Generalized Force Balance Equations

It is easy to see that the external forces generated from the variational formulation (1) must enter the force balance equation (6) as a static irrotational field.¹ To add additional flexibility to the snake model, it is possible to start from the force balance equation directly, and to add another external force ${\bf F}^2_{\rm ext}$as follows

$${\bf F}_{\rm int} + {\bf F}^1_{\rm ext} + {\bf F}^2_{\rm ext} = 0$$ (8)

Selecting ${\bf F}^2_{\rm ext}$ has a profound impact on the behavior of the snake. The simplest non-trivial choice would be to make ${\bf F}^2_{\rm ext}$ a static solenoidal field, so that the sum of ${\bf F}^1_{\rm ext}$ and ${\bf F}^2_{\rm ext}$comprise the most general static field possible (see the Helmholtz theorem in [26]). A second extension would be to make ${\bf F}^2_{\rm ext}$ a time-dependent (dynamic) field. A third extension would be to make ${\bf F}^2_{\rm ext}$ additionally depend on ${\bf x}$, the snake itself.

Balloon models [20] comprise an important example from the third class of extensions given above. These models introduce pressure (or normal) forces, which act on closed snakes -- or balloons -- in a direction normal to the curve to cause the balloon to either expand or contract. This largely eliminates the requirement that the image potential forces extend great distances from the image boundaries. Instead, pressure forces can cause large balloons to shrink on their own until they encounter stopping potential forces. Alternatively, small balloons can be made to grow until they encounter potential forces. One problem with this approach is that the balloon must be initialized to either shrink or grow. Another more subtle problem is that the strength of the pressure forces may be difficult to set. They must be chosen to be large enough to overcome weak edges and noise, but small enough so they do not overwhelm legitimate edge forces. In this paper, we consider active contour formulations that do not include pressure forces.

Without pressure forces, initialization of the active contour becomes problematic. If it is initialized too far away, the external forces may not be strong enough to attract the active contour toward the object. This could be solved by taking a multi-resolution approach, in which the active contour initially sees external forces created by a large-$\sigma$ blurring of the image. As the snake moves closer to the object, the resolution of the edges can be improved by lowering $\sigma$. With this approach, there are invariably problems with scheduling $\sigma$ decreases and with the fact that $\sigma$ might need to be different on one part of the image versus another. It is not clear that the complexity of this type of approach is worth the gain.

Without pressure forces, active contours have difficulty progressing into concave boundary regions, often leaving the contour split across the region. An example of this problem is shown in Fig. 1. Fig. 1a shows a ( $64\times 64$ pixel) line drawing of a U-shaped object having a concave region (as viewed from the outside) at the top of the figure. Fig. 1b shows the potential force field given by $-\nabla E_{\rm ext}^4$ where $\sigma = 1.0$ (pixels). Fig. 1c shows a sequence of curves depicting the iterative progression of a standard snake (no pressure forces) toward the boundary. The outermost curve depicts the initial active contour, drawn by hand. The active contours are generated using (7) where $\alpha = 0.6$ and $\beta = 0.0$. Clearly, the final solution is undesirable because it did not find the concave portion of the U-shape.

The reason for the poor convergence in Fig. 1c is revealed in Fig. 1b, where the external field is depicted. Although the field correctly points toward the object boundary, the problem is that in the concave portion, the forces are largely pointing horizontally in opposite directions. Thus, the curve is ``pulled'' apart toward the U-shape, but not made to progress downward, into the concave region. It is important to point out that the solution shown in Fig. 1c satisfies the Euler equations (6). Hence, the poor performance demonstrated above lies in the problem formulation, not its method of solution. In the following section, the GVF field is introduced, and the GVF snake is defined as the snake that uses the GVF field as its sole external forces.