FIGURE 4 (a) A square with a long, thin indentation and broken boundary,(b) GVF-I field (zoomed); (c) GVF-II field (zoomed); (d) initial contour position for both the GVF-I deformable contour and the GVF-II deformable contour; (e) final result of the GVF-I deformable contour; and (f) final result of the GVF-II deformable contour. Reprinted from Signal Processing, 71, C. Xu and J. L. Prince, Generalized gradient vector flow external forces for active contours, 131-139, © 1998 with permission from Elsevier Science.

both the GVF-I field 0 = 0.2) and the GVF-II field (.K = 0.05) were computed, as shown zoomed in Figs 4b and 4c, respectively. We note that in this experiment both the GVF-I field and the GVF-II field were normalized with respect to their magnitudes and used as external forces. Next, a deformable contour (a = 0.25, ¿6 = 0) was initialized at the position shown in Fig. 4d and allowed to converge within each of the external force fields. The GVF-I result, shown in Fig. 4e, stops well short of convergence to the long, thin, boundary indentation. On the other hand, the GVF-II result, shown in Fig. 4f, is able to converge completely to this same region. It should be noted that both GVF-I and GVF-II have wide capture ranges (which is evident because the initial deformable contour is fairly far away from the object), and they both preserve subjective contours (meaning that they cross the short boundary gaps).

The underlying formulation of GVF is valid for gray-level images as well as binary images. To compute GVF for gray-level images, the edge-map function f (x, y) must first be calculated. Two possibilities are f (1)(x, y) = |VI(x, y)| or f (2)(x, y) = |V(Gff(x, y) * I(x, y))|, where the latter is more robust in the presence of noise. Other more complicated noise-removal techniques such as median filtering, morphological filtering, and anisotropic diffusion could also be used to improve the underlying edge map. Given an edge-map function and an approximation to its gradient, GVF is computed in the usual way as in the binary case.

Figure 5a shows a gray-level image of the U-shaped object corrupted by additive white Gaussian noise; the signal-to-noise ratio is 6dB. Figure 5b shows an edge-map computed using

FIGURE 5 (a) A noisy 64 x 64-pixel image of a U-shaped object; (b) the edge map |V(Ga * I)|2 with a = 1.5; (c) the GVF external force field; and (d) convergence of the GVF deformable contour. 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.

FIGURE 5 (a) A noisy 64 x 64-pixel image of a U-shaped object; (b) the edge map |V(Ga * I)|2 with a = 1.5; (c) the GVF external force field; and (d) convergence of the GVF deformable contour. 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.

f (x, y) = f (2)(x, y) with a = 1.5 pixels, and Fig. 5c shows the computed GVF field. It is evident that the stronger edge-map gradients are retained while the weaker gradients are smoothed out. Superposed on the original image, Fig. 5d shows a sequence of GVF deformable contours (plotted in a shade of gray) and the GVF deformable contour result (plotted in white). The result shows an excellent convergence to the boundary, despite the initialization from far away, the image noise, and the boundary concavity.

Another demonstration of GVF applied to gray-scale imagery is shown in Fig. 6. Figure 6a shows a magnetic resonance image (short-axis section) of the left ventricle of a human heart, and Fig. 6b shows an edge map computed using f (x, y) =f (2)(x, y) with a = 2.5. Figure 6c shows the computed GVF, and Fig. 6d shows a sequence of GVF deformable contours ( plotted in a shade of gray) and the GVF deformable contour result ( plotted in white), both overlaid on the original image. Clearly, many details on the endocardial border are captured by the GVF deformable contour result, including the papillary muscles (the bumps that protrude into the cavity).

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