Point Flow in GVF Fields

It is interesting to examine where points would flow within a GVF field if they are not constrained to be part of an active contour. This is easily accomplished by eliminating the internal forces of the active contour formulation (e.g., by setting $\alpha = 0$ and $\beta = 0$) and using the standard numerical procedure [Equation 17] on a collection of scattered points. Fig. 7 shows a line drawing in (a) and its GVF field in (b). The regular grid of 4096 ( $64\times 64$) points shown in (c) is allowed to flow under the influence of the GVF field only, and the positions of these points are shown in (d)-(h) after 10, 20, 50, 100, and 600 iterations, respectively.

We observe in Figs. 7(d)-(f) that many points flow quite rapidly to the lines (edges) in the drawing. Points near the middle of a straight edge, however, tend to flow to the corners; this is apparent because the straight edges seem to be ``broken'' in their centers (see (f) in particular). On the other hand, the central-most points on these same edges are stable for many iterations, disappearing only in the last frame. Fig. 7(h) shows an equilibrium, in which all 4096 points have settled on the nine ``corners'' of the original figure. In that sense, the corners have been detected. But it could be argued that a great deal more information about the shape of the object was determined during the iterative process. Edges were located early on, while points of symmetry were found to be long-lived.

Another way to study the flow of points in a GVF field is to look at their streamlines -- that is, the paths over which points move (still in the absence of active contour internal forces). Fig. 8 shows the streamlines for the object and points shown in Fig. 7a and b, respectively. This figure gives an idea about the ``capture pattern'' of GVF fields. In particular, streamlines that flow apart will tend to pull apart an active contour, while those that flow together will tend to push it together. It should be noted that in light of Fig. 7h, an active contour converging to the outer boundary of this figure would always have its points pulled toward the corners. Thus, only the internal forces present in a snake will keep it wrapped around the figure.

It is interesting to compare the streamlines generated by the standard potential forces and those created by GVF. Fig. 9a gives the streamlines for the potential force field of Fig. 1b, and Fig. 9b gives the streamlines for the GVF field of Fig. 2b. Two effects are clear from this figure. First, the capture range of GVF is clearly much larger than that of the potential forces. Second, GVF provides downward forces within the concave region at the top of the U-shape, while potential forces only provide sideways forces.