In this section, we first show several examples of GVF field computations on simple objects and demonstrate several key properties of GVF deformable contours. We then show the results of applying GVF deformable contours on both a noisy image and a real MR image. We used a = 0.6 and ft = 0.0 for all deformable contours and p = 0.2 for GVF unless stated separately. The deformable contours were dynamically repar-ameterized to maintain contour point separation to within 0.5-1.5 pixels (cf. [13]). All edge maps used in GVF computations were normalized to the range [0, 1].

4.1 Convergence to Boundary Concavity

In our first experiment, we computed the GVF field for the same U-shaped object used in Figs 1 and 2. The results are shown in Fig. 3. Comparing the GVF field shown in Fig. 3b to the traditional potential force field of Fig. 1b reveals several key differences. First, like the distance potential force field (Fig. 2b), the GVF field has a much larger capture range than traditional potential forces. A second observation, which can be seen in the close-up of Fig. 3c, is that the GVF vectors within the boundary concavity at the top of the U shape have a downward component. This stands in stark contrast to both the traditional potential forces of Fig. 1c and the distance potential forces of Fig. 2c. Finally, it can be seen from Fig. 3b that the GVF field behaves in an analogous fashion when viewed from the inside of the object. In particular, the GVF vectors are pointing upward into the "fingers" of the U shape, which represent concavities from this perspective.

Figure 3a shows the initialization, progression, and final configuration of a GVF deformable contour. The initialization is the same as that of Fig. 2a, and the deformable contour parameters are the same as those in Figs 1 and 2. Clearly, the GVF deformable contour has a broad capture range and superior convergence properties. The final deformable contour configuration closely approximates the true boundary, arriving at a subpixel interpolation through bilinear interpolation of the GVF force field.

As discussed in Section 3.2, the GVF-I field tends to smooth between opposite edges when there is a long, thin indentation along the object boundary while the GVF-II field does not. Figure 4 demonstrates this performance difference. Using an edge map obtained from the original image shown in Fig. 4a,

FIGURE 3 (a) The convergence of a deformable contour using (b) GVF external forces, (c) shown close-up within the boundary concavity. Reprinted from C. Xu and J. L. Prince, Snakes, shapes, and gradient vector flow. IEEE Trans, on Image Processing, 7(3):359-369, March, 1998. ©1998 IEEE.
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