anisotropy existed in the scalar field at short times, at steady state the scalar field became nearly isotropic. A parameter study was then conducted to ascertain the effect of independently varying u'/s(L) and Re-lambda (where u' is the rms velocity, s(L) is the laminar burning velocity, and Re-lambda is the Reynolds number based on the Taylor microscale). The turbulent burning velocity increased with increases in either u'/s(L) or Re- lambda, however, the model predicted a finite turbulent burning velocity as u'/s(L) --> infinity, even though flame quenching was not accounted for. This finite asymptote for the burning velocity was traced to tile constitutive relationship used for the flame thickness and the ratio of the Markstein length to the flame thickness. It was also shown that the dominant wrinkling of the flame surface and subsequent contribution to the turbulent burning velocity occurred at smaller and smaller length scales as the inertial range of the scalar spectrum increased. Single point models will therefore have great difficulty reproducing this significant result. Scalar spectra exhibited changes over all wave numbers as either u'/s(L) or Re-lambda, was modified. Transfer spectra, which arose in the form of convolution integrals as a result of the advection and propagation processes, were also analyzed and separated into their pairwise spectral interactions to determine which nonlinear terms in each integral were dominant. (C) 1997 American Institute of Physics.

Abstract:   in this work. a mixed Eulerian-Lagrangian algorithm, called ELAFINT (Eulerian Lagrangian algorithm for interface tracking is developed further and applied to compute hows with solid- fluid and fluid-fluid interfaces. The method is capable of handling fluid Bows in the presence of both irregularly shaped solid boundaries and moving boundaries on a fixed Cartesian grid. The held equations are solved on the underlying fixed grid using a collocated variable, pressure-based formulation. The moving boundary is tracked explicitly the Lagrangian translation of marker particles. The moving boundary passes through the grid and the immersed boundary technique is used to handle its interaction with the underlying grid. The internal solid boundaries are dealt with by using a cut-cell technique. Particular attention is directed toward conservation and consistency in the vicinity of both phase boundaries. The complex geometry feature has been tested for a variety of Bow problems. The performance of the immersed boundary representation is demonstrated in the simulation of Newtonian Liquid drops. The combination of the two features is then employed in the simulation of motion of drops through constricted tubes. The capabilities developed here can be useful for solving Bow problems involving moving and stationary complex boundaries. (C) 1997 Academic Press.

Abstract:   In this paper we study the application of the Affine Morphological Scale Space (AMSS) to the analysis of singularities (corners or multiple junctions) of the shapes present in a 2-D image. We introduce a new family of travelling wave solutions of AMSS which determines the evolution of the initial shapes given by conics. We characterize the evolution of corners accross the scales according to their angle, We develop a numerical algorithm to compute AMSS accross the scales and we present some experimental results about corners and multiple junction detection.

Abstract:   A framework for object segmentation in vector-valued images is presented in this paper. The first scheme proposed is based on geometric active contours moving toward the objects to be detected in the vector-valued image. Object boundaries are obtained as geodesics or minimal weighted-distance curves, where the metric is given by a definition of edges in vector- valued data. The curve flow corresponding to the proposed active contours holds formal existence, uniqueness, stability, and correctness results. The scheme automatically handles changes in the deforming curve topology. The technique is applicable, for example, to color and texture images as well as multiscale representations. We then present an extension of these vector active contours, proposing a possible image flow for vector-valued image segmentation. The algorithm is based on moving each one of the image level sets according to the proposed vector active contours. This extension also shows the relation between active contours and a number of partial- differential-equations-based image processing algorithms as anisotropic diffusion and shock filters. (C) 1997 Academic Press.

Abstract:   A scale space for images painted on surfaces is introduced. Based on the geodesic curvature flow of the iso-gray level contours of an image painted on the given surface, the image is evolved and forms the natural geometric scale space. Its geometrical properties are discussed as well as the intrinsic nature of the proposed flow; i.e., the flow is invariant to the bending of the surface. (C) 1997 Academic Press.

Abstract:   Subpixel methods that locate curves and their singularities, and that accurately measure geometric quantities, such as orientation and curvature, are of significant importance in computer vision and graphics. Such methods often use local surface fits or structural models for a local neighborhood of the curve to obtain the interpolated curve. Whereas their performance is good in smooth regions of the curve, it is typically poor in the vicinity of singularities. Similarly, the computation of geometric quantities is often regularized to deal with noise present in discrete data. However, in the process, discontinuities are blurred over, leading to poor estimates at them and in their vicinity. In this paper we propose a geometric interpolation technique to overcome these limitations by locating curves and obtaining geometric estimates while (1) not blurring across discontinuities and (2) explicitly and accurately placing them, The essential idea is to avoid the propagation of information across singularities. This is accomplished by a one-sided smoothing technique, where information is propagated from the direction of the side with the ''smoother'' neighborhood. When both sides are nonsmooth, the two existing discontinuities are relieved by placing a single discontinuity, or shock. The placement of shacks is guided by geometric continuity constraints, resulting in subpixel interpolation with accurate geometric estimates. Since the technique was originally motivated by curve evolution applications, we demonstrate its usefulness in capturing not only smooth evolving curves, but also ones with orientation discontinuities. In particular, the technique is shown to be far better than traditional methods when multiple or entire curves are present in a very small neighborhood. (C) 1997 Academic Press.

Abstract:   Previously, Adalsteinsson and Sethian have applied the level set formulation to the problem of surface advancement in two and three-dimensional topography simulation of deposition, etching, and lithography processes in integrated circuit fabrication. The level set formulation is based on solving a Hamilton-Jacobi type equation for a propagating level set function, using techniques borrowed from hyperbolic conservation laws. Topological changes, corner, and cusp development, and accurate determination of geometric properties such as curvature and normal direction are naturally obtained in this setting. Part I presented the basic equations and algorithms for two dimensional simulations, including the effects of isotropic and uni-directional deposition and etching, visibility, reflection, and material dependent etch/deposition rates. Part II focused on the extension to three dimensions. This paper completes the series, and add the effects of redeposition, reemission, and surface diffusion. This requires the solution of the transport equations for arbitrary geometries, and leads to simulations that contain multiple simultaneous competing effects of visibility, directional and source flux functions, complex sputter yield flux functions, wide ranges of sticking coefficients for the reemission and redeposition functions, multilayered fronts and thin film layers. (C) 1997 Academic Press.

Abstract:   We intend to discuss the relations between the snore abstract notion of entropy and the ideas of information using nonlinear partial differential equations as an example. Hyperbolic equations of the first order where vanishing of the characteristics in a discontinuity may be interpreted as loss of initial information, will serve as illustration. The problem of non-physical rarefaction shocks in discussed. and with Lax' shock condition a first entropy condition is introduced. The approximation of a physical problem with friction by an inviscid problem leads to a demand for an entropy inequality characterizing the solutions of the model with friction in the limit, when friction vanishes. This demand for an additional condition, namely for an entropy inequality as mathematical pendant to the second fundamental law also dominates the numerics of the presented equations. A is only the fulfilment of a discrete entropy condition - i.e. of a discretized form of the second fundamental law - that provides for the convergence of finite difference procedures.

Abstract:   A finite-difference scheme is derived for numerical solution of the eikonal scalar-field front propagation equation. The numerical scheme is implemented for the description of the dynamics of a flame surface propagating through a vortical flow field. Attributes of the numerical scheme are its accuracy and numerical stability when large velocity fluctuations in the flow are present, and the absence of explicit artificial viscosity. (C) 1997 Elsevier Science Ltd.

Abstract:   A simple model problem is formulated to describe the coupling between premixed-flame and flow-field dynamics resulting from gas expansion within the flame. The energy conservation equation is integrated analytically across the flame in order to reduce the number of governing equations for the computational problem. A system of six equations and associated boundary conditions are formulated for computation of the time evolution of an initially prescribed three-dimensional velocity field and the flame surface. (C) 1997 Elsevier Science Ltd.

Abstract:   We present an algorithm that builds a correspondence between two arbitrary genus-0 objects and generates a sequence of inbetween objects. A warp function deforms the source object and aligns it with the target object. An iterative polygon- evolution algorithm blurs the details of the warped source and target objects into two convex objects with similar shapes that are projected to two identical circles. Merging the topologies of the projected objects and reconstructing the original objects results in two objects with identical topologies. A two-part transformation produces the morph sequence. The rigid part moves and rotates the objects to their relative positions. The elastic part establishes the position of each of the vertices forming the inbetween object.

Abstract:   Recently, Majda and Souganidis have presented a rigorous asymptotic theory governing the large-scale renormalized flame front dynamics for a reaction-diffusion-advection system involving KPP type chemistries and small scale turbulence. This theory is valid in the context of an infinitely thin reaction layer. Embid, Majda and Souganidis have explored this rigorous theory within the context of a shear layer flow geometry, and demonstrate that the enhanced burning speed is sensitively dependent upon the presence of a mean wind transverse to the direction of the flame propagation. Here, we address the effect that a thin reaction layer may have on the enhanced flame propagation for the case of a small scale shear layer with and without a transverse mean wind. We show through high resolution numerical simulations that the enhanced burning speed is sensitively dependent on the presence of a transverse mean wind in a qualitatively similar fashion to the asymptotic theory; but further, we exhibit that a finite reaction zone yields effective burning speeds which are smaller than the theoretical predictions for infinitely thin flame fronts and that the decay of these corrections depends upon the relative scale separation between the reaction layer scale, the turbulence scale, and the integral scale.

Abstract:   A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of a two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables us to compute flows with large density ratios (1000/1) and flows that are surface tension driven, with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, among others. We validate our code against experiments and theory. (C) 1998 Elsevier Science Ltd. All rights reserved.

Abstract:   A framework for contrast enhancement via image evolution hows and variational formulations is introduced in this paper. First, an algorithm for histogram modification via image evolution equations is presented. We show that the image histogram can be modified to achieve any given distribution as the steady state solution of this differential equation. We then prove that the proposed evolution equation solves an energy minimization problem. This gives a new interpretation to histogram modification and contrast enhancement in general. This interpretation is completely formulated in the image domain, in contrast with classical techniques for histogram modification which are formulated in a probabilistic domain. From this, new algorithms for contrast enhancement, including, for example, image and perception models, can be derived, Based on the energy formulation and its corresponding differential form, we show that the proposed histogram modification algorithm can be combined with image regularization schemes, This allows us to perform simulations contrast enhancement and denoising, avoiding common noise sharpening effects in classical schemes, Theoretical results regarding the existence of solutions to the proposed equations are presented. (C) 1997 Academic Press.

Abstract:   We review the problem of finding an apparent horizon in Cauchy data (Sigma,g(ab),K-ab) in three space dimensions without symmetries. We describe a family of algorithms which includes the pseudospectral apparent horizon finder of Nakamura et al. and the curvature flow method proposed by Tod as special cases. We suggest that other algorithms in the family may combine the speed of the former with the robustness of the latter. A numerical implementation for Cauchy data given on a grid in Cartesian coordinates is described, and tested on Brill- Lindquist and Kerr initial data. The new algorithm appears faster and more robust than previous ones.

Abstract:   We investigate a three-dimensional macroscopic model of wave- front propagation related to the excitation process in the left ventricular wall represented by an anisotropic bidomain. The whole left ventricle is modeled, whereas, in a previous paper, only a flat slab of myocardial tissue was considered. The direction of cardiac fibers, which affects the anisotropic conductivity of the myocardium, rotates from the epi-to the endocardium. If the ventricular wall is conceived as a set of packed surfaces, the fibers may be tangent to them or more generally may cross them obliquely; the latter case is described by an "imbrication angle." The effect of a simplified Purkinje network also is investigated. The cardiac excitation process, more particularly the depolarization phase, is modeled by a nonlinear elliptic equation, called an eikonal equation, in the activation time. The numerical solution of this equation is obtained by means of the finite element method, which includes an upwind treatment of the Hamiltonian part of the equation. By means of numerical simulations in an idealized model of the left ventricle, we try to establish whether the eikonal approach contains the essential basic elements for predicting the features of the activation patterns experimentally observed. We discuss and compare these results with those obtained in our previous papers [1,2] for a flat part of myocardium. The general rules governing the spread of excitation after local stimulations, previously delineated for the flat geometry, are extended to the present, more realistic monoventricular model. (C) 1998 Elsevier Science Inc.

Abstract:   This paper presents a method for automatic segmentation of the tibia and femur in clinical magnetic resonance images of knees. Texture information is incorporated into an active contours framework through the use of vector-valued geodesic snakes with local variance as a second value at each pixel, in addition to intensity. This additional information enables the system to better handle noise and the non-uniform intensities found within the structures to be segmented. It currently operates independently on 2D images (slices of a volumetric image) where the initial contour must be within the structure but not necessarily near the boundary. These separate segmentations are stacked to display the performance on the entire 3D structure.

Abstract:   In this paper, we review the recently published work on deformable models. We have chosen to concentrate on 2D deformable models and relate the energy minimization approaches to the Bayesian formulations. We categorize the various active contour systems according to the definition of the deformable model. We also present in detail one particular formulation for deformable templates which combines edge, texture, color and region information for the external energy and model deformations using wavelets, splines or Fourier descriptors. We explain how these models can be used for segmentation, image retrieval in a large database and object tracking in a video sequence. (C) 1998 Elsevier Science B.V. All rights reserved.

Abstract:   We introduce a model for epitaxial phenomena based on the motion of island boundaries, which is described by the level- set method. Our model treats the growing film as a continuum in the lateral direction, but retains atomistic discreteness in the growth direction. An example of such an "island dynamics" model using the level-set method is presented and compared with the corresponding rate equation description. Extensions of our methodology to more general settings are then discussed. [S1063-651X(98)50212-7].

Abstract:   The flow of two viscous liquids is investigated numerically with a volume of fluid scheme. The scheme incorporates a semi- implicit Stokes solver to enable computations at low Reynolds numbers, and a second-order velocity interpolation. The code is validated against linear theory for the stability of two-layer Couette flow, and weakly nonlinear theory for a Hopf bifurcation. Examples of long-time wave saturation are shown. The formation of fingers for relatively small initial amplitudes as well as larger amplitudes are presented in two and three dimensions as initial-value problems. Fluids of different viscosity and density are considered, with an emphasis on the effect of the viscosity difference. Results at low Reynolds numbers show elongated fingers in two dimensions that break in three dimensions to form drops, while different topological changes take place at higher Reynolds numbers. (C) 1998 American Institute of Physics. [S1070-6631(98)00612-6].

Abstract:   An accurate Semi-Lagrangian (SL) scheme for incompressible Navier-Stokes equations written in generalized coordinates was developed. The scheme explicitly calculates the advection phase of the governing equations by a newly developed SL transport model and the viscous term by a modified Crank-Nicholson method. Several numerical schemes approximating the trajectory and the unknowns at the upstream departure point have been verified to construct a well-balanced, totally accurate numerical solver. Employing the contravariant velocity, the scheme directly predicates the generalized coordinates of the upstream departure point by the four-stage fourth order Runge- Kutta method. The velocity at the departure point is interpolated with the third order accuracy. Unlike the traditional Eulerian schemes, the scheme allows a large time step length free from the CFL condition while keeping accuracy. The unsteady flows around a 2D circular cylinder are accurately predicted in 1/6 to 1/3 of the CPU times required for the traditional Eulerian solvers. The proposed scheme is expected to replace the traditional Eulerian solvers to quickly investigate the practical flow problems.

Abstract:   In this paper implicit representations of deformable models for medical image enhancement and segmentation are considered. The advantage of implicit models over classical explicit models is that their topology can he naturally adapted to objects in the scene, A geodesic formulation of implicit deformable models is especially attractive since it has the energy minimizing properties of classical models, The aim of this pager is twofold, First, a modification to the customary geodesic deformable model approach is introduced by considering all the level sets in the image as energy minimizing contours. This approach is used to segment multiple objects simultaneously and for enhancing and segmenting cardiac computed tomography (CT) and magnetic resonance images. Second, the approach is used to effectively compare implicit and explicit models for specific tasks. This shows the complementary character of implicit models since in case of poor contrast boundaries or gaps in boundaries e.g. due to partial volume effects, noise, or motion artifacts, they do not perform well, since the approach is completely data-driven.

Abstract:   This paper presents a method that uses the level sets of volumes to reconstruct the shapes of 3D objects from range data. The strategy is to formulate 3D reconstruction as a statistical problem: find that surface which is mostly likely, given the data and some prior knowledge about the application domain. The resulting optimization problem is solved by an incremental process of deformation. We represent a deformable surface as the level set of a discretely sampled scalar function of three dimensions, i.e., a volume. Such level-set models have been shown to mimic conventional deformable surface models by encoding surface movements as changes in the greyscale values of the volume. The result is a voxel-based modeling technology that offers several advantages over conventional parametric models, including flexible topology, no need for reparameterization, concise descriptions of differential structure, and a natural scale space for hierarchical representations. This paper builds on previous work in both 3D reconstruction and level-set modeling. It presents a fundamental result in surface estimation from range data: an analytical characterization of the surface that maximizes the posterior probability. It also presents a novel computational technique for level-set modeling, called the sparse-field algorithm, which combines the advantages of a level-set approach with the computational efficiency and accuracy of a parametric representation. The sparse-field algorithm is more efficient than other approaches, and because it assigns the level set to a specific set of grid points, it positions the level-set model more accurately than the grid itself. These properties, computational efficiency and subcell accuracy, are essential when trying to reconstruct the shapes of 3D objects. Results are shown for the reconstruction objects from sets of noisy and overlapping range maps.

Abstract:   One of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar held is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier-Stokes) and the Cahn-Hilliard equations. The model takes into account weak non-locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non-locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a nontrivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non-solenoidal (quasi-incompressibility). To demonstrate the effects of quasi-incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well-separated interfacial layers, an appropriately scaled quasi-incompressible Euler-Cahn-Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.

Abstract:   We consider the motion of polygons by crystalline curvature. We show that "smooth" polygon evolves by crystalline curvature "smoothly" and that it shrinks to a point in finite time. We also establish the convergence of crystalline motion to the motion by mean curvature.

Abstract:   A computational method combining the genetic algorithm (GA) and shape mutation is reported for electromagnetic imaging of a homogeneous cylinder of arbitrary shape. By measuring the scattered field, the shape location, size, and permittivity of the object are retrieved quite successfully. The forward problem is solved based on the equivalent source current and the method of moments (MoM) while the inverse problem is reformulated into an optimization one, and is solved by the proposed scheme. Numerical simulation shows that, even with a bad initial guess, good reconstruction can be obtained for a hollow object or multiple-cylinder object as long as the noise level is less than or equal to -20 dB. (C) 1998 John Wiley & Sons, Inc.

Abstract:   A numerical model is developed for the simulation of moving interfaces in viscous incompressible flows. The model is based on the finite element method with a pseudo-concentration technique to track the front. Since a Eulerian approach is chosen, the interface is advected by the flow through a fixed mesh. Therefore, material discontinuity across the interface cannot be described accurately. To remedy this problem, the model has been supplemented with a local mesh adaptation technique. This latter consists in updating the mesh at each time step to the interface position, such that element boundaries lie along the front. It has been implemented for unstructured triangular finite element meshes. The outcome of this technique is that it allows an accurate treatment of material discontinuity across the interface and, if necessary, a modelling of interface phenomena such as surface tension by using specific boundary elements. For illustration, two examples are computed and presented in this paper: the broken dam problem and the Rayleigh-Taylor instability. Good agreement has been obtained in the comparison of the numerical results with theory or available experimental data. (C) 1998 John Wiley & Sons, Ltd.

Abstract:   Borrowing from techniques developed for conservation law equations, numerical schemes which discretize the Hamilton- Jacobi (H-J), level set, and Eikonal equations on triangulated domains are presented. The first scheme is a provably monotone discretization for the H-J equations. Unfortunately, the basic scheme lacks Lipschitz continuity of the numerical Hamiltonian. By employing a "virtual" edge flipping technique local Lipschitz continuity of the numerical flux is restored on acute triangulations. Next, schemes are introduced and developed based on the weaker concept of positive coefficient approximations for homogeneous Hamiltonians. These schemes possess a discrete maximum principle on arbitrary triangulations and exhibit local Lipschitz continuity of the numerical Hamiltonian. Finally, a class of Petrov-Galerkin approximations is considered. These schemes are stabilized via a least-squares bilinear form. The Petrov-Galerkin schemes do not possess a discrete maximum principle but generalize to high order accuracy. Discretization of the level set equation also requires the numerical approximation of a mean curvature term. A simple mass-lumped Galerkin approximation is presented in Section 6 and analyzed using maximum principle analysis. The use of unstructured meshes permits several forms of mesh adaptation which have been incorporated into numerical examples. These numerical examples include discretizations of convex and nonconvex forms of the H-J equation, the Eikonal equation, and the level set equation. (C) 1998 Academic Press.

Abstract:   We introduce a new formulation for the motion of curves in R-2 (easily extendable to the motion of surfaces in R-3), when the original motion generally corresponds to an ill-posed problem such as the Cauchy-Riemann equations. This is, in part, a generalization of our earlier work in [6], where we applied similar ideas to compute flows with highly concentrated vorticity, such as vortex sheets or dipoles, for incompressible Euler equations. Our new formulation involves extending the level set method of [12] to problems in which the normal velocity is not intrinsic. We obtain a coupled system of two equations, one of which is a level surface equation. This yields a fixed-grid, Eulerian method which regularizes the ill- posed problem in a topological fashion. We also present an analysis of curvature regularizations and some other theoretical justification. Finally, we present numerical results showing the stability properties of our approach and the novel nature of the regularization, including the development of bubbles for curves evolving under Cauchy-Riemann flow.

Abstract:   In this paper, we prove that mean curvature motion can be regarded as the singular limit of the following model: [GRAPHICS] where epsilon > 0 is a Small parameter and W(u) =(1 - u(1)(2))(2)-(1-u(2)(2))(2) + 1. This model is related to the Landau-Lifshitz equation of ferromagnetism. Local existence of classical solutions of the Dirichlet problem and global existence of the travelling wave solutions are also obtained.

Abstract:   This work presents some newly developed efficient numerical schemes and techniques for the dynamics of multi-phase flows. In the first part of this paper a semi-Lagrangian method for an advection equation, as an important part of our numerical model, is introduced. The scheme is constructed from a rational function and proved to be convexity preserving. It has third- order accuracy in the smooth region and possesses an oscillation suppressing property near discontinuities or steep gradients. There is no need to calculate the slope limiter to eliminate numerical oscillation as other high resolution schemes do. The paper also discusses in the second part some numerical algorithms that prove important to the establishment of an efficient and robust code for simulating the dynamics of multi-material flows, such as the calculation of a moving boundary, the unified procedure for evaluating pressure over different materials and the equivalent volume force formulations for different types of forces. (C) 1998 Elsevier Science Ltd.

Abstract:   The problem of simulating the motion of evolving surfaces with junctions according to some curvature-dependent speed arises in a number of applications. By alternately diffusing and sharpening characteristic functions for each region, a variety of motions have been obtained which allow for topological mergings and breakings and produce no overlapping regions or vacuums. However, the usual finite difference discretization of these methods is often excessively slow when accurate solutions are sought, even in two dimensions. Vie propose a new, spectral discretization of these diffusion-generated methods which obtains greatly improved efficiency over the usual finite difference approach. These efficiency gains are obtained, in part through the use of a quadrature-based refinement technique, by integrating Fourier modes exactly and by neglecting the contributions of rapidly decaying solution transients. Indeed, numerical studies demonstrate that the new algorithm is often more than 1000 times faster than the usual finite difference discretization. Our findings are demonstrated on several examples. (C) 1998 Academic Press.

Abstract:   Characterization of pulmonary nodules plays a significant role in the differential diagnosis of lung cancer. This paper presents a method to quantify surface characteristics of small pulmonary nodules with well-defined surface based on thin- section CT images. The segmentation of the three-dimensional (3-D) nodule images are obtained by a 3-D deformable surfaces approach. The feature extraction algorithms are designed to quantify the surface characteristic parameters from 3-D nodule images by using surface curvatures and ridge lines. Experimental results of our method, applied to patients 3-D nodule images, demonstrate it performance.

Abstract:   A front tracking method to study multi-fluid flows in which a sharp interface separates incompressible fluids of different densities and viscosities is adopted to simulate the unsteady motion of an infinitely thin premixed name characterized by significant chemical heat release and hence thermal expansion. The How field is discretized by a conservative finite difference approximation on a stationary grid, and the flame surface is explicitly represented by connected marker points that move with the local flame speed, relative to the flow field. The performance of the method is tested by applying it to a steady planar name and the Darrieus-Landau instability. The numerical results are in good agreement with analytical results. The method is also applied to the interaction between a name and a vortex array. The results show that the name can destroy the vorticity originally in the unburnt gas and generate vorticity of opposite sign in the burnt gas. (C) 1998 Academic Press.

Abstract:   We study a two-parameter family of Riemann problems for the unsteady transonic small disturbance (UTSD) equation, also called the two-dimensional Burgers equation. The two parameters, a and b, which define oblique shock initial data, correspond to the slopes of the initial shock waves in the upper half-plane. For each a and b, the three constant states in the upper halfplane satisfy the Rankine-Hugoniot conditions across the shocks. This leads to a two-parameter family of oblique shock interaction problems. In this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram. Our study supplements the related theoretical results and conjectures recently obtained by S. Canic and B. L. Keyfitz. We employ a high resolution numerical method which reveals fine solution structures. Our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave interactions arising in this model.

Abstract:   Relaxation of disclination loops created during shear flow of a low molar mass and a polymer liquid crystal were monitored using a special shear stage with a videomicroscope. Loops in the polymer system generally displayed initially highly distorted contours. In the small molecule liquid crystal, the loop contours consistently exhibited very simple, generally convex shapes. In the polymer system, the complex line shape reflects the many prior loop-loop coalescence events due to the greater density of loops than in the small molecule system. Sequential images of loops were analysed to determine the velocity of the disclination loops as a function of the local curvature. Observations and simulations indicate that local disclination line curvature is a driving force in loop evolution. The reduction of regions of high loop curvature is inherently slower in the polymer liquid crystal due to the higher viscosity. In addition, the motion of the disclination contour at very high values of curvature in the polymer liquid crystal is slowed due to the presence of lower molecular weight components at the defect core which themselves must diffuse along with the line defect. (C) 1998 Elsevier Science Ltd. All rights reserved.

Abstract:   This paper describes a novel approach to surface tracking in volumetric image stacks. It draws on a statistical model of the uncertainties inherent in the characterisation of feature contours to compute an evidential field for putative inter- frame displacements. This field is computed using Gaussian density kernels which are parameterised in terms of the variance-covariance matrices for contour displacement. The underlying variance model accommodates the effects of raw image noise on the estimated surface normals. The evidential field effectively couples contour displacements to the intensity features on successive frames through a statistical process of contour tracking. Hard contours are extracted using a dictionary-based relaxation process. The method is evaluated on both MRI data and simulated data. (C) 1998 Elsevier Science B.V. All rights reserved.

Abstract:   A curvilinear level set formulation has been developed for highly deformable free surface problems. In this new scheme, the grid lines follow the irregular domain generated by the multizone adaptive grid generation (MAGG) scheme [1] and free surfaces are captured by level set functions among the curvilinear grids. Navier-Stokes equations are discretized and solved based on a multiphase curvilinear finite-volume method [2], and the level set function is solved based on a finite- difference method using the second-order essentially nonoscillatory (ENO) scheme [3]. The scheme is capable of accurately and efficiently representing the deformation, oscillation, merger, and separation of free surfaces. The effectiveness and robustness of the algorithm are demonstrated by using it for problems involving merger of bubbles, mold filling, and the spreading and solidification of molten droplets on a cold substrate where both free surface and solidification interfaces move and the mass of the liquid phase is continuously decreased.

Abstract:   We address the problem of adapting the functions controlling the material properties of 2-D snakes, and show how introducing oriented smoothness constraints results in a novel class of active contour models for segmentation, which extends standard isotropic inhomogeneous membrane/thin-plate stabilizers. These constraints, expressed as adaptive L-2 matrix norms, are defined by two second-order symmetric and positive definite tensors that are invariant with respect to rigid motions in the image plane. These tensors, equivalent to directional adaptive stretching and bending densities, are quadratic with respect to first- and second-order derivatives of the image luminance, respectively. A representation theorem specifying their canonical form is established and a geometrical interpretation of their effects is developed. Within this framework, it is shown that by achieving a directional control of regularization such nonisotropic constraints consistently relate the differential properties (metric and curvature) of the deformable model with those of the underlying luminance surface, yielding a satisfying preservation of image contour characteristics. In particular, this model adapts to nonstationary curvature variations along image contours to be segmented, thus providing a consistent solution to curvature underestimation problems encountered near high curvature contour points by classical snakes evolving with constant material parameters. Optimization of the model within continuous and discrete frameworks is discussed in detail. Finally, accuracy and robustness of the model are established on synthetic images. Its efficacy is further demonstrated on 2- D MRI sequences for which comparisons with segmentations obtained using classical snakes are provided. (C) 1998 SPIE and IS&T. [S1017-9909(98)02101-1].

Abstract:   We reproduce the general behavior of complicated bubble and droplet motions using the variational level set formulation introduced by the authors earlier. Our approach here ignores inertial effects; thus the motion is only correct as an approximation for very viscous problems. However, the steady states are true equilibrium solutions. Inertial forces will be added in future work. The problems include: soap bubbles colliding and merging, drops falling or remaining attached to a (generally irregular) ceiling, and liquid penetrating through a funnel in both two and three dimensions. Each phase is identified with a particular "level set" function. The zero level set of this function is that particular phase boundary. The level set functions all evolve in time through a constrained gradient descent procedure so as to minimize an energy functional. The functions are coupled through physical constraints and through the requirements that different phases do not overlap and vacuum regions do not develop. Both boundary conditions and inequality constraints are cast in terms of (either local or global) equality constraints. The gradient projection method leads to a system of perturbed (by curvature, if surface tension is involved) Hamilton-Jacobi equations coupled through a constraint. The coupling is enforced using the Lagrange multiplier associated with this constraint. The numerical implementation requires much of the modem level set technology; in particular, we achieve a significant speed up by using the fast localization algorithm of H.-K. Zhao, M. Kang, B. Merriman, D. Peng, and S. Osher. (C) 1998 Academic Press.

Abstract:   We are concerned with the retrieval of the unknown cross section of a homogeneous cylindrical obstacle embedded in a homogeneous medium and illuminated by time-harmonic electromagnetic line sources. The dielectric parameters of the obstacle and embedding materials are known and piecewise constant. That is, the shape (here, the contour) of the obstacle is sufficient for its full characterization. The inverse scattering problem is then to determine the contour from the knowledge of the scattered field measured for several locations of the sources and/or frequencies. An iterative process is implemented: given an initial contour, this contour is progressively evolved such as to minimize the residual in the data fit. This algorithm presents two main important points. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity field whose expression only requires the resolution of an adjoint problem at each step. The second concerns the use of a level-set function in order to represent the obstacle. This level-set function will be of great use to handle in a natural way splitting or merging of obstacles along the iterative process. The evolution of this level-set is controlled by a Hamilton-Jacobi-type equation which will be solved by using an appropriate finite-difference scheme. Numerical results of inversion obtained from both noiseless and noisy synthetic data illustrate the behaviour of the algorithm for a variety of obstacles.

Abstract:   We are concerned with the retrieval of the unknown cross section of a homogeneous cylindrical obstacle embedded in a homogeneous medium and illuminated by time-harmonic electromagnetic line sources. The dielectric parameters of the obstacle and embedding materials are known and piecewise constant. That is, the shape (here, the contour) of the obstacle is sufficient for its full characterization. The inverse scattering problem is then to determine the contour from the knowledge of the scattered field measured for several locations of the sources and/or frequencies. An iterative process is implemented: given an initial contour, this contour is progressively evolved such as to minimize the residual in the data fit. This algorithm presents two main important points. The first concerns the choice of the transformation enforced on the contour. We will show that this involves the design of a velocity field whose expression only requires the resolution of an adjoint problem at each step. The second concerns the use of a level-set function in order to represent the obstacle. This level-set function will be of great use to handle in a natural way splitting or merging of obstacles along the iterative process. The evolution of this level-set is controlled by a Hamilton-Jacobi-type equation which will be solved by using an appropriate finite-difference scheme. Numerical results of inversion obtained from both noiseless and noisy synthetic data illustrate the behaviour of the algorithm for a variety of obstacles.

Abstract:   A level set method for capturing the interface between two fluids is combined with a variable density projection method to allow for computation of a two-phase flow where the interface can merge/break and the flow can have a high Reynolds number. A distance function formulation of the level set method enables us to compute flows with large density ratios (1000/1) and flows that are surface tension driven, with no emotional involvement. Recent work has improved the accuracy of the distance function formulation and the accuracy of the advection scheme. We compute flows involving air bubbles and water drops, among others. We validate our code against experiments and theory. (C) 1998 Elsevier Science Ltd. All rights reserved.

Abstract:   This paper presents a new algorithm for planning the time- optimal motion of a robot traveling with limited velocity from a given location to a given destination on a surface in the presence of moving obstacles. Additional constraints such as space variant terrain traversability and fuel economy can be accommodated. A multivalued distance map is defined and applied in computing optimal trajectories. The multivalued distance map incorporates constraints imposed by the moving obstacles, surface topography, and terrain traversability, It is generated by an efficient numerical curve propagation technique.

Abstract:   A simple shock-capturing approach to multicomponent flow problems is developed for the compressible Euler equations with a stiffened gas equation of state in multiple space dimensions. The algorithm uses a quasi-conservative formulation of the equations that is derived to ensure the correct fluid mixing when approximating the equations numerically with interfaces. A gamma-based model and a volume-fraction model have been described, and both of them are solved using the standard high- resolution wave propagation method for general hyperbolic systems of partial differential equations. Several calculations are presented with a Roe approximate Riemann solver that show accurate results obtained using the method without any spurious oscillations in the pressure near the interfaces, Convergence of the computed solutions to the correct weak ones has been verified for a two-dimensional Richtmyer-Meshkov unstable interface problem where we have performed a mesh-refinement study and also shown front-tracking results for comparison. (C) 1998 Academic Press.

Abstract:   A local smoothing operator applied in the reverse direction is used to obtain planar shape enhancement and exaggeration. Inversion of a smoothing operator is an inherently unstable operation. Therefore, a stable numerical scheme simulating the inverse smoothing effect is introduced. Enhancement is obtained for short time spans of evolution. Carrying the evolution further yields shape exaggeration or caricaturization effect. Introducing attraction forces between the evolving shape and the initial one yields an enhancement process that converges to a steady state. These forces depend on the distance of the evolving curve from the original one and on local properties. Results of applying the unrestrained and restrained evolution on planar shapes, based on a stabilized inverse geometric heat equation, are presented showing enhancement and caricaturization effects. (C) 1998 Academic Press.

Abstract:   Recent developments in reservoir characterization and in the management of uncertainty have lead to the ability of the petroleum industry to routinely generate large, multimillion- cell detailed geologic models. This has resulted in a steadily increasing gap between flow simulation and the static model, not only because of the size of these models, but also because of our desire to obtain reservoir performance predictions for multiple realizations of such models. Three-dimensional streamline-based computation offers significant potential to meet some of these challenges, leading to a rapidly developing technology. The purpose of this paper is to review current streamline technology: its foundations (the 'time of flight' formulation), historical precedents (streamtubes and front trackers), current applications, open questions, and potential limitations. A wide range of applications will be used to demonstrate the utility of both streamline simulation and the underlying formulation. Where required, new material will be presented (analysis of field tracer response, streamline modeling in corner point cells, evaluation of grid orientation effects, discussion of open questions). Otherwise, existing results will be drawn from the literature.

Abstract:   In part I the authors reported on the design of a robust and versatile gradient-weighted moving finite element (GWMFE) code in one dimension and on its application to a variety of PDEs and PDE systems. This companion paper does the same for the two-dimensional (2D) case. These moving node methods are especially suited to problems which develop sharp moving fronts, especially problems where one needs to resolve the fine-scale structure of the fronts. The many potential pitfalls in the design of GWMFE codes and the special features of the implicit one-dimensional (1D) and 2D codes which contribute to their robustness and efficiency are discussed at length in part I; this paper concentrates on issues unique to the 2D case. Brief explanations are given of the variational interpretation of GWMFE, the geometrical-mechanical interpretation, simplified regularization terms, and the treatment of systems. A catalog of inner products which occur in GWMFE is given, with particular attention paid to those involving second-order operators. After presenting an example of the 2D phenomenon of grid collapse and discussing the need for long-time regularization, the paper reports on the application of the 2D code to several nontrivial problems - nonlinear arsenic diffusion in the manufacture of semiconductors, the drift- diffusion equations for semiconductor device simulation, the Buckley-Leverett black oil equations for reservoir simulation, and the motion of surfaces by mean curvature.

Abstract:   This paper presents a numerical method directed towards the local simulation of axisymmetric vapor bubble growth. We use an interface tracking method in conjunction with a finite volume method on a moving unstructured mesh. We allow metastable bulk slates and assume the interface exists in thermal and chemical equilibrium. The bulk fluids are viscous, conducting, and compressible. The control volume continuity, momentum and energy equations are modified in the presence of a phase interface to include surface tension and discontinuous pressure and velocity. A solid wall model is included to allow for conjugate heat transfer modes. (C) 1998 Elsevier Science Ltd. All rights reserved.

Abstract:   This paper presents a numerical method directed towards the local simulation of axisymmetric vapor bubble growth. We use an interface tracking method in conjunction with a finite volume method on a moving unstructured mesh. We allow metastable bulk slates and assume the interface exists in thermal and chemical equilibrium. The bulk fluids are viscous, conducting, and compressible. The control volume continuity, momentum and energy equations are modified in the presence of a phase interface to include surface tension and discontinuous pressure and velocity. A solid wall model is included to allow for conjugate heat transfer modes. (C) 1998 Elsevier Science Ltd. All rights reserved.

Abstract:   This paper presents a numerical method directed towards the local simulation of axisymmetric vapor bubble growth. We use an interface tracking method in conjunction with a finite volume method on a moving unstructured mesh. We allow metastable bulk slates and assume the interface exists in thermal and chemical equilibrium. The bulk fluids are viscous, conducting, and compressible. The control volume continuity, momentum and energy equations are modified in the presence of a phase interface to include surface tension and discontinuous pressure and velocity. A solid wall model is included to allow for conjugate heat transfer modes. (C) 1998 Elsevier Science Ltd. All rights reserved.

Abstract:   In this paper we present a new definition for the global-in- time propagation (motion) of fronts (hypersurfaces, boundaries) with a prescribed normal velocity, past the first time they develop singularities. We show that if this propagation satisfies a geometric maximum principle (inclusion-avoidance- type property), then the normal velocity must depend only on the position of the front, its normal direction and principal curvatures. This new approach, which is more geometric and, as it turns out, equivalent to the level-set method, is then used to develop a very general and simple method to rigorously validate the appearance of moving interne faces at the asymptotic limit of general evolving systems like interacting particles and reaction-diffusion equations. We finally present a number of new asymptotic results. Among them are the asymptotics of (i) reaction-diffusion equations with rapidly oscillating coefficients, (ii) fully nonlinear nonlocal (integral differential) equations, and (iii) stochastic Ising models with long-range anisotropic interactions and general spin-flip dynamics.

Abstract:   We consider the propagation of chemical fronts in a Hele-Shaw flow where the front is assumed to propagate with a curvature dependent velocity. The motivation is to model some recent experiments that employ aqueous autocatalytic chemical reactions in such a device. The density change across the front in such experiments is quite small so the Boussinesq approximation can be used, and the flow field generated is exclusively due to buoyancy effects. We derive a free boundary formulation based on Darcy's law and potential theory, and describe the evolution in terms of the theta-L formulation, in which the tangent angle and the perimeter of the closed front are followed in time. Numerical solutions are obtained for this formulation with a rising and expanding bubble. As observed in the experiments, a fingering phenomenon which is different from the surface tension associated phenomenon appears in our calculations. The mechanisms that control the wavelength selection of the fingers, and a comparison with the result of a linear stability analysis for flat fronts are discussed. (C) 1998 American Institute of Physics.

Abstract:   The dynamics of a thin Huygens front propagating through a turbulent medium is considered. A rigorous asymptotic expression for the effective velocity v(F) proportional to the front area is derived. The small-scale fluctuations of the front position are shown to be strongly intermittent. This intermittency prays a crucial role in establishing a steady state magnitude of the front velocity. The results are compared with experimental data.

Abstract:   A fast, second-order accurate iterative method is proposed for the elliptic equation del .(beta(x,y)del u) = f(x,y) in a rectangular region Omega in two-space dimensions. We assume that there is an irregular interface across which the coefficient beta, the solution u and its derivatives, and/or the source term f may have jumps. We are especially interested in the cases where the coefficients beta are piecewise constant and the jump in beta is large. The interface may or may not align with an underlying Cartesian grid. The idea in our approach is to precondition the differential equation before applying the immersed interface method proposed by LeVeque and Li [SIAM J. Numer. Anal., 4 (1994), pp. 1019-1044]. In order to take advantage of fast Poisson solvers on a rectangular region, an intermediate unknown function, the jump in the normal derivative across the interface, is introduced. Our discretization is equivalent to using a second-order difference scheme for a corresponding Poisson equation in the region, and a second-order discretization for a Neumann-like interface condition. Thus second-order accuracy is guaranteed. A GMRES iteration is employed to solve the Schur complement system derived from the discretization. A new weighted least squares method is also proposed to approximate interface quantities from a grid function. Numerical experiments are provided and analyzed. The number of iterations in solving the Schur complement system appears to be independent of both the jump in the coefficient and the mesh size.

Abstract:   We introduce a new paradigm, the differential invariant signature curve or manifold, for the invariant recognition of visual objects. A general theorem of E. Cartan implies that two curves are related by a group transformation if and only if their signature curves are identical. The important examples of the Euclidean and equi-affine groups are discussed in detail. Secondly, we show how a new approach to the numerical approximation of differential invariants, based on suitable combination of joint invariants of the underlying group action, allows one to numerically compute differential invariant signatures in a fully group-invariant manner. Applications to a variety of fundamental issues in vision, including detection of symmetries, visual tracking, and reconstruction of occlusions, are discussed.

Abstract:   The medial axis is an attractive shape feature; however, its high sensitivity to boundary noise hinders its use in many applications. In order to overcome the sensitivity problem some regularization has to be performed. Pruning is a family of medial axis regularization processes, incorporated in most skeletonization and thinning algorithms. Pruning algorithms usually appear in a variety of application-dependent formulations. Inconsistent terminology used until now prevented analysis and comparison of the various pruning methods. Indeed many seemingly different algorithms are in fact equivalent. In this paper we suggest the rate pruning paradigm as a standard for pruning methods. The proposed paradigm is a framework in which it is easy to analyze, compare, and tailor new pruning methods. We analyze existing pruning methods, propose two new methods, and compare the methods via a model-based analysis. The theoretical analysis is supported by simulation results of the various pruning methods. (C) 1998 Academic Press.

Abstract:   In this paper the equation of mean curvature flow (with forcing term) is modified, to account not only for surface motion but also for nucleation and other discontinuities in set evolution. Existence of a solution is proved for a weak formulation, which is written in terms of the characteristic function of the evolving set. The argument is based on implicit time- discretization, derivation of uniform estimates, and passage to the limit.

Abstract:   Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods. A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another. The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution. The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

Abstract:   Boundary integral methods to simulate interfacial flows are very sensitive to numerical instabilities. In addition, surface tension introduces nonlinear terms with high order spatial derivatives into the interface dynamics. This makes the spatial discretization even more difficult and, at the same time, imposes a severe time step constraint for stable explicit time integration methods. A proof of the convergence of a reformulated boundary integral method for two-density fluid interfaces with surface tension is presented. The method is based on a scheme introduced by Hou, Lowengrub and Shelley [J. Comp. Phys. 114 (1994), pp. 312-338] to remove the high order stability constraint or stiffness. Some numerical filtering is applied carefully at certain places in the discretization to guarantee stability. The key of the proof is to identify the most singular terms of the method and to show, through energy estimates, that these terms balance one another. The analysis is at a time continuous-space discrete level but a fully discrete case for a simple Hele-Shaw interface is also studied. The time discrete analysis shows that the high order stiffness is removed and also provides an estimate of how the CFL constraint depends on the curvature and regularity of the solution. The robustness of the method is illustrated with several numerical examples. A numerical simulation of an unstably stratified two-density interfacial flow shows the roll-up of the interface; the computations proceed up to a time where the interface is about to pinch off and trapped bubbles of fluid are formed. The method remains stable even in the full nonlinear regime of motion. Another application of the method shows the process of drop formation in a falling single fluid.

Abstract:   We propose a new definition of the total variation norm for vector-valued functions that can be applied to restore color and other vector-valued images, The new TV norm has the desirable properties of 1) not penalizing discontinuities (edges) in the image, 2) being rotationally invariant in the image space, and 3) reducing to the usual TV norm in the scalar case. Some numerical experiments on deionising simple color images in red-green-blue (RGB) color space are presented.

Abstract:   We present a novel geometric approach for solving the stereo problem for an arbitrary number of images (greater than or equal to 2). It is based upon the definition of a variational principle that must be satisfied by the surfaces of the objects in the scene and their images, The Euler-Lagrange equations that are deduced from the variational principle provide a set of partial differential equations (PDE's) that are used to deform an initial set of surfaces which then move toward the objects to he detected, The level set implementation of these PDE's potentially provides an efficient and robust way of achieving the surface evolution and to deal automatically with changes in the surface topology during the deformations, i.e., to deal with multiple objects, Results of an implementation of our theory also dealing with occlusion and vilility are presented on sydnthetic and real images.+

Abstract:   We present an accurate numerical scheme for the affine plane curve evolution and its morphological extension to grey-level images, This scheme is based on the iteration of a nonlocal, fully affine invariant and numerically stable operator, which can be exactly computed on polygons, The properties of this operator ensure that a fea iterations are sufficient to achieve a very good accuracy, unlike classical finite difference schemes that generally require a tot of iterations, Convergence results are provided, as well as theoretical examples and experiments.

Abstract:   New process simulation programs can closely model the complex structures of ULSI interconnects. Level-set methods simulate and predict the structure of evolving surfaces in three dimensions, such as that seen in thin-film deposition. Models rely upon iterative calibration using empirical results. The ramifications of process or design changes can be predicted, and reliability-related problems such as void formation during thin film deposition can be prevented.

Abstract:   We review the development of diffuse-interface models of hydrodynamics and their application to a wide variety of interfacial phenomena. These models have been applied successfully to situations in which the physical phenomena of interest have a length scale commensurate with the thickness of the interfacial region (e.g. near-critical interfacial phenomena or small-scale flows such as those occurring near contact lines) and fluid flows involving large interface deformations and/or topological changes (e.g. breakup and coalescence events associated with fluid jets, droplets, and large-deformation waves). We discuss the issues involved in formulating diffuse-interface models for single-component and binary fluids. Recent applications and computations using these models are discussed in each case. Further, we address issues including sharp-interface analyses that relate these models to the classical free-boundary problem, computational approaches to describe interfacial phenomena, and models of fully miscible fluids.

Abstract:   We discuss hypersurface motions in Riemannian manifolds whose normal velocity is a function of the induced hypersurface volume element and derive a second-order partial differential equation for the corresponding time function tau(x) at which the hypersurface passes the point x. Equivalently, these motions may be described in a Hamiltonian formulation as the singlet sector of certain diffeomorphism-invariant held theories. At least in some (infinite class of) cases, which could be viewed as a large-volume limit of Euclidean M-branes moving in an arbitrary M+1-dimensional Riemannian manifold, the models are integrable: In the time-function formulation the equation becomes linear [with tau(x) a harmonic function on the embedding Riemannian manifold]. We explicitly compute solutions to the large volume limit of Euclidean membrane dynamics in R-3 by methods used in electrostatics and point out an additional gradient how structure in R-n. In the Hamiltonian formulation we discover infinitely many hierarchies of integrable, multidimensional, N-component theories possessing infinitely many diffeomorphism invariant, Poisson commuting, conserved charges. (C) 1998 American Institute of Physics. [S0022- 2488(97)00912-2].

Abstract:   In this paper we present and briefly describe a Windows user- friendly system designed to assist with the analysis of images in general, and biomedical images in particular. The system, which is being made publicly available to the research community, implements basic 2D image analysis operations based on partial differential equations (PDE's). The system is under continuous development, and already includes a large number of image enhancement and segmentation routines that have been tested for several applications.

Abstract:   In this paper, we propose extensions to a powerful geometric shape modeling scheme introduced in [14]. The extension allows the model to automatically cope with topological changes and for the first time, introduces the concept of a global shape into geometric/geodesic snake models. The ability to characterize global shape of an object using very few parameters facilitates shape learning and recognition. In this new modeling scheme, object shapes are represented using a parameterized function - called the generator - which accounts for the global shape of an object and the pedal curve/surface of this global shape with respect to a geometric snake to represent any local detail. Traditionally, pedal curves/surfaces are defined as the loci of the feet of perpendiculars to the tangents of the generator from a fixed point called the pedal point. We introduce physics-based control for shaping these geometric models by using distinct pedal points - lying on a snake - for each point on the generator. The model dubbed as a "snake pedal" allows for interactive manipulation via forces applied to the snake. Automatic topological changes of the model may be achieved by implementing the geometric active contour in a level-set framework. We demonstrate the applicability of this modeling scheme via examples of shape estimation from a variety of medical image data.

Abstract:   We present a numerical method using the level set approach for serving incompressible two-phase flow with surface tension. In the level set approach, the free surface is represented as the zero level set of a smooth function; this has the effect of replacing the advection of density, which has steep gradients at the free surface, with the advection of the level set function, which is smooth. In addition, the free surface can merge or break up with no special treatment. We maintain the level set function as the signed distance from the free surface in order to accurately compute flows with high density ratios and stiff surface tension effects. In this work, we couple the level set scheme to an adaptive projection method for the incompressible Navier-Stokes equations, in order to achieve higher resolution of the free surface with a minimum of additional expense. We present two-dimensional axisymmetric and fully three-dimensional results of air bubble and water drop computations. (C) 1999 Academic Press.

Abstract:   This report is a feasibility study of a level-set method for the computation of moving material-void interfaces in an Eulerian formulation. The report briefly introduces level-set methods and focuses on the development of such a method, that does not just accurately resolve the geometry of the interface, but also the physical quantities at and near the interface. Results are presented for illustrative model problems. As concerns its ability to improve the geometrical resolution of free boundaries, as expected, the level-set method performs excellently. Concerning the improvement of physical (all other than merely geometrical) free-boundary properties, the method performs very well for downstream-facing fronts and is promising for upstream-facing ones. (C) 1999 Elsevier Science B.V. All rights reserved. AMS classification: 65M20; 65M99; 76M99; 76T05.

Abstract:   A numerical method for solving problems in which a moving surface of discontinuity separates regions of incompressible how is presented. The method developed is notable in that it does not introduce any artificial smoothing of the change in fluid properties across the surface of discontinuity. This results in an increase in accuracy relative to methods which introduce smoothing effects. The method was also shown to be fairly versatile; problems describing a free surface, an immiscible fluid interface, and a premixed flame discontinuity were solved. There is a limitation, however, in that the method appears to be most suitable for application to inviscid problems. The reason for this limitation and possible approaches toward resolving it are discussed. (C) 1999 Academic Press.

Abstract:   Level set techniques are numerical techniques for tracking the evolution of interfaces. They rely on two central embeddings; first, the embedding of the interface as the zero level set of a higher dimensional function, and second, the embedding (or extension) of the interface's velocity to this higher dimensional level set function. This paper applies Sethian's Fast Marching Method, which is a very fast technique for solving the eikonal and related equations, to the problem of building fast and appropriate extension velocities for the neighboring level sets. Our choice and construction of extension velocities serves several purposes. First, it provides a way of building velocities for neighboring level sets in the cases where the velocity is defined only on the front itself. Second, it provides a subgrid resolution not present in the standard level set approach. Third, it provides a way to update an interface according to a given velocity field prescribed on the front in such a way that the signed distance function is maintained, and the front is never re- initialized; this is valuable in many complex simulations. In this paper, we describe the details of such implementations, together with speed and convergence tests and applications to problems in visibility relevant to semi-conductor manufacturing and thin film physics. (C) 1999 Academic Press.

Abstract:   A novel approach for shape preserving contrast enhancement is presented in this paper. Contrast enhancement is achieved by means of a local histogram equalization algorithm which preserves the level-sets of the image. This basic property is violated by common local schemes, thereby introducing spurious objects and modifying the image information, The scheme is based on equalizing the histogram in all the connected components of the image, which are defined based both on the grey-values and spatial relations between pixels in the image, and following mathematical morphology, constitute the basic objects in the scene. We give examples for both grey-value and color images.

Abstract:   Nonlinear anisotropic diffusion algorithms provide significant improvement in image enhancement as compared to linear filters. However, the excessive computational cost of solving nonlinear PDEs precludes their use in real-time vision applications. In the present paper, we show that two orders of magnitude speed improvement is provided by a new image filtering paradigm in which an adaptively determined vector field specifies nonlocal application points for an image filter.

Abstract:   Robust segmentation of structures from an image is essential for a variety of image analysis problems. However, the conventional methods of region-based segmentation and gradient- based boundary finding are often frustrated by poor image quality. Here we propose a method to integrate the two approaches using game theory in an effort to form a unified approach that is robust to noise and poor initialization. This combines the perceptual notions of complete boundary information using edge data and shape priors with gray-level homogeneity using two computational modules. The novelty of the method is that this is a bidirectional framework, whereby both computational modules improve their results through mutual information sharing. A number of experiments were performed both on synthetic datasets and datasets of real images to evaluate the new approach and it is shown that the integrated method typically performs better than conventional gradient- based boundary finding.

Abstract:   Methods used for the modeling and numerical simulation of the plasma processes used in semiconductor integrated-circuit fabrication are reviewed. In the first part of the paper, we review continuum and kinetic methods, A model based on the drift-diffusion equations is presented as an example of a continuum model; the model and associated numerical solutions are discussed. The most widely used simulation method for kinetic modeling is the Particle-In-Cell/Monte-Carlo-Collision (PIC/MCC) method, in which the plasma is modeled by a system of charged superparticles (each of which represents a collection of a large number of ions or electrons) that move in self- consistent electromagnetic fields and collide via given collision cross sections. In the second part of the paper, we review the modeling and simulation of the evolution of surface topography in plasma etching and deposition.

Abstract:   In this paper we construct high-order weighted essentially non- oscillatory schemes on two-dimensional unstructured meshes (triangles) in the finite volume formulation. We present third- order schemes using a combination of linear polynomials and fourth-order schemes using a combination of quadratic polynomials. Numerical examples are shown to demonstrate the accuracies and robustness of the methods for shock calculations. (C) 1999 Academic Press.

Abstract:   We earlier introduced an approach to categorical shape description based on the singularities (shocks) of curve evolution equations. The approach relates to many techniques in computer vision, such as Blum's grassfire transform, but since the motivation was abstract it is not clear that it should also relate to human perception. We now report that this shock-based computational model can account for recent psychophysical data collected by Burbeck and Pizer. In these experiments subjects were asked to estimate the local centers of stimuli consisting of rectangles with 'wiggles' (sides modulated by sinusoids). Since the experiments were motivated by their 'core' model, in which the scale of boundary detail is proportional to object width, we conclude that such properties are also implicit in shock-based shape descriptions. More generally, the results suggest that significance is a structural notion, not an image- based one, and that scale should be defined primarily in terms of relationships between abstract entities, not concrete pixels. (C) 1999 Elsevier Science B.V. All rights reserved.

Abstract:   In this paper, we study the behavior of gas bubbles rising through a viscous liquid in a vertical square duct numerically. The level set formulation developed by Sussman et al. is successfully generalized for three-dimensional incompressible two-phase flows including large density and viscosity ratios as well as surface tension effect. Numerical simulations are carried out for gas-liquid flows with different ratios of density. It is shown that the effect of variation of the density ratio on the bubble shape and the flow field is extremely weak when the ratio is larger than 1:50. The simulations of flows with a single bubble in a duct are also carried out to investigate the influence of the duct walls on the flow field. It is clarified that the bubble shape and the rising velocity strongly depend upon the ratio of a duct width to an initial bubble radius, but that there is hardly the effect of the ratio on the rising velocity when it exceeds ten. Finally, we present a numerical result on the interaction of two bubbles. The result is in qualitatively agreement with previous experimental data.

Abstract:   Comparisons between direct numerical simulation (DNS) of detonation and detonation shock dynamics (DSD) is made. The theory of DSD defines the motion of the detonation shock in terms of the intrinsic geometry of the shock surface, in particular for condensed phase explosives the shock normal velocity, D-n, the normal acceleration, (D) over dot(n), and the total curvature, kappa. In particular, the properties of three intrinsic front evolution laws are studied and compared. These are (i) constant speed detonation (Huygens construction), (ii) curvature-dependent speed propagation (D-n-kappa relation) and (iii) curvature- and speed-dependent acceleration ((D) over dot(n)-D-n-kappa relation). We show that it is possible to measure shock dynamics directly from simulation of the reactive Euler equations and that subsequent numerical solution of the intrinsic partial differential equation for the shock motion (e.g. a (D) over dot(n)-D-n-kappa relation) reproduces the computed shock motion with high precision.

Abstract:   We present a fast algorithm for solving the eikonal equation in three dimensions: based on the fast marching method. The algorithm is of the order O(N log N), where N is the total number of grid points in the computational domain. The algorithm can be used in any orthogonal coordinate system and globally constructs the solution to the eikonal equation for each point in the coordinate domain. The method is unconditionally stable and constructs solutions consistent with the exact solution for arbitrarily large gradient jumps in velocity. In addition, the method resolves any overturning propagation wavefronts. We begin with the mathematical foundation for solving the eikonal equation using the fast marching method and follow with the numerical details. We then show examples of traveltime propagation through the SEG/EAGE salt model using point-source and planewave initial conditions and analyze the error in constant velocity media. The algorithm allows for any shape of the initial wavefront. While a point source is the most commonly used initial condition, initial plane waves can be used for controlled illumination or for downward continuation of the traveltime field from one depth to another or from a topographic depth-surface to another. The algorithm presented here is designed for computing first- arrival traveltimes. Nonetheless, since it exploits the fast marching method for solving the eikonal equation, we believe it is the fastest of all possible consistent schemes to compute first arrivals.

Abstract:   A computational method is proposed for the dynamics of solids capable of twinning and phase transitions. In a two- dimensional, sharp-interface model of twinning, the stored- energy function is a nonconvex potential with multiple wells. The evolution of twin interfaces is governed by held equations and jump conditions of momentum balance, and by a kinetic relation expressing the interface velocity as a function of the local driving traction and interfacial orientation. A regularized version of the model is constructed based on the level-set method. A level-set function which changes signs across the interface is introduced, The evolution of this function is described by a Hamilton-Jacobi equation, whose velocity coefficient is determined by the kinetic relation. Jump conditions are thereby eliminated, allowing finite- difference discretization. Numerical simulations exhibit complex evolution of the interface, including cusp formation, needle growth, spontaneous tip splitting, and topological changes that result in microstructure refinement. The results capture experimentally observed phenomena in martensitic crystals. (C) 1999 Academic Press.

Abstract:   We prove comparison results between viscosity sub- and supersolutions of degenerate elliptic and parabolic equations associated to, possibly nonlinear, Neumann boundary conditions. These results are obtained under more general assumptions on the equation tin particular the dependence in the gradient of the solution and they allow applications to quasilinear, possibly singular, elliptic or parabolic equations. One of the main applications is the extension of the so-called level set approach for equations set in bounded domains with nonlinear Neumann boundary conditions, In such a framework, the level set approach provides a weak notion for the motion of hypersurfaces with curvature dependent velocities and a prescribed contact angle at the boundary. (C) 1999 Academic Press.

Abstract:   In Sussman, Smereka, and Osher [J. Comp. Phys., 94 (1994), pp. 146-159], a numerical scheme was presented for computing incompressible air-water flows using the level set method. Crucial to the above method was a new iteration method for maintaining the level set function as the signed distance from the zero level set. In this paper we implement a "constraint" along with higher order difference schemes in order to make the iteration method more accurate and efficient. Accuracy is measured in terms of the new computed signed distance function and the original level set function having the same zero level set. We apply our redistancing scheme to incompressible flows with noticeably better resolved results at reduced cost. We validate our results with experiment and theory. We show that our "distance level set scheme" with the added constraint competes well with available interface tracking schemes for basic advection of an interface. We perform basic accuracy checks and more stringent tests involving complicated interfacial structures. As with all level set schemes, our method is easy to implement.

Abstract:   We demonstrate a general situation in which a hypersurface in R-n propagating by mean curvature, plus a nonzero driving force, develops an interior after finite time in the level set formulation of the problem.

Abstract:   This article addresses several issues of offset sizes. A new type of isoperimetric problems is introduced: find the shapes, which have offsets of minimal surface area change. This problem arises in some physical and chemical processes with constant energy release. A computational solution is presented for two cases: convex sets, and a subclass of non-convex sets (star- like sets). Both cases are discussed in the plane and in the space. The solution uses methods from convex theory including the theory of mixed volumes and also some optimization techniques. The formulas developed in the optimization problem can be applied to get analytical formulas of the length of the curve offsets, as well as of the surface area and the volume of the surface offsets. Evaluating properties of offsets without constructing them proves useful for preliminary design in solid modeling, for approximating offsets curves and for planning the velocity of offset paths. (C) 1999 Published by Elsevier Science Ltd. All rights reserved.

Abstract:   Fast adaptive numerical methods for solving moving interface problems are presented. The methods combine a level set approach with frequent redistancing and semi-Lagrangian time stepping schemes which are explicit yet unconditionally stable. A quadtree mesh is used to concentrate computational effort on the interface, so the methods move an interface with N degrees of freedom in O(N log N) work per time step. Efficiency is increased by taking large time steps even for parabolic curvature flows. The methods compute accurate viscosity solutions to a wide variety of difficult moving interface problems involving merging, anisotropy, faceting, and curvature. (C) 1999 Academic Press.

Abstract:   A new numerical method for solving geometric moving interface problems is presented. The method combines a level set approach and a semi-Lagrangian time stepping scheme which is explicit yet unconditionally stable. The combination decouples each mesh point from the others and the time step from the CFL stability condition, permitting the construction of methods which are efficient, adaptive, and modular. Analysis of a linear one- dimensional model problem suggests a surprising convergence criterion which is supported by heuristic arguments and confirmed by an extensive collection of two-dimensional numerical results. The new method computes comet viscosity solutions to problems involving geometry, anisotropy, curvature, and complex topological events. (C) 1999 Academic Press.

Abstract:   A numerical method for studying migration of voids driven by surface diffusion and electric current in a metal conducting line is developed. The mathematical model involves moving boundaries governed by a fourth order nonlinear partial differential equation which contains a nonlocal term corresponding to the electrical field and a nonlinear term corresponding to the curvature. Numerical challenges include efficient computation of the electrical field with sufficient accuracy to afford fourth order differentiation along the void boundary and to capture singularities arising in topological changes. We use the modified immersed interface method with a fixed Cartesian grid to solve for the electrical field, and the fast local level set method to update the position of moving voids, Numerical examples are performed to demonstrate the physical mechanisms by which voids interact under electromigration. (C) 1999 Academic Press.

Abstract:   Segmentation of MR brain images using intensity values is severely limited owing to field inhomogeneities, susceptibility artifacts and partial volume effects. Edge based segmentation methods suffer from spurious edges and gaps in boundaries. A multiscale method to MRI brain segmentation is presented which uses both edge and intensity information. First a multiscale representation of an image is created, which can be made edge dependent to favor intra-tissue diffusion over inter-tissue diffusion. Subsequently a multiscale linking model (the hyperstack) is used to group voxels into a number of objects based on intensity. It is shown that both an improvement in accuracy and a reduction in image post-processing can be achieved if edge dependent diffusion is used instead of linear diffusion. The combination of edge dependent diffusion and intensity based linking facilitates segmentation of grey matter, white matter and cerebrospinal fluid with minimal user interaction. To segment the total brain (white matter plus grey matter) morphological operations are applied to remove small bridges between the brain and cranium. If the total brain is segmented, grey matter, white matter and cerebrospinal fluid can be segmented by joining a small number of segments. Using a supervised segmentation technique and MRI simulations of a brain phantom for validation it is shown that the errors are in the order of or smaller than reported in literature.

Abstract:   New numerical approaches for moving boundary/interface applications tailored for Stefan problems and crystal growth simulation are proposed in this article. The focus is on the issues of accuracy and speed-up. A modified Crank-Nicolson method that is second-order accurate and stable is developed. The alternating directional implicit (ADI) method is also developed to speed up the simulation for a certain class of problems. The ADI method is shown to be asymptotically stable and at least first-order accurate. Numerical results, however, show that the ADI method actually provides second-order accuracy if the velocity can be calculated accurately. The level set method is used to update the moving interface so that the topological changes can be handled easily. Numerical experiments are compared to exact solutions and results in the literature.

Abstract:   Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton-Jacobi equations. Based on entropy-satisfying upwind schemes and fast sorting techniques, they yield consistent, accurate, and highly efficient algorithms. They are optimal in the sense that the computational complexity of the algorithms is O(N log N), where N is the total number of points in the domain. The schemes are of use in a variety of applications, including problems in shape offsetting, computing distances from complex curves and surfaces, shape-from-shading, photolithographic development, computing first arrivals in seismic travel times, construction of shortest geodesics on surfaces, optimal path planning around obstacles, and visibility and reflection calculations. In this paper, we review the development of these techniques, including the theoretical and numerical underpinnings; provide details of the computational schemes, including higher order versions; and demonstrate the techniques in a collection of different areas.

Abstract:   Instability and fragmentation of a core melt jet in water have been actively studied during the past 10 years. Several models, and a few computer codes, have been developed. However, there are, still, large uncertainties, both, in interpreting experimental results and in predicting reactor-scale processes. Steam explosion and debris coolability, as reactor safety issues, are related to the jet fragmentation process. A better understanding of the physics of jet instability and fragmentation is crucial for assessments of fuel-coolant interactions (FCIs). This paper presents research, conducted at the Division of Nuclear Power Safety, Royal Institute of Technology (RIT/NPS), Stockholm, concerning molten jet-coolant interactions, as a precursor for premixing. First, observations were obtained from scoping experiments with simulant fluids. Second, the linear perturbation method was extended and applied to analyze the interfacial-instability characteristics. Third, two innovative approaches to computational fluid dynamics (CFD) modeling of jet fragmentation were developed and employed for analysis. The focus of the studies was placed on (a) identifying potential factors, which may affect the jet instability, (b) determining the scaling laws, and (c) predicting the jet behavior for severe accident conditions. In particular, the effects of melt physical properties, and the thermal hydraulics of the mixing zone, on jet fragmentation were investigated. Finally, with the insights gained from a synthesis of the experimental results and analysis results, a new phenomenological concept, named 'macrointeractions concept of jet fragmentation' is proposed. (C) 1999 Elsevier Science S.A. All rights reserved.

Abstract:   Since the work by Osher and Sethian on level-sets algorithms for numerical shape evolutions, this technique has been used for a large number of applications in numerous fields. In medical imaging, this numerical technique has been successfully used for example, in segmentation and cortex unfolding algorithms. The migration from a Lagrangian implementation to a Eulerian one via implicit representations or level-sets brought some of the main advantages of the technique, i.e., topology independence and stability. This migration means also: that the evolution is parametrization free. Therefore, we do not know exactly how each part of the shape is deforming and the point- wise correspondence is lost. In this note we present a technique to numerically track regions on surfaces that are being deformed. using the level-sets method. The basic idea is to represent the region of interest as the intersection of two implicit surfaces and then track its deformation from the deformation of these surfaces. This technique then solves one of the main shortcomings of the very useful level-sets approach. Applications include lesion localization in medical images, region tracking in functional MRI (fMRI) visualization, and geometric surface mapping.

Abstract:   Level set methods for moving interface problems require efficient techniques for transforming an interface to a globally defined function whose zero set is the interface, such as the signed distance to the interface. This paper presents efficient algorithms for this "redistancing" problem. The algorithms use quadtrees and triangulation to compute global approximate signed distance functions. A quadtree mesh is built to resolve the interface and the vertex distances are evaluated exactly with a robust search strategy to provide both continuous and discontinuous interpolants. Given a polygonal interface with N elements, our algorithms run in O (N) space and O(N log N) time. Two-dimensional numerical results show they are highly efficient in practice. (C) 1999 Academic Press.

Abstract:   In this pager, we propose a new lattice Boltzmann scheme for simulation of multiphase flow in the nearly incompressible limit. The new scheme simulates fluid flows based on distribution functions. The interfacial dynamics, such as phase segregation and surface tension, are modeled by incorporating molecular interactions. The lattice Boltzmann equations are derived from the continuous Boltzmann equation with appropriate approximations suitable for incompressible flow. The numerical stability is improved by reducing the effect of numerical errors in calculation of molecular interactions. An index function is used to track interfaces between different phases. Simulations of the two-dimensional Rayleigh-Taylor instability yield satisfactory results. The interface thickness is maintained at 3-4 grid spacings throughout simulations without artificial reconstruction steps. (C) 1999 Academic Press.

Abstract:   First results of simulations are presented which compute the dynamical evolution of a Chandrasekhar-mass white dwarf, consisting of equal amounts of carbon and oxygen, from the onset of violent thermonuclear burning, by means of a new two- dimensional numerical code. Since in the interior of such a massive white dwarf nuclear burning progresses on microscopic scales as a sharp discontinuity, a so-called flamelet, which cannot be resolved by any numerical scheme, and since on macroscopic scales the burning front propagates due to turbulence, we make an attempt to model both effects explicitly in the framework of a finite-volume hydrodynamics code. Turbulence is included by a sub-grid model, following the spirit of large eddy simulations, and the well-localized burning front is treated by means of a level set, which allows us to compute the geometrical structure of the front more accurately than with previous methods. The only free parameters of our simulations are the location and the amount of nuclear fuel that is ignited as an initial perturbation. We find that models in which explosive carbon burning is ignited at the center remain bound by the time the front reaches low densities, where we stopped the computations because our description of combustion is no longer applicable. In contrast, off-center ignition models give rise to explosions which, however, are still too weak for typical Type Ia supernovae. Possible reasons for this rather disappointing result are discussed.

Abstract:   We present a new way of modeling deflagration fronts in reactive fluids, the main emphasis being on turbulent thermonuclear deflagration fronts in white dwarfs undergoing a Type Ia supernova explosion. Our approach is based on a level set method which treats the front as a mathematical discontinuity and allows full coupling between the front geometry and the flow field (Smiljanovski et al., 1997). With only minor modifications, this method can also be applied to describe contact discontinuities. Two different implementations are described and their physically correct behaviour for simple testcases is shown. First results of the method applied to the concrete problems of Type Ia supernovae and chemical hydrogen combustion are briefly discussed; a more extensive analysis of our astrophysical simulations is given in Reinecke et al. (1998).

Abstract:   The paper describes a high resolution method (CICSAM) for the accurate capturing of fluid interfaces on meshes of arbitrary topology. It is based on the finite-volume technique and is fully conservative. The motion of the interface is tracked by the solution of a scalar transport equation for a phase- indicator held that is discontinuous at the interface and uniform elsewhere; no explicit interface reconstruction, which is perceived to be difficult to implement on unstructured meshes, is needed. The novelty of the method lies in the adaptive combination of high resolution discretisation schemes which ensure the preservation of the sharpness and shape of the interface while retaining boundedness of the field. The special implicit implementation presented herein makes it applicable to unstructured meshes and an extension to such grids is presented. The method is capable of handling interface rupture and coalescence. The paper outlines the methodology of CICSAM and its validation against academic test cases used to verify its accuracy. (C) 1999 Academic Press.

Abstract:   We study the convergence of general threshold dynamics type approximation schemes to hypersurfaces moving with normal velocity depending on the normal direction and the curvature tensor. We also present results about the asymptotic shape of fronts propagating by threshold dynamics. Our results generalize and extend models introduced in the theories of cellular automaton and motion by mean curvature.

Abstract:   We study the coarsening observed in spiral-mode growth of thin films. The high-temperature superconductor YBa2Cu3O7-delta provides a suitable model system. The density of spirals at the surface decreases as the him gets thicker. In other words, the grain size coarsens with distance from the substrate. We propose a simple mechanism for this coarsening, based on geometrical competition of spirals with different vertical growth rates. The consequences of this mechanism are developed both analytically and numerically in the limit where adatom attachment is controlled by surface diffusion. In particular, we show how the time-evolution of spiral density, film thickness, and surface roughness depend on the spiral growth rate statistics. (C)1999 Elsevier Science B.V. All rights reserved.

Abstract:   A finite-difference formulation is applied to track solid- liquid boundaries on a fixed underlying grid. The interface is not of finite thickness but is treated as a discontinuity and is explicitly tracked. The imposition of boundary conditions exactly on a sharp interface that passes through the Cartesian grid is performed using simple stencil readjustments in the vicinity of the interface. Attention is paid to formulating difference schemes that are globally second-order accurate in x and t. Error analysis and grid refinement studies are performed for test problems involving the diffusion and convection- diffusion equations, and for stable solidification problems. Issues concerned with stability and change of phase of grid points in the evolution of solid-liquid phase fronts are also addressed. It is demonstrated that the field calculation is second-order accurate while the position of the phase front is calculated to first-order accuracy. Furthermore, the accuracy estimates hold for the cases where there is a property jump across the interface. Unstable solidification phenomena are simulated and an attempt is made to compare results with previously published work. The results indicate the need to begin an effort to benchmark computations of instability phenomena. (C) 1999 Academic Press.

Abstract:   This paper is devoted to a new method which allows to compute the convex envelope of a given function, by an evolution equation and techniques of curve evolution. We study the problem in the context of viscosity solutions and we propose numerical algorithms, to convexify a function, in one and two dimensions. In the end, we validate the model by presenting various numerical results.

Abstract:   We call "natural" image any photograph of an outdoor or indoor scene taken by a standard camera. We discuss the physical generation process of natural images as a combination of occlusions, transparencies and contrast changes. This description fits to the phenomenological description of Gaetano Kanizsa according to which visual perception tends to remain stable with respect to these basic operations. We define a contrast invariant presentation of the digital image, the topographic map, where the subjacent occlusion-transparency structure is put into evidence by the interplay of level lines. We prove that each topographic map represents a class of images invariant with respect to local contrast changes. Several visualization strategies of the topographic map are proposed and implemented and mathematical arguments are developed to establish stability properties of the topographic map under digitization.

Abstract:   We call "natural" image any photograph of an outdoor or indoor scene taken by a standard camera. We discuss the physical generation process of natural images as a combination of occlusions, transparencies and contrast changes. This description fits to the phenomenological description of Gaetano Kanizsa according to which visual perception tends to remain stable with respect to these basic operations. We define a contrast invariant presentation of the digital image, the topographic map, where the subjacent occlusion-transparency structure is put into evidence by the interplay of level lines. We prove that each topographic map represents a class of images invariant with respect to local contrast changes. Several visualization strategies of the topographic map are proposed and implemented and mathematical arguments are developed to establish stability properties of the topographic map under digitization.

Abstract:   The evolution of plane curves obeying the equation v = beta (k), where v is normal velocity and k curvature of the curve is studied. Morphological image and shape multiscale analysis of Alvarez, Guichard, Lions and Morel and affine invariant scale space of curves introduced by Sapiro and Tannenbaum as well as isotropic motions of plane phase interfaces studied by Angenent and Gurtin are included in the model. We introduce and analyze a numerical scheme for solving the governing equation and present numerical experiments, (C) 1999 Elsevier Science B.V. and IMACS. All rights reserved.

Abstract:   The level set method for multiphase compressible flows is simple to implement, especially in the presence of topological changes. However, this method was shown to suffer from large spurious oscillations. A new Ghost Fluid Method (GFM) was developed to remove these spurious oscillations by minimizing the numerical smearing in the entropy field with the help of an Isobaric Fix technique. The GFM was designed for traditional contact discontinuities where the interface moves with the fluid velocity only. In this paper, the GFM is extended to treat multimaterial interfaces where the interface velocity includes a regression rate due to the presence of chemical reactions converting one material into another. Specifically, interface models for deflagration and detonation discontinuities are considered. The resulting numerical method is robust and easy to implement. (C) 1999 Academic Press.

Abstract:   We study the highly nonlinear stages of the Rayleigh-Taylor instability (RTI) for three-dimensional flow. The proposed numerical and analytical methods are original approaches to the problem. They validate each other and the obtained results agree well. (C) 1999 American Institute of Physics. [S1070- 6631(99)00311-6].

Abstract:   The geometry of a space curve is described in terms of a Euclidean invariant frame field, metric, connection, torsion and curvature. Here the torsion and curvature of the connection quantify the curve geometry. In order to retain a stable and reproducible description of that geometry, such that it is slightly affected by non-uniform protrusions of the curve, a linearised Euclidean shortening flow is proposed. (Semi)- discretised versions of the flow subsequently physically realise a concise and exact (semi-)discrete curve geometry. Imposing special ordering relations the torsion and curvature in the curve geometry can be retrieved on a multi-scale basis not only for simply closed planar curves but also for open, branching, intersecting and space curves of non-trivial knot type. In the context of the shortening flows we revisit the maximum principle, the semi-group property and the comparison principle normally required in scale-space theories. We show that our linearised flow satisfies an adapted maximum principle, and that its Green's functions possess a semi-group property. We argue that the comparison principle in the case of knots can obstruct topological changes being in contradiction with the required curve simplification principle. Our linearised flow paradigm is not hampered by this drawback; all non-symmetric knots tend to trivial ones being infinitely small circles in a plane. Finally, the differential and integral geometry of the multi-scale representation of the curve geometry under the flow is quantified by endowing the scale- space of curves with an appropriate connection, and calculating related torsion and curvature aspects. This multi-scale modern geometric analysis forms therewith an alternative for curve description methods based on entropy scale-space theories.

Abstract:   Recently, Caselles et al. have shown the equivalence between a classical snake problem of Kass et al. and a geodesic active contour model. The PDE derived from the geodesic problem gives an evolution equation for active contours which is very powerfull for image segmentation since changes of topology are allowed using the level set implementation. However in Caselles' paper the equivalence with classical snake is only shown for 2D images and 1D curves, by using concepts of Hamiltonian theory which have no meanings for active surfaces. This paper propose to examine the notion of equivalence and to revisite Caselles et al. arguments. Then a notion equivalence is introduced and shown for classical snakes and geodesic active contours in the 2D (active contour) and 3D (active surface) case.

Abstract:   In this paper, using certain conformal mappings from uniformization theory, we give an explicit method far flattening the brain surface in a way which preserves angles. From a triangulated surface representation of the cortex, we indicate how the procedure may be implemented using finite elements. Further, we show how the geometry of the brain surface may be studied using this approach.

Abstract:   In this paper we study analytically and numerically a novel relaxation approximation for front evolution according to a curvature-dependent local law. In the Chapman-Enskog expansion, this relaxation approximation leads to the level-set equation for transport-dominated front propagation, which includes the mean curvature as the next-order term. This approach yields a new and possibly attractive way of calculating numerically the propagation of curvature-dependent fronts. Since the relaxation system is a symmetrizable, semilinear, and linearly convective hyperbolic system without singularities, the relaxation scheme captures the curvature-dependent front propagation without discretizing directly the complicated yet singular mean curvature term. (C) 1999 John Wiley & Sons, Inc.

Abstract:   We discuss a procedure for synthesizing controllers for safety specifications for hybrid systems. The procedure depends on the construction of the set of states of a continuous dynamical system that can be driven to a subset of the state space, avoiding another subset of the state space (the Reach-Avoid set). We present a new characterization of the Reach-Avoid set in terms of the solution of a pair of coupled Hamilton-Jacobi partial differential equations. We also discuss a computational algorithm for solving such partial differential equations and demonstrate its effectiveness on numerical examples.

Abstract:   We develop a fast method to localize the level set method of Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address two important issues that are intrinsic to the level set method: (a) how to extend a quantity that is given only on the interface to a neighborhood of the interface; (b) how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface. This fast local level set method reduces the computational effort by one order of magnitude, works in as much generality as the original one, and is conceptually simple and easy to implement. Our approach differs from previous related works in that we extract all the information needed from the level set function (or functions in multiphase flow) and do not need to find explicitly the location of the interface in the space domain. The complexity of our method to do tasks such as extension and distance reinitialization is O (N), where N is the number of points in space, not O(N log N) as in works by Sethian (1996, Proc. Not. Acad. Sci. 93, 1591) and Helmsen and co-workers (1996, SPIE Microlithography IX, p. 253). This complexity estimation is also valid for quite general geometrically based front motion for our localized method. (C) 1999 Academic Press.

Abstract:   We develop a fast method to localize the level set method of Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address two important issues that are intrinsic to the level set method: (a) how to extend a quantity that is given only on the interface to a neighborhood of the interface; (b) how to reset the level set function to be a signed distance function to the interface efficiently without appreciably moving the interface. This fast local level set method reduces the computational effort by one order of magnitude, works in as much generality as the original one, and is conceptually simple and easy to implement. Our approach differs from previous related works in that we extract all the information needed from the level set function (or functions in multiphase flow) and do not need to find explicitly the location of the interface in the space domain. The complexity of our method to do tasks such as extension and distance reinitialization is O (N), where N is the number of points in space, not O(N log N) as in works by Sethian (1996, Proc. Not. Acad. Sci. 93, 1591) and Helmsen and co-workers (1996, SPIE Microlithography IX, p. 253). This complexity estimation is also valid for quite general geometrically based front motion for our localized method. (C) 1999 Academic Press.

Abstract:   A 3D numerical model for computing large rigid objects suspended in fluid flow has been developed. Rather than calculating the surface pressure upon the solid body, we evaluate the net force and torque based on a volume force formulation. The total effective force is obtained by summing up the forces at the Eulerian grids occupied by the rigid body. The effects of the moving bodies are coupled to the fluid flow by imposing the velocity field of the bodies to the fluid. A Poisson equation is used to compute the pressure over the whole domain. The objects are identified by color functions and calculated by the PPM scheme and a tangent function transformation which scales the transition region of the computed interface to a compact thickness. The model is then implemented on a parallel computer of distributed memory and validated with Stokes and low Reynolds number hows. (C) 1999 Academic Press.

Abstract:   In this paper, we present a discontinuous Galerkin finite element method for solving the nonlinear Hamilton-Jacobi equations. This method is based on the Runge-Kutta discontinuous Galerkin finite element method for solving conservation laws. The method has the flexibility of treating complicated geometry by using arbitrary triangulation, can achieve high-order accuracy with a local, compact stencil, and is suited for efficient parallel implementation. One- and two- dimensional numerical examples are given to illustrate the capability of the method. At least kth order of accuracy is observed for smooth problems when kth degree polynomials are used, and derivative singularities are resolved well without oscillations, even without limiters.

Abstract:   This paper is devoted to the construction of numerical methods for the approximation of nonclassical solutions to multidimensional Hamilton{Jacobi equations for both scalar and vectorial problems. Recent theoretical results have yielded existence of solutions in many cases for which the usual viscosity approach was ill-suited or not applicable. The selection criterion used here is based on a viscoelasticity/capillarity approach, common in solid mechanics. Numerical methods adapted to this framework are built. Consistency of the model equation with the given selection criterion is essential. It is achieved here through the use of high-order finite difference schemes. By considering applications to potential well problems, the convergence of the methods are investigated.

Abstract:   Bounding contours of physical objects are often fragmented by other occluding objects. Long-distance perceptual grouping seeks to join fragments belonging to the same object. Approaches to grouping based on invariants assume objects are in restricted classes, while those based on minimal energy continuations assume a shape for the missing contours and require this shape to drive the grouping process. While these assumptions may be appropriate for certain specific tasks or when contour gaps are small, in general occlusion can give rise to large gaps, and thus long-distance contour fragment grouping is a different type of perceptual organization problem. We propose the long-distance principle that those fragments should be grouped whose fragmentation could have arisen from a shared, simple occluder. The gap skeleton is introduced as a representation of this virtual occluder, and an algorithm for computing it is given. Finally, we show that a view of the virtual occluder as a disk can be interpreted as an equivalence class of curves interpolating the fragment endpoints. (C) 1999 Academic Press.

Abstract:   R. Hamilton in [Ham1] proved that a planar region on a convex hypersurface does not "instantly bend", and so instantly vanish, under Gauss curvature flow. We demonstrate that if the surface is smooth, the planar region in fact does not move at all for some positive time. This is a sort of geometric analogue of "waiting time" phenomena for the porous medium equation.

Abstract:   Past studies using G-equation successfully described the effect of flame stretch on the laminar flame propagation. In those studies, flames were regarded as a passive interface that did not influence the flow field. The experimental evidences, however, suggested that flow field was significantly modified by the flames as the burned gas expanded at the flame. A method using G-equation and Biot-Savart law to approximate induced velocity field is described to estimate the effect of volume expansion. Present method was applied to initially wrinkled and planar flames propagating in an imposed velocity field and the average flame speed was evaluated from the ratio of simulated flame surface area and projected area of unburned stream channel. It was found that the initial wrinkling of flame could not sustain itself without Velocity disturbance but decayed into planar flame. The rate of decay of the wrinkles increased as the volume expansion ratio increased, The asymptotic change in the average burning speed occurred only in a disturbed velocity field. The average burning speed was always affected by the volume expansion that directly influenced the velocity field. With relatively small expansion ratio of 3, the average flame speed increased 10%. The comparison of the relative significance of volume expansion and flame stretch suggested that the effect of volume expansion was no less important than that of flame stretch and warranted that both of the effects should be taken into account in the simulation of flame propagation.

Abstract:   We have been developing a theory for the generic representation of 2-D shape, where structural descriptions are derived from the shocks (singularities) of a curve evolution process, acting on bounding contours. We now apply the theory to the problem of shape matching. The shocks are organized into a directed, acyclic shock graph, and complexity is managed by attending to the most significant (central) shape components first. The space of all such graphs is highly structured and can be characterized by the rules of a shock graph grammar. The grammar permits a reduction of a shock graph to a unique rooted shock tree. We introduce a novel tree matching algorithm which finds the best set of corresponding nodes between two shock trees in polynomial time. Using a diverse database of shapes, we demonstrate our system's performance under articulation, occlusion, and moderate changes in viewpoint.

Abstract:   Deformable models, which include deformable contours (the popular snakes) and deformable Surfaces, are a powerful model- based medical image analysis technique. We develop a new class of deformable models by formulating deformable surfaces in terms of an affine cell image decomposition (ACID). Our approach significantly extends standard deformable surfaces, while retaining their interactivity and other desirable properties. In particular, the ACID induces an efficient reparameterization mechanism that enables parametric deformable surfaces to evolve into complex geometries, even modifying their topology as necessary. We demonstrate that our new ACID- based deformable surfaces, dubbed T-surfaces, can effectively segment complex anatomic structures from medical volume images.

Abstract:   In this paper we undertake a systematic investigation of affine invariant object detection and image denoising. Edge detection is first presented from the point of view of the affine invariant scale-space obtained by curvature based motion of the image level-sets. In this case, affine invariant maps are derived as a weighted difference of images at different scales. We then introduce the affine gradient as an affine invariant differential function of lowest possible order with qualitative behavior similar to the Euclidean gradient magnitude. These edge detectors are the basis for the extension of the affine invariant scale-space to a complete affine flow for image denoising and simplification, and to define affine invariant active contours for object detection and edge integration. The active contours are obtained as a gradient flow in a conformally Euclidean space defined by the image on which the object is to be detected. That is, we show that objects can be segmented in an affine invariant manner by computing a path of minimal weighted affine distance, the weight being given by functions of the affine edge detectors. The gradient path is computed via an algorithm which allows to simultaneously detect any number of objects independently of the initial curve topology. Based on the same theory of affine invariant gradient flows we show that the affine geometric heat flow is minimizing, in an affine invariant form, the area enclosed by the curve.

Abstract:   We consider a second-order finite difference scheme to solve the eikonal equation. Upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives, whereas centered differences improve the accuracy of the computed traveltime, A second-order upwind essentially non- oscillatory (ENO) scheme satisfies these requirements. It is implemented with a dynamic down 'n' out (DNO) marching, an expanding box approach. To overcome the instability of such an expanding box scheme, the algorithm incorporates an efficient post sweeping (PS), a correction-by-iteration method. Near the source, an efficient and accurate mesh-refinement initialization scheme is suggested for the DNO marching, The resulting algorithm, ENO-DNO-PS, turns out to be unconditionally stable, of second-order accuracy, and efficient; for various synthetic and real velocity models having large contrasts, two PS iterations produce traveltimes accurate enough to complete the computation.

Abstract:   In this paper, a linear CNN template class is studied with a symmetric feedback matrix capable of generating trigger-waves, a special type of binary traveling-wave. The qualitative properties of these waves are examined and some simple control strategies are derived based on modifying the bias and feedback terms in a CNN template. It is shown that a properly controlled wave-front can be efficiently used in segmentation, shape and structure detection/recovery tasks. Shape is represented by the contour of an evolving front. An algorithmic framework is discussed that incorporates bias controlled trigger-waves in tracking the active contour of the objects during rigid and non-rigid motion. The object skeleton (structure) is obtained as a composition of stable annihilation lines formed during the collision of trigger wave-fronts. The shortest path problem in a binary labyrinth is also formulated as a special type of skeletonization task and solved by combined trigger-wave based techniques.

Abstract:   In this paper we summarize the main features of a new time dependent model to approximate the solution to the nonlinear total variation optimization problem for deblurring and noise removal introduced by Rudin, Osher and Fatemi. Our model is based on level set motion whose steady state is quickly reached by means of an explicit procedure based on an ENO Hamilton- Jacobi version of Roe's scheme. We show numerical evidence of the speed, resolution and stability of this simple explicit procedure in two representative 1D and 2D numerical examples.

Abstract:   In this paper we develop partial differential equations (PDEs) that model the generation of a large class of morphological filters, the levelings and the openings/closings by reconstruction. These types of filters are very useful in numerous image analysis and vision tasks ranging from enhancement, to geometric feature detection, to segmentation. The developed PDEs are nonlinear functions of the first spatial derivatives and model these nonlinear filters as the limit of a controlled growth starting from an initial seed signal. This growth is of the multiscale dilation or erosion type and the controlling mechanism is a switch that reverses the growth when the difference between the current evolution and a reference signal switches signs. We discuss theoretical aspects of these PDEs, propose discrete algorithms for their numerical solution and corresponding filter implementation, and provide insights via several experiments. Finally, we outline the use of these PDEs for improving the Gaussian scale-space by using the latter as initial seed to generate: multiscale levelings that have a superior preservation of image edges and boundaries.

Abstract:   The classical morphological segmentation paradigm is based on the watershed transform, constructed by flooding the gradient image seen as a topographic surface. For flooding a topographic surface, a topographic distance is defined from which a minimum distance algorithm is derived for the watershed. In a continuous formulation, this is modeled via the eikonal PDE, which can be solved using curve evolution algorithms. Various ultrametric distances between the catchment basins may then be associated to the flooding itself. To each ultrametric distance is associated a multiscale segmentation; each scale being the closed balls of the ultrametric distance.

Abstract:   Since the work by Osher and Sethian on level-sets algorithms for numerical shape evolutions, this technique has been used for a large number of applications in numerous fields. In medical imaging, this numerical technique has been successfully used for example in segmentation and cortex unfolding algorithms. The migration from a Lagrangian implementation to an Eulerian one via implicit representations or level-sets brought some of the main advantages of the technique, mainly, topology independence and stability. This migration means also that the evolution is parametrization free, and therefore we do not know exactly how each part of the shape is deforming, and the point-wise correspondence is lost. In this note we present a technique to numerically track regions on surfaces that are being deformed using the level-sets method. The basic idea is to represent the region of interest as the intersection of two implicit surfaces, and then track its deformation from the deformation of these surfaces. This technique then solves one of the main shortcomings of the very useful level-sets approach. Applications include lesion localization in medical images, region tracking in functional MRI visualization, and geometric surface mapping.

Abstract:   We present a supervised classification model based on a variational approach. This model is devoted to find an optimal partition compound of homogeneous classes with regular interfaces. We represent the regions of the image defined by the classes and their interfaces by level set functions, and we define a functional whose minimum is an optimal partition. The coupled Partial Differential Equations (PDE) related to the minimization of the functional axe considered through a dynamical scheme. Given an initial interface set (zero level set), the different terms of the PDE's are governing the motion of interfaces such that, at convergence, we get an optimal partition as defined above. Each interface is guided by internal forces (regularity of the interface), and external ones (data term, no vacuum, no regions overlapping). Several experiments were conducted on both synthetic an real images.

Abstract:   The partial differential equations describing the propagation of (wave) fronts in space are closely connected with the morphological erosion and dilation. Strangely enough this connection has not been explored in the derivation of numerical schemes to solve the differential equations. In this paper the morphological facet model is introduced in which an analytical function is locally fitted to the data. This function is then dilated analytically with an infinitesimal small structuring element. These sub-pixel dilations form the core of the numerical solution schemes presented in this paper. One of the simpler morphological facet models leads to a numerical scheme that is identical with a well known classical upwind finite difference scheme. Experiments show that the morphological facet model provides stable numerical solution schemes for these partial differential equations.

Abstract:   In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. The model is a combination between more classical active contour models using mean curvature motion techniques, and the Mumford-Shah model for segmentation. We minimize an energy which can be seen as a particular case of the so-called minimal partition problem. In the level set formulation, the problem becomes a "mean- curvature flow" -like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the image, as in the classical active contour models, but is instead related to a particular segmentation of the image. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable.

Abstract:   This paper is concerned with the simulation of the Partial Differential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as when do we have to "reinitialize" the distance function? How do we "reinitialize" the distance function? Etc... which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in two applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces [26], (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface.

Abstract:   Level set methods provide a robust way to implement geometric flows, but they suffer from two problems which are relevant when using smoothing flows to unfold the cortex: the lack of point-correspondence between scales and the inability to implement tangential velocities. In this paper, we suggest to solve these problems by driving the nodes of a mesh with an ordinary differential equation. We state that this approach does not suffer from the known problems of Lagrangian methods since all geometrical properties axe computed on the fixed (Eulerian) grid. Additionally, tangential velocities can be given to the nodes, allowing the mesh to follow general evolution equations, which could be crucial to achieving the final goal of minimizing local metric distortions. To experiment with this approach, we derive area and volume preserving mean curvature flows and use them to unfold surfaces extracted from MRI data of the human brain.

Abstract:   A method for deforming curves in a given image to a desired position in a second image is introduced in this paper. The algorithm is based on deforming the first image toward the second one via a partial differential equation, while tracking the deformation of the curves of interest in the first image with an additional, coupled, partial differential equation. The tracking is performed by projecting the velocities of the first equation into the second one. In contrast with previous PDE based approaches, both the images and the curves on the frames/slices of interest axe used for tracking. The technique can be applied to object tracking and sequential segmentation. The topology of the deforming curve can change, without any special topology handling procedures added to the scheme. This permits for example the automatic tracking of scenes where, due to occlusions, the topology of the objects of interest changes from frame to frame. In addition, this work introduces the concept of projecting velocities to obtain systems of coupled partial differential equations for image analysis applications. We show examples for object tracking and segmentation of electronic microscopy. We also briefly discuss possible uses of this framework for three dimensional morphing.

Abstract:   We use an unconditionally stable numerical scheme to implement a fast version of the geodesic active contour model. The proposed scheme is useful for object segmentation in images, like tracking moving objects in a sequence of images. The method is based on the Weickert-Romeney-Viergever [33] AOS scheme. It is applied at small regions, motivated by Adalsteinsson-Sethian [1] level set narrow band approach, and uses Sethian's fast marching method [26] for re-initialization. Experimental results demonstrate the power of the new method for tracking in color movies.

Abstract:   Tn the Focusing problem, we study a solution of the porous medium equation u(t) = Delta(u(m)) whose initial distribution is positive in the exterior of a closed noncircular two- dimensional region, and zero inside. We implement a numerical scheme that renormalizes the solution each time that the average size of the empty region reduces by a half. The initial condition is a function with circular level sets distorted with a small sinusoidal perturbation of wave number k > 3. We find that for nonlinearity exponents m smaller than a critical value which depends on k, the solution tends to a self-similar regime, characterized by rounded polygonal interfaces and similarity exponents that depend on m and on the discrete rotational symmetry number k. For m greater than the critical value, the final form of the interface is circular. (C) 2000 Elsevier Science B.V. All rights reserved.

Abstract:   We numerically study the spiral mode of crystal growth using a theory developed by Burton, Cabrera and Frank using a level set method. This method is novel in that it can handle not only closed curves but open curves as well. We use our method to compute interacting spirals and make estimates of growth rates. We also propose a possible coarsening mechanism for a large number of interacting spirals. (C) 2000 Elsevier Science B.V. All rights reserved.

Abstract:   Many moving boundary problems that are driven in some way by the curvature of the free boundary are gradient flows for the area of the moving interface. Examples are the Mullins-Sekerka flow, the Hele-Shaw flow, flow by mean curvature, and flow by averaged mean curvature. The gradient flow structure suggests an implicit finite differences approach to compute numerical solutions. The proposed numerical scheme will allow us to treat such free boundary problems in both R-2 and R-3. The advantage of such an approach is the reusability of much of the setup for all of the different problems. As an example of the method, we compute solutions to the averaged mean curvature flow that exhibit the formation of a singularity.

Abstract:   A physical interface can often be modeled as a surface that moves with a velocity determined by the local geometry. Accordingly, there is great interest in algorithms that generate such geometric interface motion. In this paper we unify and generalize two simple algorithms for constant and mean curvature based interface motion: the classical Huygens' principle and diffusion-generated motion. We show that the resulting generalization can be viewed both geometrically as a type of Huygens' principle and algebraically as a convolution- generated motion. Using the geometric-algebraic duality from the unification, we construct specific convolution-generated motion algorithms for a common class of anisotropic, curvature- dependent motion laws. We validate these algorithms with numerical experiments and show that they can be implemented accurately and efficiently with adaptive resolution and fast Fourier transform techniques.

Abstract:   A new method for generating adaptive moving grids is formulated based on physical quantities. Level set functions are used to construct the adaptive grids, which are solutions of the standard level set evolution equation with the Cartesian coordinates as initial values. The intersection points of the level sets of the evolving functions form a new grid at each time. The velocity vector in the evolution equation is chosen according to a monitor function and is equal to the node velocity. A uniform grid is then deformed to a moving grid with desired cell volume distribution at each time. The method achieves precise control over the Jacobian determinant of the grid mapping as the traditional deformation method does. The new method is consistent with the level set approach to dynamic moving interface problems. (C) 2000 Academic Press.

Abstract:   Hamilton-Jacobi equations are frequently encountered in applications, e.g., calculus of variations, control theory and differential games. in this paper a discontinuous Galerkin finite element method for nonlinear Hamilton-Jacobi equations (first proposed by Hu and Shu (to appear)) is investigated. This method handles the complicated geometry by using arbitrary triangulation, achieves the high order accuracy in smooth regions and the high resolution of the derivatives discontinuities. Theoretical results on accuracy and stability properties of the method are proved for certain cases and related numerical examples are presented. (C) 2000 IMACS. Published by Elsevier Science B.V. Ail rights reserved.

Abstract:   This paper presents a new variational framework for detecting and tracking multiple moving objects in image sequences. Motion detection is performed using a statistical framework for which the observed interframe difference density function is approximated using a mixture model. This model is composed of two components, namely, the static (background) and the mobile (moving objects) one. Both components are zero-mean and obey Laplacian or Gaussian law. This statistical framework is used to provide the motion detection boundaries. Additionally, the original frame is used to provide the moving object boundaries. Then, the detection and the tracking problem are addressed in a common framework that employs a geodesic active contour objective function. This function is minimized using a gradient descent method, where a flow deforms the initial curve towards the minimum of the objective function, under the influence of internal and external image dependent forces. Using the level set formulation scheme, complex curves can be detected and tracked white topological changes for the evolving curves are naturally managed. To reduce the computational cost required by a direct implementation of the level set formulation scheme, a new approach named Hermes is proposed. Hermes exploits aspects from the well-known front propagation algorithms (Narrow Band. Fast Marching) and compares favorably to them. Very promising experimental results are provided using real video sequences.

Abstract:   Interfaces have a variety of boundary conditions (or jump conditions) that need to be enforced. The Ghost Fluid Method (GFM) was developed to capture the boundary conditions at a contact discontinuity in the inviscid Euler equations and has been extended to treat more general discontinuities such as shocks, detonations, and deflagrations and compressible viscous flows. In this paper, a similar boundary condition capturing approach is used to develop a new numerical method for the variable coefficient Poisson equation in the presence of interfaces where both the variable coefficients and the solution itself may be discontinuous. This new method is robust and easy to implement even in three spatial dimensions. Furthermore, the coefficient matrix of the associated linear system is the standard symmetric matrix for the variable coefficient Poisson equation in the absence of interfaces allowing for straightforward application of standard "black box" solvers. (C) 2000 Academic Press.

Abstract:   In this paper we propose a new idea of tracking material interface. Since the regions filled with different materials at the initial time are merely transported by the velocity field, the material type at present is determined by its original location. me introduce the advection equation of the base coordinates to specify the material type, and solve this equation in the Euler framework. Thanks to the initial linear distribution, this method is free of numerical diffusion for the problems with a constant or a rigid body rotation velocity field and can produce accurate results for the general case. Moreover, it is applicable to the advection function of arbitrary distribution, for example, problems with move than two types of fluids. The new method is incorporated into a newly developed flow solver employing the semi-Lagrangian model to successfully solve the flow problems with multiple types of fluids. [S0098-2202(00)02001-0].

Abstract:   We propose a new active contour model for shape extraction of objects in grey-valued two-dimensional images based on an energy-minimization formulation. The energy functional that we consider takes into account the two requirements of object isolation and smoothness of the contour. After deriving the Euler-Lagrange equations corresponding to the energy functional, we bring out some important geometric properties of a solution to these equations. The discussion on our solution method-with the help of which we try to minimize the energy functional by evolving an initial curve-also focuses on how to prescribe the initial curve fully automatically. The effectiveness of our algorithms is demonstrated with the help of experimental results. Copyright (C) 2000 John Wiley & Sons, Ltd.

Abstract:   We introduce a method which enables us to isolate different sources of fluctuations during a typical aggregation process. As an example, we focus on the evolution of islands during irreversible submonolayer epitaxy. We show that only spatial fluctuations in island nucleation are required to produce the scaling of their size distribution as determined by Monte Carlo simulations. In particular, once the islands are seeded, their growth can be described in a purely deterministic manner.

Abstract:   We present a new way of modeling turbulent thermonuclear deflagration fronts in white dwarfs undergoing a type Ia supernova explosion. Our approach is based on a level set method which treats the front as a mathematical discontinuity and allows for full coupling between the front geometry and the flow field. First results of the method applied to the problem of type Ia supernovae are discussed. It will be shown that even in 2D and even with a physically motivated sub-grid model numerically "converged" results are difficult to obtain. (C) 2000 Published by Elsevier Science B.V. All rights reserved.

Abstract:   The behaviour of a premixed turbulent flame is numerically studied in this paper. The numerical model is based on solving turbulent flow field by the discrete vortex method. The flame is considered to be of zero thickness boundary which separates burnt and unburnt regions with different constant density and propagates into the fresh mixture at a local curvature- dependent flame speed. The flame front is located by means of level-set algorithm. The flow turbulence is simulated through the unsteady vortex-shedding mechanism. The computed velocity fields, turbulence scalar statistics as well as flame brush thickness for the turbulent V-flame are well comparable to experimental results. The computed Reynolds stresses in the flame brush region based on unconditioned velocities are substantial, but the two conditioned Reynolds stresses are negligible. These results show that the intermittency effect is a major influence on turbulent statistics in premixed flame and should require careful consideration in numerical models. Copyright (C) 2000 John Wiley & Sons, Ltd.

Abstract:   In this paper we present a nonlinear scale-space representation based on a general class of morphological strong filters, the levelings, which include the openings and closings by reconstruction. These filters are very useful for image simplification and segmentation. From one scale to the next, details vanish, but the contours of the remaining objects are preserved sharp and perfectly localized. Both the lattice algebraic and the scale-space properties of levelings are analyzed and illustrated. We also develop a nonlinear partial differential equation that models the generation of levelings as the limit of a controlled growth starting from an initial seed signal. Finally, we outline the use of levelings in improving the Gaussian scale-space by using the latter as an initial seed to generate multiscale levelings that have a superior preservation of image edges. (C) 2000 Academic Press.

Abstract:   This paper is concerned with the simulation of the partial differential equation driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian have proposed to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as When do we have to reinitialize the distance function? How do we reinitialize the distance function?, which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in three applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces (X. Zeng, et al., in Proceedings of the International Conference on Computer Vision and Pattern Recognition, June 1998), (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface, (iii) the reconstruction of the surface of 3D objects through stereo (O. Faugeras and R. Keriven, Lecture Notes in Computer Science, Vol. 1252, pp. 272- 283). (C) 2000 Academic Press.

Abstract:   The method of curve evolution is a popular method for recovering shape boundaries. However, isotropic metrics have always been used to induce the how of the curve and potential steady states tend to be difficult to determine numerically, especially in noisy or tow-contrast situations. Initial curves shrink past the steady slate and soon vanish. In this paper, anisotropic metrics are considered to remedy the situation by taking the orientation of the feature gradient into account. The problem of shape recovery or segmentation is formulated as the problem of finding minimum cuts of a Riemannian manifold. Approximate methods, namely anisotropic geodesic flows and the solution of an eigenvalue problem, are discussed. (C) 2000 Academic Press.

Abstract:   In this paper, we propose an active contour algorithm for object detection in vector-valued images (such as RGB or multispectral). The model is an extension of the scalar Chan- Vese algorithm to the vector-valued case [1]. The model minimizes a Mumford-Shah functional over the length of the contour, plus the sum of the fitting error over each component of the vector-valued image. Like the Chan-Vese model, our vector-valued model can detect edges both with or without gradient. We show examples where our model detects vector- valued objects which are undetectable in any scalar representation. For instance, objects with different missing parts in different channels are completely detected (such as occlusion). Also, in color images, objects which are invisible in each channel or in intensity can be detected by our algorithm. Finally, the model is robust with respect to noise, requiring no a priori denoising step. (C) 2000 Academic Press.

Abstract:   The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in R-n and on manifolds and connections to more general static Hamilton- Jacobi equations.

Abstract:   The Fast Marching Method is a numerical algorithm for solving the Eikonal equation on a rectangular orthogonal mesh in O(M log M) steps, where M is the total number of grid points. The scheme relies on an upwind finite difference approximation to the gradient and a resulting causality relationship that lends itself to a Dijkstra-like programming approach. In this paper, we discuss several extensions to this technique, including higher order versions on unstructured meshes in R-n and on manifolds and connections to more general static Hamilton- Jacobi equations.

Abstract:   We present a model and algorithm for segmentation of images with missing boundaries. In many situations. the human visual system fills in missing gaps in edges and boundaries, building and completing information that is not present This presents a considerable challenge in computer vision, since most algorithms attempt to exploit existing data. Completion models, which postulate how to construct missing data, are popular but are often trained and specific to particular images. In this paper, we take the following perspective: We consider a reference point within an image as given and then develop an algorithm that tries to build missing information on the basis of the given point of view and the available information as boundary data to the algorithm. We test the algorithm on some standard images, including the classical triangle of Kanizsa and low signal:noise ratio medical images.

Abstract:   In this paper, we construct second-order central schemes for multidimensional Hamilton Jacobi equations and we show that they are nonoscillatory in the sense of satisfying the maximum principle. Thus, these schemes provide the rst examples of nonoscillatory second-order Godunov-type schemes based on global projection operators. Numerical experiments are performed; L-1/L-infinity-errors and convergence rates are calculated. For convex Hamiltonians, numerical evidence con rms that our central schemes converge with second-order rates, when measured in the L-1-norm advocated in our recent paper [Numer. Math, to appear]. The standard L-infinity-norm, however, fails to detect this second-order rate.

Abstract:   In this paper, we present a weighted ENO (essentially nonoscillatory) scheme to approximate the viscosity solution of the Hamilton Jacobi equation: phi(t) + H (x(1),...,x(d), t, phi, phi(x1),...,phi(xd)) = 0. This weighted ENO scheme is constructed upon and has the same stencil nodes as the third order ENO scheme but can be as high as fifth order accurate in the smooth part of the solution. In addition to the accuracy improvement, numerical comparisons between the two schemes also demonstrate that the weighted ENO scheme is more robust than the ENO scheme.

Abstract:   A level-set formulation for the motion of faceted interfaces is presented. The evolving surface of a crystal is represented as the zero-level of a phase function. The crystal is identified by its orientation and facet speeds. Accuracy is tested on a single crystal by comparison with the exact evolution. The method is extended to study the evolution of a polycrystal. Numerical examples in two and three dimensions are presented.

Abstract:   A three-dimensional adaptive level set method has Been developed for deformable free surface problems with or without solidification. In the new scheme, a three-dimensional multizone adaptive grid generation (MAGG) scheme is employed to track the moving boundaries and a level set method is used to capture the free surface deformation. The effectiveness and robustness of the algorithm are demonstrated by solving the droplet spreading and solidification problem, in which both free surface and solidification interface movements are important.

Abstract:   The push toward higher specific fuel consumption and smaller, lighter packaging for reduced-cost aerospace gas turbine engines has resulted in large increases in engine operating pressures and temperatures, as well as major efforts to reduce gas path losses and increase component Efficiencies. This is a tread that is expected to continue, and as a result, thermal management of the hot engine section, including the fuel nozzle, combustor, and turbine, has emerged as a critical technology area requiring further research and development. For the fuel injection system, nozzle thermal management, turndown ratio, and atomization performance while maintaining correct combustor aerodynamics and low pollutant emissions are the most important performance features that necessitate optimization. Complex and expensive heat-shielded designs are often required to reduce nettle wetted-wall temperatures and prevent the formation of carbonaceous deposits within the fuel delivery passages. Optimization of designs using current computational methods is limited in capability and expensive. Significant advances infuel injection concepts, physical understanding, and computational methods are required to merl these increasingly demanding combustor requirements, with configurations at or below current cost levels. Five injector designs are presented, which include an advanced hybrid air blast (HAB) atomizer, a ball direct-injection (LDI) concept and three lean prevaporized premixer (LPP) designs that exemplify advanced fuel injection technology and ideas to address the challenges of next- generation gas turbine combustors.

Abstract:   Modeling of microstructural evolution during thin-film deposition requires a knowledge of several key activation energies (surface diffusion, island edge atom diffusion, adatom migration over descending step edges, etc.). These and other parameters must be known as a function of crystal orientation. In order to generate values for these parameters, we have developed a numerical simulation in tandem with physical experiments. By tuning the simulation to the results from experiments Lye can extract and verify approximate values for these parameters. The numerical method we use is based upon the level set method. Our model is a continuum model in directions parallel to the crystal facet, and resolves each discrete atomic layer in the normal direction. The model includes surface diffusion, step edge dynamics, and attachment/detachment rates all of which may depend upon the local geometry of the step edge. The velocity field for advancing the island edges in the level set framework is generated by computing the equilibrium adatom density on the flat terraces resulting in Laplace's equation with mixed boundary conditions at the step edges. We have turned to the finite element method for solving this equation, which results in very good agreement with analytically known solutions and with experiment, (C) 2000 Academic Press.

Abstract:   Seismic techniques incorporating high frequency asymptotic representation of the 3D elastic Green's function require efficient solution methods for the computation of traveltimes. For finite difference eikonal solvers, upwind differences are requisite to sharply resolve discontinuities in the traveltime derivatives. In anisotropic media, the direction of energy propagation is not in general tangent to the wavefront normal, while finite difference eikonal solvers compute the solution based on the traveltime gradients and wavefront normal. Local convexity of the wavefronts in transverse isotropic (TI) media is proved to shaw that wavefront normal determines the upwind direction of the energy propagation. The eikonal equations for the traveltimes in TI media of a generally inclined symmetry axis (ITI) are derived in a way that the eikonal solvers fit conveniently. A stable, second-order, shock-capturing, upwind finite difference scheme is suggested for solving ITI eikonal equations in regular grids in 3D. Numerical experiments are presented to demonstrate the efficiency of the algorithm. (C) 2000 Elsevier Science B.V. All rights reserved.

Abstract:   Reachability analysis is frequently used to study the safety of control systems. We present an implementation of an exact reachability operator for nonlinear hybrid systems. After a brief review of a previously presented algorithm for determining reachable sets and synthesizing control laws - upon whose theory the new implementation rests - an equivalent formulation is developed of the key equations governing the continuous state reachability. The new formulation is implemented using level set methods, and its effectiveness is shown by the numerical solution of three examples.

Abstract:   We present a coupled level set/volume-of-fluid (CLSVOF) method for computing 3D and axisymmetric incompressible two-phase flows. This method combines some of the advantages of the volume-of-fluid method with the level set method to obtain a method which is generally superior to either method alone. We present direct comparisons between computations made with the CLSVOF method and computations made with the level set method, the volume-of-fluid method, and the boundary integral method. We also compare our computations to the exact solution for an oscillating ellipse due to Lamb and experimental results obtained for a rising gas bubble in liquid obtained by Hnat and Buckmaster. Our computational examples focus on flows in which surface tension forces and changes in topology are dominant features Of the flow. (C) 2000 Academic Press.

Abstract:   A general method for modeling ionized physical vapor deposition is presented. As an example, the method is applied to growth of an aluminum film in the presence of an ionized argon flux. Molecular dynamics techniques are used to examine the surface adsorption, reflection, and sputter reactions taking place during ionized physical vapor deposition. We predict their relative probabilities and discuss their dependence on energy and incident angle. Subsequently, we combine the information obtained from molecular dynamics with a line of sight transport model in a two-dimensional feature, incorporating all effects of reemission and resputtering. This provides a complete growth rate model that allows inclusion of energy- and angular- dependent reaction rates. Finally, a level-set approach is used to describe the morphology of the growing film. We thus arrive at a computationally highly efficient and accurate scheme to model the growth of thin films. We demonstrate the capabilities of the model predicting the major differences on Al film topographies between conventional and ionized sputter deposition techniques studying thin film growth under ionized physical vapor deposition conditions with different Ar fluxes.

Abstract:   We compute time-dependent solutions of the sharp-interface model of dendritic solidification in two dimensions by using a level set method. The steady-state results are in agreement with solvability theory. Solutions obtained from the level set algorithm are compared with dendritic growth simulations performed using a phase-field model and the two methods are found to give equivalent results. Furthermore, we perform simulations with unequal diffusivities in the solid and liquid phases and find reasonable agreement with the available theory.

Abstract:   A method for deforming curves in a given image to a desired position in the second image is introduced in this paper. The algorithm is based on deforming the first image toward the second one via a Partial Differential Equation (PDE), while tracking the deformation of the curves of interest in the first image with an additional, coupled, PDE. The tracking is performed by projecting the velocities of the first equation into the second one. In contrast with previous PDE-based approaches, both the images and the curves on the frames/slices of interest are used for tracking. The technique can be applied to object tracking and sequential segmentation. The topology of the deforming curve can change without any special topology handling procedures added to the scheme. This permits, for example, the automatic tracking of scenes where, due to occlusions, the topology of the objects of interest changes from frame to frame. In addition, this work introduces the concept of projecting velocities to obtain systems of coupled PDEs for image analysis applications We show examples for object tracking and segmentation of electronic microscopy.

Abstract:   In this paper we formulate a time dependent model to approximate the solution to the nonlinear total variation optimization problem for deblurring and noise removal introduced by Rudin and Osher [ Total variation base image restoration with free local constraints, in Proceedings IEEE Internat. Conf. Imag. Proc., IEEE Press, Piscataway, NJ, ( 1994), pp. 31-35] and Rudin, Osher, and Fatemi [ Phys. D, 60 ( 1992), pp. 259-268], respectively. Our model is based on level set motion whose steady state is quickly reached by means of an explicit procedure based on Roe's scheme [ J. Comput. Phys., 43 ( 1981), pp. 357-372], used in fluid dynamics. We show numerical evidence of the speed of resolution and stability of this simple explicit procedure in some representative 1D and 2D numerical examples.

Abstract:   Moving surfaces that self-intersect arise naturally in the geometric optics model of wavefront motion. Standard ray tracing techniques can be used to compute these motions, but they lose resolution as rays diverge. In this paper we develop numerical methods that maintain uniform spatial resolution of the front at all times. Our approach is a fixed grid, wavefront capturing formulation based on the Dynamic Surface Extension method of Steinhoff and Fan (Technical report, University of Tennessee Space Institute). The new methods can treat arbitrarily complicated self intersecting fronts, as well as refraction, reflection, and focusing. We also develop methods fur curvature-dependent front motions and the motion of filaments. We validate our methods with numerical experiments. (C) 2000 Academic Press.

Abstract:   An improved numerical solver for the unified solution of hows involving an interface between either compressible or incompressible fluids is proposed. This method is based on the CIP-CUP (Cubic Interpolated Propagation/Combined, Unified Procedure) which is a semi-implicit solver for the Euler equations of fluid flows. In the CIP-CUP method, each of convection and acoustic parts of the Euler equations are treated individually by a splitting manner. Namely, the convection part is solved by CTP method and the acoustic part is solved by CUP method. As Part I of this series of articles, we propose an improved scheme for the convection part. The ability of the CIP method to capture interfaces is highly improved by replacing the cubic interpolation function used in the CIP scheme with a quadratic-type extrapolation function only around the interface. With this improvement, oscillation and diffusion in the solution at interfaces (or phase boundaries) are removed, which are the most significant problems on free-surface flow simulations especially in the case where we treat some materials which have quite different properties. By giving some constraints on the extrapolation function, its stability was guaranteed. Furthermore, we propose a simple scheme for recognizing interface location and motion with a color function or a level set function. This scheme is very useful for the extrapolation process. Effectiveness, accuracy and stability of the improved method were demonstrated with some examples. (C) 2000 Published by Elsevier Science B.V.

Abstract:   Eye tracking is one of the key problems in controlled active vision. Because of modelling uncertainty and noise in the signals, it becomes a challenging problem for robust control. In this paper, we outline some of the key issues involved as well as some possible solutions. We will need to make contact with techniques from machine vision and multi-scale image processing in carrying out this task. In particular, we will sketch some of the necessary methods from computer vision and image processing including optical flow, active contours ('snakes'), and geometric driven flows. The paper will thus have a tutorial flavor as well. Copyright (C) 2000 John Wiley & Sons, Ltd.

Abstract:   We have performed two-dimensional (2D) and three-dimensional (3D) (axisymmetric) numerical simulations of physical vapor deposition into high aspect ratio trenches and vias used for modern very large-scale integration interconnects. The topographic evolution is modeled using (continuum) level set methods. The level set approach is a powerful computational technique for accurately tracking moving interfaces or boundaries, where the advancing front is embedded as the zero level set (isosurface) of a higher dimensional mathematical function. We have validated both codes against analytic formulas for step coverage. First, we study the 2D case of long rectangular trenches including 3D out-of-plane target flux. The 3D flux can be obtained from molecular dynamics computations, and hence our approach represents a hybrid atomistic/continuum model. Second, we report results of axisymmetric 3D simulations of high aspect ratio vias, which we compare with experimental data for Ti/TiN barrier layers. We find that the simulations (using a cosine angular distribution for the flux from the target) overpredict bottom coverage in some cases by approximately 20%-30% for both collimated and uncollimated deposition, but in other cases provide a reasonably accurate comparison with experiment. (C) 2000 American Institute of Physics. [S0021-8979(00)09120-9].

Abstract:   We present a fast marching level set method for reservoir simulation based on a fractional flow formulation of two-phase, incompressible, immiscible flow in two or three space dimensions. The method uses a fast marching approach and is therefore considerably faster than conventional finite difference methods. The fast marching approach compares favorably with a front tracking method as regards both efficiency and accuracy. In addition, it maintains the advantage of being able to handle changing topologies of the front structure.

Abstract:   A class of fourth-order partial differential equations (PDEs) are proposed to optimize the trade-off between noise removal and edge preservation. The time evolution of these PDEs seeks to minimize a cost functional which is an increasing function of the absolute value of the Laplacian of the image intensity function. Since the Laplacian of an image at a pixel is zero if the image is planar in its neighborhood, these PDEs attempt to remove noise and preserve edges by approximating an observed image with a piecewise planar image. Piecewise planar images look more natural than step images which anisotropic diffusion (second order PDEs) uses to approximate an observed image. So the proposed PDEs are able to avoid the blocky effects widely seen in images processed by anisotropic diffusion, while achieving the degree of noise removal and edge preservation comparable to anisotropic diffusion. Although both approaches seem to be comparable in removing speckles in the observed images, speckles are more visible in images processed by the proposed PDEs, because piecewise planar images are less likely to mask speckles than step images and anisotropic diffusion tends to generate multiple false edges. Speckles can be easily removed by simple algorithms such as the one presented in this paper.

Abstract:   We develop and test an algorithmic approach to the boundary design of elastic structures. The goal of our approach is two- fold: first, to develop a method which allows one to rapidly solve the two-dimensional Lame equations in arbitrary domains and compute, for example, the stresses, and second, to develop a systematic way of modifying the design to optimize chosen properties. At the core, our approach relies on two distinct steps. Given a design, we first apply an explicit jump immersed interface method to compute the stresses for a given design shape. We then use a narrow band level set method to perturb this shape and progress towards an improved design. The equations of 2D linear elastostatics in the displacement formulation on arbitrary domains are solved quickly by domain embedding and the use of fast elastostatic solvers. This effectively reduces the dimensionality of the problem by one. Once the stresses are found, the level set method, which represents the design structure through an embedded implicit function, is used in the second step to alter the shape, with velocities depending on the stresses in the current design, Criteria are provided for advancing the shape in an appropriate direction and fur correcting the evolving shape when given constraints are violated. (C) 2000 Academic Press.

Abstract:   In this paper we analyse the non-stationary iterative Tikhonov- Morozov method analytically and numerically for the stable evaluation of differential operators and for denoizing images. A relationship between non-stationary iterative Tikhonov- Morozov regularization and a filtering technique based on a differential equation of third order is established and both methods are shown to be effective for denoizing images and for the stable evaluation of differential operators. The theoretical results are verified numerically on model problems in ultrasound imaging and numerical differentiation. Copyright (C) 2000 John Wiley & Sons, Ltd.

Abstract:   We present a method to design controllers for safety specifications in hybrid systems. The hybrid system combines discrete event dynamics with nonlinear continuous dynamics: the discrete event dynamics model linguistic and qualitative information and naturally accommodate mode switching logic, and the continuous dynamics model the physical processes themselves, such as the continous response of an aircraft to the forces of aileron and throttle. Input variables model both continuous and discrete control and disturbance parameters. We translate safety specifications into restrictions on the system's reachable sets of states. Then, using analysis based on optimal control and game theory for automata and continuous dynamical systems, we derive Hamilton-Jacobi equations whose solutions describe the boundaries of reachable sets. These equations are the heart of our general controller synthesis technique for hybrid systems, in which we calculate feedback control laws for the continuous and discrete variables, which guarantee that the hybrid system remains in the "safe subset" of the reachable set. We discuss issues related to computing solutions to Hamilton-Jacobi equations. Throughout, we demonstrate our techniques on examples of hybrid automata modeling aircraft conflict resolution, autopilot flight mode switching, and vehicle collision avoidance.

Abstract:   As a new method, the Level Set method had been developed to compute the interface of two-phase flow. The basic mathematical theory and the detailed method to solve the free surface hydrodynamic problem had been investigated. By using the Level Set method, the transformation of a solitary wave over a front step was simulated. The results were in good agreement with laboratory experiments.

Abstract:   A recently developed spectral model for premixed turbulent combustion in the flamelet regime (based on the EDQNM turbulence theory) has been compared with both direct numerical simulations (DNS) and experimental data. The 128(3) DNS is performed at a Reynolds number of 223 based on the integral length scale. Good agreement is observed for both single- and two-point quantities (i.e. ratio of the turbulent to laminar burning velocities, scalar autocorrelation. dissipation and scalar-velocity cross correlation spectral for the two different values of u'/s(LO) considered. The model also predicts the rapid transient behaviour of the flame at early times. An experimental set-up is then described for generating a lean methane-ah flame and measuring two- point spatial correlations along the midpoint of the flame brush (i.e. along the (C) over bar = 0.5 contour). The experimental measurements in the flamelet regime take the form of a discontinuous or 'telegraph' signal. The EDQNM model, in contrast, describes an 'ensemble' of flames, and thus is based solely on continuous variables. A theoretical relationship between the correlation obtained from the EDQNM model and the equivalent correlation for a discontinuous (experimental) flame is derived. The relationship is used to enable a meaningful comparison between experimentally observed and model correlations. In general, the agreement is good for the three different cases considered in this study, with most of the error occurring at the lowest Reynolds number (Re-L = 22). Furthermore, it is shown that considerably more error would result if no attempt is made to convert the ensemble representation in the model to an equivalent single-flame or 'telegraph' signal.

Abstract:   In this paper, a partial-differential equations (PDE)-based system for detecting the boundary of skin lesions in digital clinical skin images is presented. The image is first preprocessed via contrast-enhancement and anisotropic diffusion. If the lesion is covered by hairs, a PDE-based continuous morphological filter that removes them is used as an additional preprocessing step. Following these steps, the skin lesion is segmented either by the geodesic active contours model or the geodesic edge tracing approach. These techniques are based on computing, again via PDEs, a geodesic curve in a space defined by the image content. Examples showing the performance of the algorithm are given.

Abstract:   This paper presents a novel method for blending images, Image blending refers to the process of creating a set of discrete samples of a continuous, one-parameter family of images that connects a pair of input images. Image blending has uses in a variety of computer graphics and image processing applications. In particular, it can be used for image morphing, which is a method for creating video streams that depict transformations of objects in scenes based solely on pairs of images and sets of user-defined fiducial points. Image blending also has applications for video compression and image-based rendering. The proposed method for image blending relies on the progressive minimization of a difference metric which compares the level sets between two images. This strategy results in an image blend which is the solution of a pair of coupled, nonlinear, first-order, partial differential equations that model multidimensional level-set propagations. When compared to interpolation this method produces more natural appearances of motion because it manipulates the shapes of image contours rather than simply interpolating intensity values. This strategy results in a process that has the qualitative property of deforming greyscale objects in images rather than producing a simple fade from one object to another. This paper presents the mathematics that underlie this new method, a numerical implementation, and results on real images that demonstrate its effectiveness.

Abstract:   High-speed planar laser-induced fluorescence (PLIF) and 3-D large eddy simulations (LES) are used to study turbulent flame kernel growth, wrinkling and the formation of separated flame pockets in methane/air mixtures. Turbulence was effected by a set of rotary fans situated in a cylindrical enclosure. Flame wrinkling was followed on sequential 2-D OH images captured at kHz repetition rates. Under stoichiometric conditions and low turbulence levels the flame kernel remains singly connected and close to spherical in shape. By increasing turbulence or reducing the stoichiometry of the mixture the formation of separated pockets could be observed and studied. The mechanisms behind these phenomena are investigated qualitatively by LES of a level-set G-equation describing the flame surface propagation in turbulent flows.

Abstract:   In this paper, we use partial-differential-equation-based filtering as a preprocessing add post processing strategy for computer-aided cytology, We wish to accurately extract and classify. the shapes of nuclei from confocal microscopy images, which is a prerequisite to an accurate quantitative intranuclear (genotypic and phenotypic) and internuclear (tissue structure) analysis of tissue and cultured specimens. First, we study the use of a geometry-driven edge-preserving image smoothing mechanism before nuclear segmentation. We show how this biter outperforms other widely-used filters in that it provides higher edge fidelity. Then we apply the same filter,,vith a different initial condition, to smooth nuclear surfaces and obtain sub-pixel accuracy. Finally we use another instance of the geometrical filter to correct for misinterpretations of the nuclear surface by the segmentation algorithm. Our prefiltering and post filtering nicely complements our initial segmentation strategy, in that it provides substantial and measurable improvement in the definition of the nuclear surfaces.

Abstract:   The performance of a dynamic subgrid model for the turbulent burning speed of a premixed flame is investigated for a series of idealized test cases where the flame front is wrinkled by a multiple-scale shear flow; a rigorous asymptotic subgrid model is also implemented for comparison. Explicit formulae for the flame wrinkled shape and turbulent speed are available to generate a reference database. The role of the subgrid wrinkling models is to achieve the same overall flame shape and propagation speed in a simulation where only the largest scales of the flow are explicitly accounted for. Very good results are obtained when the subgrid burning speed enhancement is estimated using the asymptotic subgrid model. On the other hand, the dynamic model attempts to exploit the scaling observable in the simulation to extrapolate the turbulent burning speed enhancement in the original system The performance of this strategy is adequate for some regimes but poor for others; the source of the problem is traced back to the existence of a scaling transition that occurs as the flame propagating speed is adjusted during the large-eddy simulation. A modification to the scaling of the enhanced burning is implemented to account for the existence of the two distinct scaling ranges; it improves significantly the predictions of the dynamic model away from the transition, but results in the near-critical range remain predictably very poor compared with the rigorous asymptotic model results. These conclusions based on apriori performance for the reference steady data are confirmed by comparing unsteady large-eddy and direct simulations. Results based on rigorous mathematical tools are possible here because of the separation of length scales in the special class of idealized flow fields used in this study: their relevance to more realistic flows is also discussed.

Abstract:   We report the development of the first apparent horizon locator capable of finding multiple apparent horizons in a "generic" numerical black hole spacetime. We use a level-how method which, starting from a single arbitrary initial guess surface, can undergo topology changes as it flows towards disjoint apparent horizons. The level flow method has two advantages: (1) The solution is independent of changes in the initial guess and (2) the solution can have multiple components. We illustrate our method of locating apparent horizons in a short Kerr-Schild binary black hole grazing collision.

Abstract:   We discuss the origin, use and implementation of ray tracing methods for nonlinear inverse modelling problems associated with wave propagation phenomena. These methods have a long tradition in acoustic and elastodynamic wave propagation problems for Various important applications, and they can surely be helpful in other realms, including electromagnetic wave propagation and diffusion dominated phenomena. The subjacent models used for forward simulation have increased in complexity and dimension as computer power has enabled us to solve such problems in an acceptable amount of time. Thus, we will devote part of this paper to discuss modelling issues and the parallel implementation of algorithms for applications in earth sciences.

Abstract:   In this paper we consider a fundamental visualization problem: shape reconstruction from an unorganized data set. A new minimal-surface-like model and its variational and partial differential equation (PDE) formulation are introduced. In our formulation only distance to the data set is used as our input. Moreover, the distance is computed with optimal speed using a new numerical PDE algorithm. The data set can include points, curves, and surface patches. Our model has a natural scaling in the nonlinear regularization that allows flexibility close to the data set while it also minimizes oscillations between data points. To find the final shape, we continuously deform an initial surface following the gradient flow of our energy functional. An offset (an exterior contour) of the distance function to the data set is used as our initial surface. We have developed a new and efficient algorithm to find this initial surface. We use the level set method in our numerical computation in order to capture the deformation of the initial surface and to find an implicit representation (using the signed distance function) of the final shape on a fixed rectangular grid. Our variational/PDE approach using the level set method allows us to handle complicated topologies and noisy or highly nonuniform data sets quite easily. The constructed shape is smoother than any piecewise linear reconstruction. Moreover, our approach is easily scalable for different resolutions and works in any number of space dimensions. (C) 2000 Academic Press.

Abstract:   In the Eulerian approach for simulating interfaces in two-phase Rows, the main difficulties arise from the tired character of the mesh which does not follow the interface. Therefore. near the interface there are computational cells containing both fluids which require a suitable modelling of the mixture. Furthermore. most numerical algorithms, such as the volume of fluid or the level set method, involve the transport of a function indicating the localization of each phase. Due to unavoidable numerical diffusion. they have the tendency to thicken this mixture layer around the interface. It is thus necessary to model correctly the two-phase mixture. In the context of compressible gas dynamics we propose such a model. valid for any type of state laws, which satisfies an important property of pressure stability through the interface. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.

Abstract:   We present a supervised classification model based on a variational approach. This model is devoted to find an optimal partition composed of homogeneous classes with regular interfaces. The originality of the proposed approach concerns the definition of a partition by the use of level sets. Each set of regions and boundaries associated to a class is defined by a unique level set function. We use as many level sets as different classes and all these level sets are moving together thanks to forces which interact in order to get an optimal partition. We show how these forces can be defined through the minimization of a unique fonctional. The coupled Partial Differential Equations (PDE) related to the minimization of the functional are considered through a dynamical scheme. Given an initial interface set (zero level set), the different terms of the PDE's are governing the motion of interfaces such that, at convergence, we get an optimal partition as defined above. Each interface is guided by internal forces (regularity of the interface), and external ones (data term, no vacuum, no regions overlapping). Several experiments were conducted on both synthetic and real images.

Abstract:   In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. We minimize an energy which can he seen as a particular case of the minimal partition problem, In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the. image, as in the classical active contour models, hut is instead related to a particular segmentation of the image. We will give a numerical algorithm using finite differences. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.

Abstract:   Segmentation of the skull in medical imagery is an important stage in applications that require the construction of realistic models of the head. Such models are used, for example, to simulate the behavior of electro-magnetic fields in the head and to model the electrical activity of the cortex in EEG and MEG data. in this paper, we present a new approach for segmenting regions of bone in MRI volumes using deformable models. Our method takes into account the partial volume effects that occur with MRI data, thus permitting a precise segmentation of these bone regions. At each iteration of the propagation of the model, partial volume is estimated in a narrow band around the deformable model, Our segmentation method begins with a pre-segmentation stage, in which a preliminary segmentation of the skull is constructed using a region-growing method. The surface that bounds the pre- segmented skull region offers an automatic 3D initialization of the deformable model. This surface is then propagated (in 3D) in the direction of its normal. This propagation is achieved using level set method, thus permitting changes to occur in the topology of the surface as it evolves, an essential capability for our problem. The speed at which the surface evolves is a function of the estimated partial volume. This provides a sub- voxel accuracy in the resulting segmentation. (C) 2000 Elsevier Science B.V. All rights reserved.

Abstract:   This paper presents a literature survey of automatic 3D surface registration techniques emphasizing the mathematical and algorithmic underpinnings of the subject. The relevance of surface registration to medical imaging is that there is much useful anatomical information in the form of collected surface points which originate from complimentary modalities and which must be reconciled. Surface registration can be roughly partitioned into three issues: choice of transformation, elaboration of surface representation and similarity criterion, and matching and global optimization. The first issue concerns the assumptions made about the nature of relationships between the two modalities, e.g. whether a rigid-body assumption applies, and if nor, what type and how general a relation optimally maps one modality onto the other. The second issue determines what type of information we extract from the 3D surfaces, which typically characterizes their local or global shape, and how we organize this information into a representation of the surface which will lead to improved efficiency and robustness in the last stage. The last issue pertains to how we exploit this information to estimate the transformation which best aligns local primitives in a globally consistent manner or which maximizes a measure of the similarity in global shape of two surfaces. Within this framework, this paper discusses in detail each surface registration issue and reviews the state-of-the-art among existing techniques. (C) 2000 Elsevier Science BN. All rights reserved.

Abstract:   We present a new class of deformable contours (snakes) and apply them to the segmentation of medical images. Our snakes are defined in terms of an affine cell image decomposition (ACID). The 'snakes in ACID' framework significantly extends conventional snakes, enabling topological flexibility among other features. The resulting topology adaptive snakes, or 'T- snakes', can be used to segment some of the most complex-shaped biological structures from medical images in an efficient and highly automated manner. (C) 2000 Elsevier Science BN. All rights reserved.

Abstract:   The paper addresses Rayleigh-Taylor instability (RTI) problems and presents a method of describing an interface with markers on a Eulerian mesh, which is implemented in the MAH-3 code. The proposed method allows a more accurate description of the evolution of the interface caused by Rayleigh-Taylor perturbations and presences symmetry of the interface under appropriate symmetry of the problem (planar, cylindrical, and spherical). The method employs an unstructured triangular mesh of makers. Method capabilities are demonstrated on 2D and 3D Rayleigh-Taylor instability problems.

Abstract:   A new method is described to compute short acoustic or electromagnetic pulses that propagate according to geometrical optics. The pulses are treated as zero thickness sheets that can propagate over long distances through inhomogeneous media with multiple reflections. The method has many of the advantages of Lagrangian ray tracing, but is completely Eulerian, typically using a uniform Cartesian grid, Accordingly, it can treat arbitrary configurations of pulses that can reflect from surfaces and pass through each other without requiring special computational marker arrays for each pulse, Also, information describing the pulses, which are treated as continuous surfaces, can be available throughout the computational grid, rather than only at isolated individual markers. The method uses a new type of representation, which we call "Dynamic Surface Extension." The basic idea is to propagate or "broadcast" defining fields from each pulse surface through a computational grid. These fields carry information about a nearby pulse surface that is used at each node to compute the location of the pulse surfaces and other attributes, such as amplitude. Thus the emphasis is on the dynamics of these propagating defining fields, which obey only local Eulerian equations at each node. The Dynamic Surface Extension representation can be thought of as dual to level set representation: The defining fields involve single valued variables which are constant at each time along lines that are normal to the evolving surface, whereas level set techniques involve a function which has constant values on the evolving surface and neighboring surfaces. In this way the new method overcomes the inability of level set or Eikonal methods to treat intersecting pulses that obey a wave equation and can pass through each other, while still using only single-valued variables. Propagating thin pulse surfaces in 1-D, 2-D, and 3-D that can reflect from boundaries and pass through each other are computed using the new method. The method was first presented as a new, general representation of surfaces, filaments, and particles by J. Steinhoff and M. Fan (1998, Eulerian computation of evolving surfaces, curves and discontinuous fields, UTSI preprint). (C) 2000 Academic Press.

Abstract:   We investigated numerically the relation between a roller and the pressure distribution to clarify the dynamics of the roller in circular hydraulic jumps. We found that a roller which characterizes a type II jump is associated with two high pressure regions after the jump, while a type I jump (without the roller) is associated with only one high pressure region. Our numerical results show that building up an appropriate pressure field is essential for a roller.

Abstract:   Image denoising and segmentation are fundamental problems in the field of image processing and computer vision with numerous applications. In this paper, we present a nonlinear PDE-based model for image denoising and segmentation which unifies the popular model of Alvarez, Lions and Morel (ALM) for image denoising and the Caselles, Kimmel and Sapiro model of geodesic "snakes". Our model includes nonlinear diffusive as well as reactive terms and leads to quality denoising and segmentation results as depicted in the experiments presented here. We present a proof for the existence, uniqueness, and stability of the viscosity solution of this PDE-based model. The proof is in spirit similar to the proof of the ALM model; how ever, there are several differences which arise due to the presence of the reactive terms that require careful treatment/consideration. A fast implementation of our model is realized by embedding the model in a scale space and then achieving the solution via a dynamic system governed by a coupled system of first-order differential equations. The dynamic system finds the solution at a coarse scale and tracks it continuously to a desired fine scale. We demonstrate the smoothing and segmentation results on several real images. (C) 2000 Elsevier Science Ltd. All rights reserved.

Abstract:   A new anisotropic diffusion model is proposed for image restoration and segmentation, which is closely related to the minimization problems for the unconstrained total variation E(u) = integral(Omega) alpha(x)\del u\ + (beta/2)\u - I\(2). Existence, uniqueness, and stability of the viscosity solutions of the equation are proved. The experimental results are given and compared with the existing models in the framework of image restoration. The improvement on preserving sharp edges by using the new model is visible. (C) 2000 Elsevier Science Ltd. All rights reserved.

Abstract:   This paper analyzes the problem of shape modeling using the principle of active geometric deformable models. While the basic modeling technique already exists in the literature, we highlight many of its drawbacks and discuss their source and steps to overcome them. We propose a new stopping criterion to address the stopping problem. We also propose to apply level set algorithm to implement the active geometric deformable models, thereby handling topology changes automatically. To alleviate the numerical problems associated with the implementation of the level set algorithm, we propose a new adaptive multigrid narrow band algorithm. All the proposed new changes have been illustrated with experiments with synthetic images and medical images.

Abstract:   The analysis of medical images has been woven into the fabric of the Pattern Analysis and Machine Intelligence (PAMI) community since the earliest days of these Transactions. Initially, the efforts in this area were seen as applying pattern analysis and computer vision techniques to another interesting dataset. However, over the last two to three decades, the unique nature of the problems presented within this area of study have led to the development of a new discipline in its own right. Examples of these include: the types of image information that are acquired, the fully three- dimensional image data, the nonrigid nature of object motion and deformation, and the statistical variation of both the underlying normal and abnormal ground truth. In this paper, we look at progress in the field over the last 20 years and suggest some of the challenges that remain for the years to come.

Abstract:   We derive macroscopic governing laws of growth velocity, surface tension, mobility, critical nucleus size, and morphological evolution of clusters, from microscopic scale master equations for a prototype surface reaction system with long range adsorbate-adsorbate interactions.

Abstract:   The goal of the Caltech Center is to construct a Virtual Test Facility-a problem-solving environment for full 3D parallel simulation of the dynamic response of materials undergoing compression due to shock waves.

Abstract:   The fully three-dimensional interactions of one or two buoyant drops with a fluid-fluid interface are studied numerically at the moderate Reynolds number and Bond number. This complex interaction process is successfully simulated using the three- dimensional level set approach to deal with the deformable inter face and drops. The characteristics of vortex ring generated at the coalescence between the interface and the drop depend upon the viscosity ratio between two fluids, and they have much to do with a jet formation process. In the case of two drops, the motion of the trailing drop is influenced hy the deformation of interfaces through the coalescence of the leading drop with its homophase. The behavior of the trailing drop is also classified into two types with respect to the viscosity ratio. These studies show that the investigation in the viewpoint of vorticity is effective on the understanding of the interaction between drops and an interface.

Abstract:   We present a directional boundary representation which deals locally and consistently with the boundary's "inside." We show that collision and wave propagation are reduced to addition on the spectrum of directions and we derive transformation laws for differential geometrical properties such as directed curvature.

Abstract:   In this paper we study the front propagation with constant speed and small curvature viscosity. We first investigate two related problems of conservation laws, one of which is on the nonlinear viscosity methods for the conservation laws, and the other one is on the structure of solutions to conservation laws with L-1 initial data. We show that the nonlinear viscosity methods approaching the piecewise smooth solutions with finitely many discontinuity for convex conservation laws have the first-order rate of L-1-convergence. The solutions of conservation laws with L-1 initial data are shown to be bounded after t > 0 if all singular points of initial data are from shocks. These results suggest that the front propagation with constant speed and a small curvature viscosity will approach the front movements with a constant speed, as the small parameter goes to zero. After the front breaks down, the cusps will disappear promptly and corners will be formed. (C) 2000 Academic Press.

Abstract:   This paper is concerned with the simulation of the Partial Differential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately the solution to this equation is not a distance function. As a consequence, the practical application of the level set method is plagued with such questions as when do we have to "reinitialize" the distance function? How do we "reinitialize" the distance function? Etc... which reveal a disagreement between the theory and its implementation. This paper proposes an alternative to the use of Hamilton-Jacobi equations which eliminates this contradiction: in our method the implicit representation always remains a distance function by construction, and the implementation does not differ from the theory anymore. This is achieved through the introduction of a new equation. Besides its theoretical advantages, the proposed method also has several practical advantages which we demonstrate in three applications: (i) the segmentation of the human cortex surfaces from MRI images using two coupled surfaces [27], (ii) the construction of a hierarchy of Euclidean skeletons of a 3D surface, (iii) the reconstruction of the surface of 3D objects through stereo [13].

Abstract:   This paper introduces the use of spatially adaptive components into the geodesic active contour segmentation method for application to volumetric medical images. These components are derived from local structure descriptors and are used both in regularization of the segmentation and in stabilization of the image-based vector field which attracts the contours to anatomical structures in the images. They are further used to incorporate prior knowledge about spatial location of the structures of interest. These components can potentially decrease the sensitivity to parameter settings inside the contour evolution system while increasing robustness to image noise. We show segmentation results on blood vessels in magnetic resonance angiography data and bone in computed tomography data.

Abstract:   In this paper, we introduce a curve evolution approach to image magnification based on a generalization of the Mumford-Shah functional. This work is a natural extension of the curve evolution implementation of the Mumford-Shah functional presented by the authors in previous work. In particular, by considering the image magnification problem as a structured case of the missing data problem, we generalize the data fidelity term of the original Mumford-Shah energy functional by incorporating a spatially varying penalty to accommodate those pixels with missing measurements. This generalization leads us to a PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford-Shah functional which only considers simultaneous segmentation and smoothing. This novel approach for image magnification is more global and much less susceptible to blurring or blockiness artifacts as compared to other more traditional magnification techniques, and has the additional attractive denoising capability.

Abstract:   This paper is concerned with the use of the level set formalism to segment anatomical structures in 3D medical images (ultrasound or magnetic resonance images.). A closed 3D surface propagates towards the desired boundaries through the iterative evolution of a 4D implicit function. The major contribution of this work is the design of a robust evolution model based on adaptive parameters depending on the data. First the iteration step and the external propagation force, both usually constant, are automatically computed at each iteration. Additionally, region-based information rather than the gradient is used, via an estimation of intensity probability density functions over the image. As a result, the method can be applied to various kinds of data. Quantitative and qualitative results on brain MR images and 3D echographies of carotid arteries are discussed.

Abstract:   We earlier introduced an approach to categorical shape description based on the singularities (shocks) of curve evolution equations. We now consider the simplest compositions of shocks, and show that they lead to three classes of parametrically ordered shape sequences, organized along the sides of a shape triangle. By conducting several psychophysical experiments we demonstrate that shock-based descriptions are predictive of performance in shape perception. Most significantly, the experiments reveal a fundamental difference between perceptual effects dominated by when shocks form with respect to one another, versus those dominated by where they form. The shock-based theory provides a foundation for unifying tasks as diverse as shape bisection, recognition, and categorization. (C) 2001 Elsevier Science Ltd. All rights reserved.

Abstract:   A new anisotropic nonlinear diffusion model incorporating time- delay regularization into curvature-based diffusion is proposed for image restoration and edge detection. A detailed mathematical analysis of the proposed model in the form of the proof of existence, uniqueness and stability of the "viscosity" solution of the model is presented. Furthermore, implementation issues and computational methods for the proposed model are also discussed in detail. The results obtained from testing our denoising and edge detection algorithm on several synthetic and real images showed the effectiveness of the proposed model in prserving sharp edges and fine structures while removing noise.

Abstract:   The PSC (Propagation of Surfaces under Curvature) algorithm is adapted to the simulation of a flame propagation in a premixed medium including the effect of volume expansion across the flame front due to exothermicity. The algorithm is further developed to incorporate the flame anchoring scheme. This methodology is successfully applied to numerically simulate the response of an anchored V-flame to two strong free stream vortices, in accord with experimental observations of a passage of Karman vortex street through a flame. The simulation predicts flame cusping when a strong vortex pair interacts with flame front. In other words, this algorithm handles merging and breaking of the flame front and provides an accurate calculation of the flame curvature which is needed for flame propagation computation and estimation of curvature-dependent flame speeds.

Abstract:   In this paper, we develop new methods for de-noising and edge detection in images by the solution of nonlinear diffusion partial differential equations. Many previous methods in this area obtain a de-noising u of the noisy image I as the solution of an equation of the form partial derivative (t)u = L(g(\del upsilon\), delu, u - I), when g controls the speed of the diffusion and defines the edge map. The usual choice for g(s) is (1 + ks(2))(-1) and the function upsilon is always some smoothing of u. Previous choices include upsilon = u, upsilon = G(sigma) * u, and upsilon = G sigma * I. Numerical results indicate that the choice of upsilon plays a very important role in the quality of the images obtained. Notice that all these choices involve an isotropic smoothing of u, which sometimes fails to presence important corners and junctions, and this may also fail to resolve small features which are closely grouped together. This paper obtains u as the solution of a nonlinear diffusion equation which depends on u. The equation can be obtained as the energy descent equation for the total variation of upsilon penalized by the mean squared error between u and upsilon. The parameters in this energy descent equation are regarded as functions of time rather than constants, to allow for a reduction in the amount of smoothing as time progresses. Numerical tests indicate that our new method is faster and able to resolve small details and junctions better than standard methods. (C) 2001 Academic Press.

Abstract:   A new minimal path active contour model for boundary extraction is presented. Implementing the new approach requires four steps 1) users place some initial end points on or near the desired boundary through an interactive interface; 2) potential searching window is defined between two end points; 3) graph search method based on conic curves is used to search the boundary; 4) "wriggling" procedure is used to calibrate the contour and reduce sensitivity of the search results on the selected initial end points. The last three steps are performed automatically. In the proposed approach, the potential window systematically provides a new node connection for the later graph search, which is different from the row-by-row and column-by-column methods used in the classical graph search. Furthermore, this graph search also suggests ways to design a "wriggling" procedure to evolve the contour in the direction nearly perpendicular to itself by creating a list of displacement vectors in the potential window. The proposed minimal path active contour model speeds up the search and reduces the "metrication error" frequently encountered in the classical graph search methods e,g,, the dynamic programming minimal path (DPMP) method.

Abstract:   A direct numerical simulation has been carried out in order to clarify the effects of the high viscosity and the interfacial tension of a droplet on the interaction between the droplet and near-wall turbulence. A liquid turbulent plane Couette flow with an immiscible droplet of the same fluid density as that of the: continuous-phase has been used. The diameter of the droplet is fixed at one-fourth of the wall distance. which is nearly equal to 41 wall units. The droplet has been assigned in the range of 20-60 wall units from one moving wall initially. The modified volume of fluid (VOF) algorithm and local grid refinement are used for tracking the phase interface. The velocities for the fine grid are decided so that the equation of continuity is satisfied in the fine cell. It is found that the deformation of the droplet due to the surrounding fluid Row is suppressed by the effect of the interfacial tension of the droplet. The streamwise vortex is attenuated by the existence of the droplet with the interfacial tension. The small vortex is generated in the wake region of the droplet. The Reynolds- shear stress product becomes higher in a wide region around the droplet. (C) 2001 Elsevier Science Inc. All rights reserved.

Abstract:   A propagating interface can develop corners and discontinuities as it advances. Level set algorithms have been extensively applied for the problems in which the solution has advancing fronts. One of the most popular level set algorithms is the so- called fast marching method (FMM), which requires total O(N log(2)N) operations, where N is the number of grid points. The article is concerned with the development of an O(N) level set algorithm called the group marching method (GMM). The new method is based on the narrow band approach as in the FMM. However, it is incorporating a correction-by-iteration strategy to advance a group of grid points at a time, rather than sorting the solution in the narrow band to march forward a single grid point. After selecting a group of grid points appropriately, the GMM advances the group in tw iterations for the cost of slightly larger than one iteration. Numerical results are presented to show the efficiency of the method, applied to the eikonal equation in tw and three dimensions.

Abstract:   In the focusing problem, we study solutions to the porous medium equation partial derivative (t)u = Delta (u(m)) whose initial distributions are positive in the exterior of a compact 2D region and zero inside. We assume that the initial interface is elongated (i.e., has an aspect ratio > I), and possesses reflectional symmetry with respect to both the x- and y-axes. We implement a numerical scheme that adapts the numerical grid around the interface so as to maintain a high resolution as the interface shrinks to a point. We find that as t tends to the focusing time T, the interface becomes oval-like with lengths of the major and minor axes O(rootT - t) and O(T - t), respectively. Thus the aspect ratio is O(1/rootT - t). By scaling and formal asymptotic arguments we derive an approximate solution which is valid for all m. This approximation indicates that the numerically observed power law behavior for the major and minor axes is universal for all m > 1. (C) 2001 Elsevier Science B.V. All rights reserved.

Abstract:   This paper presents a new approach to 3D shape metamorphosis. We express the interpolation of two shapes as a process where one shape deforms to maximize its similarity with another shape. The process incrementally optimizes an objective function while deforming an implicit surface model. We represent the deformable surface as a level set (iso-surface) of a densely sampled scalar function of three dimensions. Such level-set models have been shown to mimic conventional parametric deformable surface models by encoding surface movements as changes in the grayscale values of a volume data set. Thus, a well-founded mathematical structure leads to a set of procedures that describes how voxel values can be manipulated to create deformations that are represented as a sequence of volumes. The result is a 3D morphing method that offers several advantages over previous methods, including minimal need for user input, no model parameterization, flexible topology, and subvoxel accuracy.

Abstract:   We present a review of the constrained interpolation profile (CIP) method that is known as a general numerical solver for solid, liquid, gas, and plasmas. This method is a kind of semi- Lagrangian scheme and has been extended to treat incompressible flow in the framework of compressible fluid. Since it uses primitive Euler representation, it is suitable for multiphase analysis. The recent version of this method guarantees the exact mass conservation even in the framework of a semi- Lagrangian scheme. We provide a comprehensive review of the strategy of the CIP method, which has a compact support and subcell resolution, including a front-capturing algorithm with functional transformation, a pressure-based algorithm, and other miscellaneous physics such as the elastic-plastic effect an;l surface tension. Some practical applications are also reviewed, such as milk crown or coronet, laser-induced melting, and turbulent mixing layer of liquid-gas interface. (C) 2001 Academic Press.

Abstract:   A variety of numerical techniques are available for tracking moving interfaces. In this review, we concentrate on techniques that result from the link between the partial differential equations that describe moving interfaces and numerical schemes designed for approximating the solutions: to hyperbolic conservation laws. This link gives rise to computational techniques for tracking moving interfaces in two and three space dimensions under complex speed laws. We discuss the evolution of these techniques, the fundamental numerical approximations, involved. implementation details, and applications. Tn particular, we review some work on three aspects of materials sciences: semiconductor process simulations. seismic processing, and optimal structural topology design. (C) 2001 Academic Press.

Abstract:   The level set method was devised by S. Osher and J. A. Sethian (1988, J. Comput, Phys. 79, 12-49) as a simple and versatile method for computing and analyzing the motion of an interface Gamma in two or three dimensions, Gamma bounds a (possibly multiply connected) region Omega. The goal is to compute and analyze the subsequent motion of Gamma under a velocity field v. This velocity can depend on position, time. the geometry of the interface, and the external physics. The interface is captured for later time as the zero level set of a smooth (at least Lipschitz continuous) function phi (x. t); i.e., Gamma (t) = {x \ phi (x, t) = 0}. phi is positive inside Omega, negative outside Omega. and is zero on Gamma (t). Topological merging and breaking are well defined and easily performed. In this review article we discuss recent variants and extensions, including the motion of curves in three dimensions, the dynamic surface extension method. fast methods for steady state problems, diffusion generated motion, and the variational level set approach. We also give a user's guide to the level set dictionary and technology and couple the method to a wide variety of problems involving external physics, such as compressible and incompressible (possibly reacting) flow, Stefan problems. kinetic crystal growth, epitaxial growth of thin films, vortex-dominated flows, and extensions to multiphase motion, We conclude with a discussion of applications to computer vision and image processing. (C) 2001 Academic Press.

Abstract:   The simulation of feature profile evolution in high-density plasma-etching processes has been carried out using a level-set technique. The main feature of this work is the inclusion of sheath dynamics, angular distribution of ions and reemission of neutrals in the trench, etch kinetics, and a level set equation for tracking a moving front of the feature profile. Sheath dynamics showed that the damped potential was somewhat shifted to the right and smaller than the applied potential. Etch profile simulations were performed for etching of silicon in inductively coupled plasmas of Cl-2 and CF4 under various conditions. In dry etching of Si with CF4 discharges, polymer deposition was dominant at p(CFx) > 10mTorr, while surface fluorination (or ion-enhanced etching) was a main mechanism at p(CFx) < 10 mTorr. The predicted etch profiles showed a slight bowing on the sidewalls and substantial tapering near the bottom, depending on the plasma parameters. (C) 2001 American Vacuum Society.

Abstract:   A level set approach for computing solutions to inviscid compressible flow with moving solid surface is presented. The solid surface is considered to be sharp and is described as the zero level set of a smooth explicit function of space and time. The finite volume TVD-MacCormack's two-step procedure is used. The boundary conditions on the solid surface are easily implemented by defining the smooth level set function. The present treatment of the level set method allows the handling of fluid flows in the presence of irregularly shaped solid boundaries, escaping from the bookkeeping complexity in the so- called 'surface-tracking' method. Using the proposed numerical techniques, a two-dimensional numerical simulation is made to investigate the aerodynamic phenomena induced by two high-speed trains passing by each other in a tunnel. Copyright (C) 2001 John Wiley & Sons, Ltd.

Abstract:   This paper presents a new variational method for the segmentation of a moving object against a still background, over a sequence of [two-dimensional or three-dimensional (3-D)] image frames. The method is illustrated in application to myocardial gated single photon emission computed tomography (SPECT) data, and incorporates a level set framework to handle topological changes while providing closed boundaries. The key innovation is the introduction of a geometrical constraint into the derivation of the Euler-Lagrange equations, such that the segmentation of each individual frame can be interpreted as a closed boundary of an object (an isolevel of a set of hyper- surfaces) while integrating information over the entire sequence. This results in the definition of an evolution velocity normal to the object boundary. Applying this method to 3-D myocardial gated SPECT sequences, the left ventricle endocardial and epicardial limits can be computed in each frame. This space-time segmentation method was tested on simulated and clinical 3-D myocardial gated SPECT sequences and the corresponding ejection fractions were computed.

Abstract:   An algorithm which couples the level set method (LSM)with the extended finite element method (X-FEM) to model crack growth is described. The level set method is used to represent the crack location, including the location of crack tips. The extended finite element method is used to compute the stress and displacement fields necessary for determining the rate of crack growth. This combined method requires no remeshing as the crack progresses, making the algorithm very efficient. The combination of these methods has a tremendous potential for a wide range of applications. Numerical examples are presented to demonstrate the accuracy of the combined methods. Copyright (C) 2001 John Wiley & Sons, Ltd.

Abstract:   We present a new numerical scheme for planar curve evolution with a normal velocity equal to F (k), where k is the curvature and F is a nondecreasing function such that F (0) = 0 and either x bar right arrow F (x(3)) is Lipschitz with Lipschitz constant less than or equal to 1 or F (x) = x(gamma) for gamma greater than or equal to 1/3. The scheme is completely geometrical and avoids some drawbacks of finite difference schemes. In particular, no special parameterization is needed and the scheme is monotone ( that is, if a curve initially surrounds another one, then this remains true during their evolution), which guarantees numerical stability. We prove consistency and convergence of this scheme in a weak sense. Finally, we display some numerical experiments on synthetic and real data.

Abstract:   We establish a rate of convergence for a semidiscrete operator splitting method applied to Hamilton Jacobi equations with source terms. The method is based on sequentially solving a Hamilton Jacobi equation and an ordinary differential equation. The Hamilton Jacobi equation is solved exactly while the ordinary differential equation is solved exactly or by an explicit Euler method. We prove that the L-infinity error associated with the operator splitting method is bounded by O(Deltat), where Deltat is the splitting (or time) step. This error bound is an improvement over the existing O (root Deltat) bound due to Souganidis [Nonlinear Anal., 9 (1985), pp. 217- 257]. In the one-dimensional case, we present a fully discrete splitting method based on an unconditionally stable front tracking method for homogeneous Hamilton Jacobi equations. It is proved that this fully discrete splitting method possesses a linear convergence rate. Moreover, numerical results are presented to illustrate the theoretical convergence results.

Abstract:   The fast marching method published by Sethian [ Proc. Natl. Acad. Sci. USA, 93 ( 1996), pp. 1591-1595] is an optimally efficient algorithm for solving problems of front volution where the front speed is monotonic. It has been used in a wide variety of applications such as robotic path planning [R. Kimmel and J. Sethian, Fast Marching Methods for Computing Distance Maps and Shortest Paths, Tech. Report 669, CPAM, University of California, Berkeley, 1996], crack propagation [M. Stolarska et al., Internat. J. Numer. Methods Engrg., 51 ( 2001), pp. 943-960; N. Sukumar, D. L. Chopp, and B. Moran, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, J. Comput. Phys., submitted], seismology [ J. Sethian and A. Popovici, Geophysics, 64 (1999), pp. 516-523], photolithography [ J. Sethian, Fast marching level set methods for three-dimensional photolithography development, in Proceedings of the SPIE 1996 International Symposium on Microlithography, Santa Clara, CA, 1996], and medical imaging [ R. Malladi and J. Sethian, Proc. Natl. Acad. Sci. USA, 93 ( 1996), pp. 9389-9392]. It has also been a valuable tool for the implementation of modern level set methods where it is used to efficiently compute the distance to the front and/or an extended velocity function. In this paper, we improve upon the second order fast marching method of Sethian [SIAM Rev., 41 ( 1999), pp. 199-235] by constructing a second order approximation of the interface generated from local data on the mesh. The data is interpolated on a single box of the mesh using a bicubic approximation. The distance to the front is then calculated by using a variant of Newtons method to solve both the level curve equation and the orthogonality condition for the nearest point to a given node. The result is a second order approximation of the distance to the interface which can then be used to produce second order accurate initial conditions for the fast marching method and a third order fast marching method.

Abstract:   A numerical investigation of grain-boundary grooving by means of a level set method is carried out. An idealized polycrystalline interconnect which consists of grains separated by parallel grain boundaries aligned normal to the average orientation of the surface is considered. Initially. the surface diffusion is the only physical mechanism assumed. The surface diffusion is driven by surface-curvature gradients. while a tired surface slope and zero atomic flux are assumed at the groove root. The corresponding mathematical system is an initial boundary value problem For a two-dimensional equation of Hamilton-Jacobi type. The results obtained are in good agreement with both Mullins analytical "small-slope" solution of the linearized problem (W. W. Mullins. 1957. J. Appl. Phys. 28. 333) (for the case of an isolated grain boundary) and with the solution for a periodic array of grain boundaries (S. A. Hackney, 1988. Scripta Metall. 22. 1731). Incorporation of an electric field changes the problem to one of electromigration. Preliminary results of electromigration drift velocity simulations in copper lines art: presented and discussed. (C) 2001 Academic Press.

Abstract:   The level set method was originally designed for problems dealing with codimension one objects. where it has been extremely succesful. especially when topological changes in the interface, i.e., merging and breaking, occur. Attempts have been made to modify it to handle objects of higher codimension, such as vortex filaments, while preserving the merging and breaking property. We present numerical simulations of a level set based method for moving curves in R-3. the model problem for higher codimension, that allows for topological changes. A vector valued level set function is used with the zero level set representing the curve. Our results show that this method can handle many types of curves moving under all types of geometrically based flows while automatically enforcing merging and breaking. (C) 2001 Academic Press.

Abstract:   In this work, we first address the problem of simultaneous image segmentation and smoothing by approaching the Mumford- Shah paradigm from a curve evolution perspective. In particular, we let a set of deformable contours define the boundaries between regions in an image where we model the data via piecewise smooth functions and employ a gradient flow to evolve these contours. Each gradient step involves solving an optimal estimation problem for the data within each region, connecting curve evolution and the Mumford-Shah functional with the theory of boundary-value stochastic processes. The resulting active contour model offers a tractable implementation of the original Mumford-Shah model (i.e., without resorting to elliptic approximations which have traditionally been favored for greater ease in implementation) to simultaneously segment and smoothly reconstruct the data within a given image in a coupled manner. Various implementations of this algorithm are introduced to increase its speed of convergence. We also outline a hierarchical implementation of this algorithm to handle important image features such as triple points and other multiple junctions. Next, by generalizing the data fidelity term of the original Mumford-Shah functional to incorporate a spatially varying penalty, we extend our method to problems in which data quality varies across the image and to images in which sets of pixel measurements are missing. This more general model leads us to a novel PDE-based approach for simultaneous image magnification, segmentation, and smoothing, thereby extending the traditional applications of the Mumford-Shah functional which only considers simultaneous segmentation and smoothing.

Abstract:   In this paper, we address two problems crucial to motion analysis: the detection of moving objects and their localisation. Statistical and level set approaches are adopted in formulating these problems. For the change detection problem, the inter-frame difference is modelled by a mixture of two zero-mean Laplacian distributions. At first, statistical tests using criteria with negligible error probability are used for labelling as changed or unchanged as many sites as possible. All the connected components of the labelled sites are used thereafter as region seeds, which give the initial level sets for which velocity fields for label propagation are provided, We introduce a new multi-label fast marching algorithm for expanding competitive regions. The solution of the localisation problem is based on the map of changed pixels previously extracted. The boundary of the moving object is determined by a level set algorithm, which is initialised by two curves evolving in converging opposite directions. The sites of curve contact determine the position of the object boundary. Experimental results using real video sequences are presented, illustrating the efficiency of the proposed approach. (C) 2001 Elsevier Science B.V. All rights reserved.

Abstract:   The morphological evolution of an epitaxial film results from atomistic processes such as adatom motion and step-adatom interactions asserting their influence over the macroscopic length and time scales of the growth front. Modelling epitaxial phenomena thus necessitates making a compromise between the detailed information provided by first-principles methods and the computational flexibility afforded by methods such as Monte-Carlo simulations and continuum equations of motion, in which atomistic processes are replaced by coarse-grained effective kinetics. We will review the various approaches that are available for modelling epitaxial phenomena using the (001) surfaces of III-V compound semiconductors as a case study, with a view to making direct comparisons with experimental measurements and to establishing a methodology that is capable of incorporating all pertinent length and time scales. Copyright (C) 2001 John Wiley & Sons, Ltd.

Abstract:   Many problems in engineering design involve optimizing the geometry to maximize a certain design objective. Geometrical constraints are often imposed. In this paper, we use the level set method devised in (Osher and Sethian, J. Comput. Phys. 79. 12 ( 1988)), the variational level set calculus presented in (Zhao et al.. J. Comput. Phys. 127, 179 (1996)), and the projected gradient method. as in (Rudin et al.. Physica D. 60. 259 (1992)), to construct a simple numerical approach for problems of this type. We apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry. Our numerical results are quite promising. We expect to use this approach to deal with a wide class of optimal design problems in the future. (C) 2001 Academic Press.

Abstract:   Many problems in engineering design involve optimizing the geometry to maximize a certain design objective. Geometrical constraints are often imposed. In this paper, we use the level set method devised in (Osher and Sethian, J. Comput. Phys. 79. 12 ( 1988)), the variational level set calculus presented in (Zhao et al.. J. Comput. Phys. 127, 179 (1996)), and the projected gradient method. as in (Rudin et al.. Physica D. 60. 259 (1992)), to construct a simple numerical approach for problems of this type. We apply this technique to a model problem involving a vibrating system whose resonant frequency or whose spectral gap is to be optimized subject to constraints on geometry. Our numerical results are quite promising. We expect to use this approach to deal with a wide class of optimal design problems in the future. (C) 2001 Academic Press.

Abstract:   Topography simulations are widely used in the microelectronics industry to study the evolution of surface profiles during such processes as deposition or etching. Comparisons between simulations and experiments are used to test proposed transport and chemistry models. The method used to move the surface (the moving algorithm) should not interfere with this testing process; i.e., it should not introduce artifacts. The reference method, shown to be accurate by several groups in many studies, is conservation law based "front tracking." Level set approaches are being increasingly used, largely for their robustness to topological changes. They have not been tested against front tracking to determine their accuracy. In this article, we present guidelines on the use of level set methods for two-dimensional surface evolutions as commonly used. Specifically, we deal with two major issues with level set algorithms: the need for "extension velocities" and the rounding of sharp corners due to contouring. We also deal with a specific approach to velocity extension that is called "fast marching." Although all methods discussed can provide the same results within the limit of small enough time steps, we demonstrate that our proposed "Riemann based" extension velocities can improve overall simulation efficiency by approximately a factor of 2, depending upon the complexity of the process being simulated. We also show that as the grid size used in the level set method decreases, extracted surface profiles can approach those calculated by front tracking, and hence to experiments. (C) 2001 American Vacuum Society.

Abstract:   The growth of crystalline silicon from the amorphous phase in the presence of an applied stress is modelled using advanced numerical methods. The crystal region is modelled as a linear elastic solid and the amorphous as a viscous fluid with a time- dependent viscosity to reflect structural relaxation. Appropriate coupling conditions across the boundary are defined, and both problems are solved using a symmetric- Galerkin boundary integral method. The interface is advanced in time using the level set technique. The results match well with experiments and support the proposed kinetic mechanism for the observed interface growth instability.

Abstract:   Continuum modeling of many physical systems typically assumes that the spatial extent of an atom is small compared to the quantities of interest and can therefore be neglected. We show that this is valid only asymptotically. For many applications of practical interest, the spatial extent of a discrete atom cannot be neglected. We have developed a model for the description of epitaxial growth based on the levelset method, and find that we can accurately predict quantities such as the island densities, if we implement boundary conditions in a region with atomic width, rather than just on a line without any spatial extent. Only in the limit of very large islands and island spacings can this be neglected.

Abstract:   In this study, a computational optimal system for the generation of curves on triangulated surfaces representing 3D brains is described. The algorithm is based on optimally computing geodesics on the triangulated surfaces following; Kimmel and Sethian ([1998]: Proc Natl Acad Sci 95:15). The system can be used to compute geodesic curves for accurate distance measurements as well as to detect sulci and gyri. These curves are defined based on local surface curvatures that are computed following a novel approach presented in this study. The corresponding software is available to the research community. Hum. Brain Mapping 14:1-15, 2001. (C) 2001 Wiley- Liss, Inc.

Abstract:   A numerical method has been developed for direct simulation of bubble dynamics with large liquid-to-vapor density ratio and phase change. The numerical techniques are based on a fixed- grid, finite volume method capable of treating the interface as a sharp discontinuity. The unsteady, axisymmetric Navier-Stokes equations and energy equation in both liquid and vapor phases are computed. The mass, momentum, and energy conditions are explicitly matched at the phase boundary to determine the interface shape and movement. The cubic B-spline is used in conjunction with a fairing algorithm to yield smooth and accurate information of curvatures. Nondimensional parameters including Reynolds, Weber, and Jakob numbers are varied to offer insight into the physical and numerical characteristics of the bubble dynamics. Based on the present sharp interface approach, bubble dynamics for density ratio of 1600 or higher, with and without phase change, can be successfully computed. (C) 2001 Elsevier Science.

Abstract:   Peskin's Immersed Boundary Method has been widely used for simulating many fluid mechanics and biology problems. One of the essential components of the method is the usage of certain discrete delta functions to deal with singular forces along one or several interfaces in the fluid domain, However, the Immersed Boundary Method is known to be first-order accurate and usually smears out the solutions. In this paper, we propose an immersed interface method for the incompressible Navier- Stokes equations with singular forces along one or several interfaces in the solution domain. The new method is based on a second-order projection method with modifications only at grid points near or on the interface. From the derivation of the new method, we expect fully second-order accuracy for the velocity and nearly second-order accuracy for the pressure in the maximum norm including those grid points near or on the interface. This has been confirmed in our numerical experiments. Furthermore, the computed solutions are sharp across the interface. Nontrivial numerical results are provided and compared with the Immersed Boundary Method. Meanwhile, a new version of the Immersed Boundary Method using the level set representation of the interface is also proposed in this paper. (C) 2001 Academic Press.

Abstract:   The volume of fluid (VOF) method, which uses an interface tracking algorithm for the simulation of the two-phase flow, is coupled with the "two-fluid" model, which is based on time and space averaged equations and cannot track the interface explicitly. The idea of the present work is to use the VOF method in the parts of the computational domain where the grid density allows surface tracking, In the pam of the domain where the flow is too dispersed to be described by the interface tracking algorithms, the two-fluid model is used, The equations of the two-fluid model are less accurate than the VOF model due to the empirical closures required in the averaged equations. However, in the case of the sufficiently dispersed flow, the two-fluid model results are still much closer to the real world than the results of the VOF method, which do not have any physical meaning when the grid becomes too coarse. Each model in the present work uses a separate set of equations suitable for description of two-dimensional, incompressible, viscous two-phase flow. Similar discretization techniques are used for both sets of equations and solved with the same numerical method. Coupling of both models is achieved via the volume fraction of one of the fluids, which is used in both models. A special criterion for the transition between the models is derived from the interface reconstruction function in the VOF method. An idealized vortical flow and the Rayleigh-Taylor instability are used as tests of the coupling. In both cases the time development causes mixing of the fluids and dispersion of the interface that is beyond the capabilities of the model based on the VOF method. Therefore the two-fluid model gradually replaces the inter-face tracking model. In the final stages of the Rayleigh-Taylor instability, when both fluids are approaching their final positions and the tractable interface appears again, the two-fluid model is gradually replaced by the VOF method. (C) 2001 Academic Press.

Abstract:   A novel framework for solving variational problems and partial differential equations for scalar and vector-valued data defined on surfaces is introduced in this paper. The key idea is to implicitly represent the surface as the level set of a higher dimensional function and to solve the surface equations in a fixed Cartesian coordinate system using this new embedding function. The equations are then both intrinsic to the surface and defined in the embedding space. This approach thereby eliminates the need for performing complicated and inaccurate computations on triangulated surfaces, as is commonly done in the literature. We describe the framework and present examples in computer graphics and image processing applications, including texture synthesis, flow field visualization, and image and vector field intrinsic regularization for data defined on 3D surfaces. (C) 2001 Elsevier Science.

Abstract:   In this paper, we propose a new model for active contours to detect objects in a given image, based on techniques of curve evolution, Mumford-Shah functional for segmentation and level sets. Our model can detect objects whose boundaries are not necessarily defined by gradient. We minimize an energy which can he seen as a particular case of the minimal partition problem, In the level set formulation, the problem becomes a "mean-curvature flow"-like evolving the active contour, which will stop on the desired boundary. However, the stopping term does not depend on the gradient of the. image, as in the classical active contour models, hut is instead related to a particular segmentation of the image. We will give a numerical algorithm using finite differences. Finally, we will present various experimental results and in particular some examples for which the classical snakes methods based on the gradient are not applicable. Also, the initial curve can be anywhere in the image, and interior contours are automatically detected.

Abstract:   A high-energy-density laser beam-material interaction process has been simulated considering a self-evolving liquid-vapour interface profile. A mathematical scheme called the level-set technique has been adopted to capture the transient Liquid- vapour interface. Inherent to this technique are: the ability to simulate merger and splitting of the liquid-vapour interface and the simultaneous updating of the surface normal and the curvature. Unsteady heat transfer and fluid flow phenomena are modelled, considering the thermo-capillary effect and the recoil pressure. A kinetic Knudsen layer has been considered to simulate evaporation phenomena at the liquid-vapour interface. Also, the homogeneous boiling phenomenon near the critical point is implemented. Energy distribution inside the vapour cavity is computed considering multiple reflection phenomena. The effect of laser power on the material removal mode, liquid layer thickness, surface temperature and the evaporation speed are presented and discussed.

Abstract:   An unsteady Navier-Stokes solver for incompressible fluid is coupled with a level set approach to describe free surface motions. The two-phase now of air and water is approximated by the flow of a single fluid whose properties, such as density and viscosity, change across the interface. The free surface location is captured as the zero level of a distance function convected by the flow field. To validate the numerical procedure, two classical two-dimensional free surface problems in hydrodynamics, namely the oscillating flow in a tank and the waves generated by the flow over a bottom bump, are studied in non-breaking conditions, and the results are compared with those obtained with other numerical approaches. To check the capability of the method in dealing with complex free surface configurations, the breaking regime produced by the flow over a high bump is analyzed. The analysis covers the successive stages of the breaking phenomenon: the steep wave evolution, the falling jet, the splash-up and the air entrainment. In all phases, numerical results qualitatively agree with the experimental observations. Finally, to investigate a flow in which viscous effects are relevant, the numerical scheme is applied to study the wavy flow past a submerged hydrofoil. Copyright (C) 2001 John Wiley & Sons, Ltd.

Abstract:   This paper presents a method to solve two-phase flows using the finite element method. On one hand, the algorithm used to solve the Navier-Stokes equations provides the neccessary stabilization for using the efficient and accurate three-node triangles for both the velocity and pressure fields. On the other hand, the interface position is described by the zero- level set of an indicator function. To maintain accuracy, even for large-density ratios, the pseudoconcentration function is corrected at the end of each time step using an algorithm successfully used in the finite difference context. Coupling of both problems is solved in a staggered way. As demonstrated by the solution of a number of numerical tests, the procedure allows dealing with problems involving two interacting fluids with a large-density ratio. Copyright (C) 2001 John Wiley & Sons, Ltd.

Abstract:   A kinetic method for hyperbolic-elliptic equations is presented in this paper. In the mixed type system, the coexistence of liquid and gas and the phase transition between them are described by the van der Waals-type equation of state (EOS). Because the fluid is unstable in the elliptic region, the interface between liquid and gas can be captured naturally through condensation and evaporation processes. which continuously remove any "averaged" numerical fluid away from the elliptic region at the interfaces. As a result, a sharp liquid-gas interface can be constructed from the competition between the numerical diffusion and phase transition. The numerical examples presented in this paper include both phase transition and multifluid interface problems. (C) 2001 Academic Press.

Abstract:   Three-dimensional (3-D) imaging of the heart is a rapidly del eloping area of research in medical imaging, Advances in hardware and methods for fast spatio-temporal cardiac imaging are extending the frontiers of clinical diagnosis and research on cardiovascular diseases. In the last few Sears, many approaches hare been proposed to analyze images and extract parameters of cardiac shape and function from a variety of cardiac imaging modalities. In particular, techniques based on spatio-temporal geometric models have received considerable attention. This paper surveys the literature of tno decades of research on cardiac modeling. The contribution of the paper is three-fold: 1) to serve as a tutorial of the field for both clinicians and technologists, 2) to provide an extensive account of modeling techniques in a comprehensive and systematic manner, and 3) to critically review these approaches in terms of their performance and degree of clinical evaluation with respect to the final goal of cardiac functional analysis, From this review it is concluded that whereas 3-D model-based approaches have the capability. to improve the diagnostic value of cardiac images, issues as robustness, 3-D interaction, computational complexity and clinical validation still require significant attention.

Abstract:   We propose a numerical method for modeling two-phase flow consisting of separate compressible and incompressible regions. This is of interest, for example, when the combustion of fuel droplets or the shock-induced mixing of liquids is numerically modeled. We use the level set method to track the interface between the compressible and incompressible regions, as well as the Ghost Fluid Method (GFM) to create accurate discretizations across the interface. The GFM is particularly effective here since the equations differ in both number and type across the interface. The numerical method is presented in two spatial dimensions with numerical examples in both one and two spatial dimensions, while three-dimensional extensions are straightforward. (C) 2001 Academic Press.

Abstract:   We present a level set based numerical algorithm for simulating a model of epitaxial growth. The island dynamics model is a continuum model for the growth of thin films. In this paper, we emphasize the details of the numerical method used to simulate the island dynamics model. (C) 2001 Academic Press.

Abstract:   A level-set algorithm for tracking discontinuities in hyperbolic conservation laws is presented. The algorithm uses a simple finite difference approach, analogous to the method of lines scheme presented in C.-W. Shu and S. Osher (1988, J. Comput. Phys. 77, 439). The zero of a level-set function is used to specify the location of the discontinuity. Since a level-set function is used to describe the front location, no extra data structures are needed to keep track of the location of the discontinuity. Also, two solution states are used at all computational nodes, one corresponding to the ''real" state, and one corresponding to a "ghost node" state, analogous to the "Ghost Fluid Method" of R. P. Fedkiw et al. (1999, J. Comput. phys. 154, 459). High-order pointwise convergence is demonstrated for linear and nonlinear conservation laws, even at discontinuities and in multiple dimensions. The solutions are compared to standard high-order shock-capturing schemes. This paper focuses on scalar conservation laws. An example is given for shock tracking in the one-dimensional Euler equations. Level-set tracking for systems of conservation laws in multidimensions will be presented in future work.

Abstract:   A technique to simulate the flow field near a moving material interface is developed for multi-material compressible flow, in particular, for compressible gas-water flow. This technique can be conveniently applied with a well-established conservative scheme to solve for the regions away from the interface. Material interfaces are captured using the level set technique with minimum or no smearing. To treat wave interaction with the interface, an implicit characteristic method is developed. In this paper, the method is described in detail and tested extensively for several one-dimensional gas-gas and gas-water cases. Application to multi-dimensional shock-free surface interaction and shock-gas bubble interaction are presented in Part II [Liu TG, Khoo BC, Yeo KS. The simulation of compressible multi-medium flow. Part II: Applications to 2D underwater shuck refraction. Comp. and Fluids 2000;30:315-37]. (C) 2001 Elsevier Science Ltd. All rights reserved.

Abstract:   A technique to simulate the flow field near a moving material interface is developed for multi-material compressible flow, in particular, for compressible gas-water flow. This technique can be conveniently applied with a well-established conservative scheme to solve for the regions away from the interface. Material interfaces are captured using the level set technique with minimum or no smearing. To treat wave interaction with the interface, an implicit characteristic method is developed. In this paper, the method is described in detail and tested extensively for several one-dimensional gas-gas and gas-water cases. Application to multi-dimensional shock-free surface interaction and shock-gas bubble interaction are presented in Part II [Liu TG, Khoo BC, Yeo KS. The simulation of compressible multi-medium flow. Part II: Applications to 2D underwater shuck refraction. Comp. and Fluids 2000;30:315-37]. (C) 2001 Elsevier Science Ltd. All rights reserved.

Abstract:   An Eulerian numerical scheme for the direct simulation of bubble dynamics in an acoustic field is proposed. The compressible Navier-Stokes equation with a surface tension term is used as a governing equation. The convection terms in the equation are solved using the hybrid interpolation- extrapolation scheme which stably provides a nonsmoothed solution of density interface between bubbles and a surrounding material. The acoustic terms in the equation are solved by the generalized Crank-Nicholson method which is an implicit method and has a temporally second-order accuracy under the low- Courant-number condition and is applicable to high-Courant- number computation. Using this scheme, some typical results of single- and multibubble phenomena are given.

Abstract:   In this paper, we propose a new numerical method for treating two-phase incompressible flow where one phase is being converted into the other, e.g., the vaporization of liquid water. We consider this numerical method in the context of treating discontinuously thin flame fronts for incompressible flow. This method was designed as an extension of the Ghost Fluid Method (1999, J. Comput. Phys. 152, 457) and relies heavily on the boundary condition capturing technology developed in Liu et al. (2000, J. Comput. Phys. 154, 15) for the variable coefficient Poisson equation and in Kang et al. (in press J. Comput. Phys.) for multiphase incompressible flow. Our new numerical method admits a sharp interface representation similar to the method proposed in Helenbrook et al. (1999, J. Comput. Phys. 148, 366). Since the interface boundary conditions are handled in a simple and straightforward fashion, the code is very robust, e.g. no special treatment is required to treat the merging of flame fronts. The method is presented in three spatial dimensions, with numerical examples in one, two, and three spatial dimensions. (C) 2001 Academic Press.

Abstract:   This paper presents a strategy for the segmentation of brain from volumetric MR images which integrates 3D segmentation and 3D registration processes. The segmentation process is based on the level set formalism. A closed 3D surface propagates towards the desired boundaries through the iterative evolution of a 4D implicit function. In this work, the propagation relies on a robust evolution model including adaptive parameters. These depend on the input data and on statistical distribution models. The main contribution of this paper is the use of an automatic registration method to initialize the surface, as an alternative solution to manual initialization, The registration is achieved through a robust multiresolution and multigrid minimization scheme. This coupling significantly improves the quality of the method, since the segmentation is faster, more reliable and fully automatic. Quantitative and qualitative results on both synthetic and real volumetric brain MR images are presented and discussed. (C) 2001 Elsevier Science B.V. All rights reserved.

Abstract:   A phase-field formulation is introduced to simulate quantitatively microstructural pattern formation in alloys. The thin-interface limit of this formulation yields a much less stringent restriction on the choice of interface thickness than previous formulations and permits one to eliminate nonequilibrium effects at the interface. Dendrite growth simulations with vanishing solid diffusivity show that both the interface evolution and the solute profile in the solid are accurately modeled by this approach.

Abstract:   We present a powerful morphing technique based on level set methods, that can be combined with a variety of scan conversion/model processing techniques. Bringing these techniques together creates a general morphing approach that allows a user to morph a number of geometric model types in a single animation. We have developed techniques for converting several types of geometric models (polygonal meshes, CSG models and MRI scans) into distance volumes, the volumetric representation required by our level set morphing approach. The combination of these two capabilities allows a user to create a morphing sequence regardless of the model type of the source and target objects, freeing him/her to use whatever model type is appropriate for a particular animation.

Abstract:   We use an unconditionally stable numerical scheme to implement a fast version of the geodesic active contour model. The proposed scheme is useful for object segmentation in images, like tracking moving objects in a sequence of images. The method is based on the Weickert-Romeney-Viergever (additive operator splitting) AOS scheme. It is applied at small regions, motivated by Adalsteinsson-Sethian level set narrow band approach, and uses Sethian's fast marching method for re- initialization. Experimental results demonstrate the power of the new method for tracking in color movies.

Abstract:   The notion of spreadability was introduced by El Jai et al. in 1994 to describe expansion phenomena. The principle consists in exploring the evolution of sets where a given property is satisfied. An extension (the A spreadability) was given in 1998 by Bernoussi who considered the measure of such sets. In some cases, the sets can be described by their boundaries and the aim of this paper is to explore some interesting connections of spreadability with wave fronts and level sets. The results are illustrated by a variety of examples.

Abstract:   A methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed. The numerical method couples the level set method (S. Osher, J.A. Sethian, J. Comput. Phys. 79 (1) (1988) 12) to the extended finite-element method (X-FEM) (N. Moes, J. Dolbow, T. Belytschko, Int. J. Numer. Methods Engrg. 46 (1) (1999) 131). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of holes and material interfaces, and in addition, the level set function is used to develop the local enrichment for material interfaces. Numerical examples in two-dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique. (C) 2001 Elsevier Science BY. All rights reserved.

Abstract:   A methodology to model arbitrary holes and material interfaces (inclusions) without meshing the internal boundaries is proposed. The numerical method couples the level set method (S. Osher, J.A. Sethian, J. Comput. Phys. 79 (1) (1988) 12) to the extended finite-element method (X-FEM) (N. Moes, J. Dolbow, T. Belytschko, Int. J. Numer. Methods Engrg. 46 (1) (1999) 131). In the X-FEM, the finite-element approximation is enriched by additional functions through the notion of partition of unity. The level set method is used for representing the location of holes and material interfaces, and in addition, the level set function is used to develop the local enrichment for material interfaces. Numerical examples in two-dimensional linear elastostatics are presented to demonstrate the accuracy and potential of the new technique. (C) 2001 Elsevier Science BY. All rights reserved.

Abstract:   The first-arrival quasi-P wave travel-time field in an anisotropic elastic solid solves a first-order nonlinear partial differential equation, the qP eikonal equation, which is a stationary Hamilton-Jacobi equation. The solution of the paraxial quasi-P eikonal equation, an evolution Hamilton-Jacobi equation in depth. gives the first-arrival travel time along downward propagating rays. We devise nonlinear numerical algorithms to compute the paraxial Hamiltonian for quasi-P wave propagation ill general anisotropic media. A second-order essentially nonoscillatory (ENO) Runge-Kutta scheme solves this paraxial eikonal equation with a point source as an initial condition in O(N) floating point operations, where N is the number of grid points, Numerical experiments using 2-D transversely isotropic models with inclined symmetry axes demonstrate the accuracy of the algorithms. (C) 2001 Academic Press.

Abstract:   It is demonstrated that the G-equation for premixed combustion admits a diversity of symmetries properties, i.e. invariance characteristics under certain transformations. Included are those of classical mechanics such as Galilean invariance, rotation invariance and others. Also a new generalized scaling symmetry has been established. It is shown that the generalized scaling symmetry defines the physical property, of the G- equation precisely. That is to say the value of G at a given flame front is arbitrary. It is proven that beside the symmetries of classical mechanics, particularly the generalized scaling symmetry uniquely defines the basic structure of the G- equation. It is also proven that the generalized scaling symmetry precludes the application of classical Reynolds ensemble averaging usually employed in statistical turbulence theory in order to avoid non unique statistical quantities such as for the mean flame position. Finally, a new averaging scheme of the G-field is presented which is fully consistent with all symmetries of the G-equation. Equations for the mean G-field and flame brush thickness are derived and a route to consistent invariant modelling of other quantities derived from the G- field is illustrated. Examples of statistical quantities derived from the G-field both in the context of Reynolds- averaged models as well as subgrid-scale models for large-eddy simulations taken from the literature are investigated as to whether they are compatible with the important generalized scaling symmetry.

Abstract:   Segmenting individual cell nuclei from microscope images normally involves volume labelling of the nuclei with a DNA stain. However, this method often fails when the nuclei are tightly clustered in the tissue, because there is little evidence from the images on where the borders of the nuclei are. In this paper we present a method which solves this limitation and furthermore enables segmentation of whole cells. Instead of using volume stains, we used stains that specifically label the surface of nuclei or cells: lamins for the nuclear envelope and alpha-6 or beta-1 integrins for the cellular surface. The segmentation is performed by identifying unique seeds for each nucleus/cell and expanding the boundaries of the seeds until they reach the limits of the nucleus/cell, as delimited by the lamin or integrin staining, using gradient- curvature flow techniques. We tested the algorithm using computer-generated objects to evaluate its robustness against noise and applied it to cells in culture and to tissue specimens. In all the cases that we present the algorithm gave accurate results.

Abstract:   This paper is devoted to the solution of shape reconstruction problems by a level set method. The basic motivation for the setup of this level set algorithm is the well-studied method of asymptotic regularization, which has been developed for ill- posed problems in Hilbert spaces. Using analogies to this method, the convergence analysis of the proposed level set method is established and it is shown that the evolving level set converges to a solution in the symmetric difference metric as the artificial time evolves to infinity. Furthermore, the regularizing properties of the level set method are shown, if the discrepancy principle is used as a stopping rule. The numerical implementation of the level set method is discussed and applied to some examples in order to compare the numerical results with theoretical statements. The numerical results demonstrate the power of the level set method, in particular for examples where the number of connected components the solution consists of is not known a priori.

Abstract:   A model problem in electrical impedance tomography for the identification of unknown shapes from data in a narrow strip along the boundary of the domain is investigated. The representation of the shape of the boundary and its evolution during an iterative reconstruction process is achieved by the level set method. The shape derivatives of this problem involve the normal derivative of the potential along the unknown boundary. Hence an accurate resolution of its derivatives along the unknown interface is essential. It is obtained by the immersed interface method.

Abstract:   We introduce a numerical method based on the level-set technique to compute capture numbers used in mean-field rate equations that describe epitaxial growth. In our level-set approach, islands grow with a velocity that is computed from solving the diffusion equation for the adatom concentration. The capture number for each island is then calculated by integrating the growth velocity of an island around the island boundary. Thus, our method by construction includes all spatial correlations between islands. The functional form of the capture numbers a, is, to first approximation, affinely dependent on the island sizes. Integration of a completely deterministic set of mean-field rate equations for the first time properly reproduces the correct island densities and cluster size distribution.

Abstract:   This paper presents an approach for the extraction of vasculature from angiography images by using a wave propagation and traceback mechanism. We discuss both the theory and the implementation of the approach. Using a dual-sigmoidal filter, we label each pixel in an angiogram with the likelihood that it is within a vessel. Representing the reciprocal of this likelihood image as an array of refractive indexes, we propagate a digital wave through the image from the base of the vascular tree. This wave "washes" over the vasculature, ignoring local noise perturbations. The extraction of the vasculature becomes that of tracing the wave along the local normals to the waveform, While the approach is inherently single instruction stream multiple data stream (SIMD), we present an efficient sequential algorithm for the wave propagation and discuss the traceback algorithm, We demonstrate the effectiveness of our integer image neighborhood-based algorithm and its robustness to image noise.

Abstract:   In this paper we study evolution of plane curves satisfying a geometric equation v = beta (k,v), where v is the normal velocity and k and are the curvature and tangential angle of a plane curve. We follow the direct approach and we analyze the so-called intrinsic heat equation governing the motion of plane curves obeying such a geometric equation. The intrinsic heat equation is modi ed to include an appropriate nontrivial tangential velocity functional. We show how the presence of a nontrivial tangential velocity can prevent numerical solutions from forming various instabilities. From an analytical point of view we present some new results on short time existence of a regular family of evolving curves in the degenerate case when beta (k,v) = gamma (v)k(m), 0 < m 2, and the governing system of equations includes a nontrivial tangential velocity functional.

Abstract:   This paper considers the level-set equation for a general planar anisotropic curvature flow equation when the interfacial energy is very singular so that the anisotropic curvature effect is nonlocal. A new notion of solutions is introduced to establish an analytic foundation of the level-set method including a comparison principle and stability results. The main idea behind the proofs is to convert the level-sets of solutions into graph-like functions. This new procedure is called slicing and it is not limited to nonlocal curvature flow equations. Our theory is useful for establishing the convergence of a crystalline algorithm as well as for justifying the crystalline flow as a limit of anisotropic curvature flow with smooth interfacial energy.

Abstract:   In this paper, we propose significant extensions to the "snake pedal" model, a powerful geometric shape modeling scheme introduced in (Vemuri and Guo, 1998). The extension allows the model to automatically cope with topological changes and for the first time, introduces the concept of a compact global shape into geometric active models. The ability to characterize global shape of an object using very few parameters facilitates shape learning and recognition. In this new modeling scheme, object shapes are represented using a parameterized function- called the generator-which accounts for the global shape of an object and the pedal curve (surface) of this global shape with respect to a geometric snake to represent any local detail. Traditionally, pedal curves (surfaces) are defined as the loci of the feet of perpendiculars to the tangents of the generator from a fixed point called the pedal point. Local shape control is achieved by introducing a set of pedal points-lying on a snake-for each point on the generator. The model dubbed as a "snake pedal" allows for interactive manipulation via forces applied to the snake. In this work, we replace the snake by a geometric snake and derive all the necessary mathematics for evolving the geometric snake when the snake pedal is assumed to evolve as a function of its curvature. Automatic topological changes of the model may be achieved by implementing the geometric snake in a level-set framework. We demonstrate the applicability of this modeling scheme via examples of shape recovery from a variety of 2D and 3D image data.

Abstract:   An algorithm for the computationally optimal construction of intrinsic weighted distance functions on implicit hyper- surfaces is introduced in this paper. The basic idea is to approximate the intrinsic weighted distance by the Euclidean weighted distance computed in a band surrounding the implicit hyper-surface in the embedding space, thereby performing all the computations in a Cartesian grid with classical and efficient numerics. Based on work on geodesics on Riemannian manifolds with boundaries, we bound the error between the two distance functions. We show that this error is of the same order as the theoretical numerical error in computationally optimal, Hamilton-Jacobi-based, algorithms for computing distance functions in Cartesian grids. Therefore, we can use these algorithms, modified to deal with spaces with boundaries, and obtain also for the case of intrinsic distance functions on implicit hyper-surfaces a computationally efficient technique. The approach can be extended to solve a more general class of Hamilton-Jacobi equations defined on the implicit surface, following the same idea of approximating their solutions by the solutions in the embedding Euclidean space. The framework here introduced thereby allows for the computation, to be performed on a Cartesian grid with computationally optimal algorithms, in spite of the fact that the distance and Hamilton-Jacobi equations are intrinsic to the implicit hyper-surface. (C) 2001 Academic Press.

Abstract:   We present a novel mathematical, physical and logical framework for describing an input image of the dynamics of physical fields, in particular the optic field dynamics. Our framework is required to be invariant under a particular gauge group, i.e., a group or set of transformations consistent with the symmetries of that physical field dynamics enveloping renormalisation groups. It has to yield a most concise field description in terms of a complete and irreducible set of equivalences or invariants. Furthermore, it should be robust to noise, i.e., unresolvable perturbations (morphisms) of the physical field dynamics present below a specific dynamic scale, possibly not covered by the gauge group, do not affect Lyapunov or structural stability measures expressed in equivalences above that dynamic scale. The related dynamic scale symmetry encompasses then a gauge invariant similarity operator with which similarly prepared ensembles of physical field dynamics are probed and searched for partial equivalences coming about at higher scales. The framework of our dynamic scale-space paradigm is partly based on the initialisation of joint (non)local equivalences for the physical field dynamics external to, induced on and stored in a vision system and represented by an image, possibly at various scales. These equivalences are consistent with the scale-space paradigm considered and permit a faithful segmentation and interpretation of the dynamic scale-space at initial scale. Among the equivalences are differential invariants, integral invariants and topological invariants not affected by the considered gauge group. These equivalences form a quantisation of the external, induced and stored physical field dynamics, and are associated to a frame field, co-frame field, metric and/or connection invariant under the gauge group. Examples of these equivalences are the curvature and torsion two-forms of general relativity, the Burgers and Frank vector density fields of crystal theory (in both disciplines these equivalences measure the inhomogeneity of translational and (affine) rotation groups over space-time), and the winding numbers and other topological charges popping up in electromagnetism and chromodynamics. Besides based on a gauge invariant initialisation of equivalences the framework of our dynamic scale-space paradigm assumes that a robust, i.e. stable and reproducible, partially equivalent representation of the physical field dynamics is acquired by a multi-scale filtering technique adapted to those initial equivalences. Effectively, the hierarchy of nested structures of equivalences, by definition too invariant under the gauge group, is obtained by applying an exchange principle for a free energy of the physical field dynamics (represented through the equivalences) that in turn is linked to a statistical partition function. This principle is operationalised as a topological current of free energy between different regions of the physical field dynamics. It translates for each equivalence into a process governed by a system of integral and/or partial differential equations (PDES) with local and global initial-boundary conditions (IBC). The scaled physical field dynamics is concisely classified in terms of local and nonlocal equivalences, conserved densities or curvatures of the dynamic scale-space paradigm that in generally are not coinciding with all initial equivalences. Our dynamic scale-space paradigm distinguishes itself intrinsically from the standard ones that are mainly developed for scalar fields. A dynamic scale-space paradigm is also operationalised for non-scalar fields like curvature and torsion tensor fields and even more complex nonlocal and global topological fields supported by the physical field dynamics. The description of the dynamic scale- spaces are given in terms of again equivalences, and the paradigms, in terms, of symmetries, curvatures and conservation laws. The topological characteristics of the paradigm form then a representation of the logical framework. A simple example of a dynamic scale-space paradigm is presented for a time-sequence of two-dimensional satellite images in the visual spectrum. The segmentation of the sequence in fore- and background dynamics at various scales is demonstrated together with a detection of ridges, courses and inflection lines allowing a concise triangulation of the image. Furthermore, the segmentation procedure of a dynamic scale-space is made explicit allowing a true hierarchically description in terms of nested equivalences. How to unify all the existing scale-space paradigms using our frame work is illustrated. This unification comes about by a choice of gauge and renormalisation group, and setting up a suitable scale-space paradigm that might be user- defined. How to extend and to generalise the existing scale- space paradigm is elaborated on. This is illustrated by pointing out how to retain a pure topological or covariant scale-space paradigm from an initially segmented image that instead of a scalar field also can represent a density field coinciding with dislocation and disclination fields capturing the cutting and pasting procedures underlying the image formation.

Abstract:   The normalform of a space curve is introduced analogously to the normalform of a plane curve and a surface, i.e. an implicit representation h (x) = 0 with \ \ delh \ \ = 1. The normalform function h is (unlike the latter cases) not differentiable at curve points. Despite of this disadvantage the normalform is a suitable tool for designing surfaces which can be treated as common implicit surfaces. Many examples (bisector surfaces, constant distance sum/product surfaces, metamorphoses, blending surfaces, smooth approximation surfaces) demonstrate applications of the normalform to surface design.

Abstract:   In this paper, we present a fast and simple discrete approach for active contours. It is based on discrete contour evolution, which operates on the boundary of digital shape, by iterative growth processes on the boundary of the shape. We consider a curve to be the boundary of a discrete shape, We attach at each point of the boundary a cost function and deform this shape according to that cost function. The method presents some advantages. It is a discrete method, which takes an implicit representation and uses discrete algorithm with a simple data structure.

Abstract:   A Cartesian grid method for computing flows with complex immersed, moving boundaries is presented. The flow is computed on a fixed Cartesian mesh and the solid boundaries are allowed to move freely through the mesh. A mixed Eulerian-Lagrangian framework is employed, which allows us to treat the immersed moving boundary as a sharp interface. The incompressible Navier-Stokes equations are discretized using a second-order- accurate finite-volume technique, and a second-order-accurate fractional-step scheme is employed for time advancement. The fractional-step method and associated boundary conditions are formulated in a manner that property accounts for the boundary motion. A unique problem with sharp inter-face methods is the temporal discretization of what are termed "freshly cleared" cells, i.e., cells that are inside the solid at one time step and emerge into the fluid at the next time step. A simple and consistent remedy for this problem is also presented. The solution of the pressure Poisson equation is usually the most time-consuming step in a fractional step scheme and this is even more so for moving boundary problems where the flow domain changes constantly. A multigrid method is presented and is shown to accelerate the convergence significantly even in the presence of complex immersed boundaries. The methodology is validated by comparing it with experimental data on two cases: (1) the flow in a channel with a moving indentation on one wall and (2) vortex shedding from a cylinder oscillating in a uniform free-stream. Finally, the application of the current method to a more complicated moving boundary situation is also demonstrated by computing the flow inside a diaphragm-driven micropump with moving valves. (C) 2001 Elsevier Science.

Abstract:   We generalize the level set approach to model epitaxial growth to include thermal detachment of atoms from island edges. This means that islands do not always grow and island dissociation can occur. We make no assumptions about a critical nucleus. Excellent quantitative agreement is obtained with kinetic Monte Carlo simulations for island densities and island size distributions in the submonolayer regime.

Abstract:   Deformable models have raised much interest and found various applications in the fields of computer vision and medical imaging. They provide an extensible framework to reconstruct shapes. Deformable surfaces, in particular, are used to represent 3D objects. They have been used for pattern recognition [Computer Vision and Image Understanding 69(2) (1998) 201; IEEE Transactions on Pattern Analysis and Machine Intelligence 19(10) (1997) 1115], computer animation [ACM Computer Graphics (SIGGRAPH'87) 21(4) (1987) 205], geometric modelling [Computer Aided Design (CAD) 24(4) (1992) 178], simulation [Visual Computer 16(8) (2000) 437], boundary tracking [ACM Computer Graphics (SIGGRAPH'94) (1994) 185], image segmentation [Computer Integrated Surgery, Technology and Clinical Applications (1996) 59; IEEE Transactions on Medical Imaging 14 (1995) 442; Joint Conference on Computer Vision, Virtual Reality and Robotics in Medicine (CVRMed-MRCAS'97) 1205 (1997) 13; Medical Image Computing and Computer-Assisted Intervention (MICCAI'99) 1679 (1999) 176; Medical Image Analysis 1(1) (1996) 19], etc. In this paper we propose a survey on deformable surfaces. Many surface representations have been proposed to meet different 3D reconstruction problem requirements. We classify the main representations proposed in the literature and we study the influence of the representation on the model evolution behavior, revealing some similarities between different approaches. (C) 2001 Elsevier Science B.V. All rights reserved.

Abstract:   We prove the convergence of an approximation scheme for computing evolutions of sets having various normal velocities depending on the curvature. This scheme is an extension of the so-called Bence-Merriman-Osher scheme for computing mean curvature motions. In order to obtain the convergence, we make use of the new weak notion of fronts motion introduced by Barles and Souganidis [Arch. Ration. Mech. Anal., 141 (1998), pp. 237-296], which is equivalent, under the no-interior condition, to the well-known level-set evolution.