Equations (11a) and (11b) can be solved by treating u and v as functions of time and solving The steady-state solution (as ) of these linear parabolic equations is the desired solution of the Euler equations (11a) and (11b). Note that these equations are decoupled, and therefore can be solved as separate scalar partial differential equations in u and v. The equations in (12) are known as generalized diffusion equations, and are known to arise in such diverse fields as heat conduction, reactor physics, and fluid flow . For us, they have appeared from our description of desirable properties of external fields for active contours. Diffusion is a natural outcome given the desired ``filling in'' property.
For convenience, we rewrite Equation (12) as follows
To set up the iterative solution, let the indices i, j, and
n correspond to x, y, and t, respectively, and let the spacing
between pixels be
and the time step for each
iteration be .
Then the required partial derivatives can be
Convergence of the above iterative process is guaranteed by a standard
result in the theory of numerical methods (cf. ). Provided
that b, c1, and c2 are bounded, (14) is
stable whenever the Courant-Friedrichs-Lewy step-size restriction
is maintained. Since normally ,
are fixed, using the definition of r in (15) we find
that the following restriction on the time-step
maintained in order to guarantee convergence of GVF: