## The Geometric GGVF Snake

In this section we briefly introduce the geometric GGVF snake and consider its advantages and shortcomings. Later in the chapter, the GGVF snake will be used along with the standard geometric snake to make comparisons to the performance of RAGS.

The gradient vector flow (GVF) active contour was first introduced by Xu et al.  in a parametric framework. The authors proposed a new external force: a diffusion of the gradient vectors of a gray level or binary edge map derived from the original image. The GVF goes some way toward forcing a snake into boundary concavities while providing a larger capture range due to its diffused gradient vector field. Figure 10.9 (right) shows the diffused gradient vectors for a simple object in Fig. 10.9 (left). The traditional potential force is shown in Fig. 10.9 (center).

The same authors have also introduced the GGVF, a generalized GVF snake model. The GGVF improves the GVF by replacing the constant weighting factor with two spatially varying weighting functions, resulting in a new external force field. The weighting factors provide a trade-off between the smoothness of the GVF field and its conformity to the gradient of the edge map. The result is Figure 10.9: GVF field compared to traditional potential force vector field. From left: a simple line-drawing U shape (binary) image, the traditional potential force vector field, and GVF field (from , (§2003 IEEE). Figure 10.10: Concavity convergence comparison. From left: initial snake, GVF snake result, and GGVF snake result, from .

that contours can converge into long, thin boundary indentations. The GGVF preserves clearer boundary information while performing vector diffusion, while the GVF will diffuse everywhere within image. As shown in Fig. 10.10, the GGVF snake shows clear ability to reach concave regions.

Later in , Xu et al. showed the GGVF equivalence in a geometric framework. A simple bimodal region force generated as a two-class fuzzy membership function was added to briefly demonstrate weak-edge leakage handling. The geometric GGVF snake is useful when dealing with boundaries with small gaps. However, it is still not robust to weak edges, especially when a weak boundary is close to a strong edge, the snake readily steps through the weak edge and stops at the strong one. This is illustrated in Fig. 10.11 (left).

A further problem with the GGVF snake is that it does not always allow the detection of multiple objects. These topological problems arise, even though  Figure 10.11: GGVF weaknesses. Left: The GGVF snake steps through a weak edge toward a neighboring strong one (final snake in white). Right: It also can encounter topological problems (final snake in black). The evolving snake is shown in a lighter color in both cases.

the GGVF snake was specified in the geometric model, when the vector field is tangent to the snake contour. In such cases there would be no force to push or pull it in the perpendicular direction (to the vectors). This effect is shown in Fig. 10.11 (right).