Convergence to a Concave Region

In our first experiment, we computed the GVF field for the line drawing of Fig. 2a using $\mu = 0.2$. Comparing the resulting field, shown in Fig. 2b, to the potential force field of Fig. 1b, reveals several key differences. First, the GVF field has a much larger capture range. It is clear that in order to get this extent using traditional potential force fields, one would have to use a large $\sigma$ in the Gaussian filter. But this would have the effect of blurring (or perhaps even obliterating) the edges, which is not happening in the GVF field. A second observation is that the GVF vectors are pointing somewhat downward into the top of the U-shape, which should cause an active contour to move farther into this concave region. Finally, it is clear that the GVF field behaves in an analogous fashion when viewed from the inside. That is, the vectors are pointing toward the boundary from as far away as possible and are pointing upward into the concave regions (the fingers of the U-shape) as viewed from the inside.

Fig. 2c shows the result of applying a GVF snake with parameters $\alpha = 0.6$ and $\beta = 0.0$ to the line drawing shown in Fig. 2a (using the external GVF field of Fig. 2b). In this case, the snake was initialized farther away from the object than the initialization in Fig. 1c, and yet it converges very well to the boundary of the U-shape. It should be noted that the blocky appearance of the U-shape results from the fact that the image is only $64\times 64$ pixels. The snake itself moves through the continuum (using bilinear interpolation to derive external field forces which are not at grid points) to arrive at a sub-pixel interpolation of the boundary.