On Curvature-Dependent Surface Evolution

Computer vision applications require representations for surfaces that are intrinsic and re ect the hierarchical nature of their geometric structure. Curvature-dependent ows have been successfully used for shape representation in two dimensions, principally due to a theorem on curve shortening ow stating that embedded curves evolving proportional to their curvature do not self-intersect and smoothly converge to a circular point. In this paper, we investigate properties of curvature-driven ows for surfaces and derive necessary conditions for the ows to avoid self-intersections. Since known curvature ows, such as mean and Gaussian ows, lead to self-intersections, we impose geometric constraints on the direction and magnitude of arbitrary ows to narrow down the space of candidates. Our main result is to establish necessary conditions for the direction of movement in order to avoid self-intersections: 1) convex elliptic points should move in, while concave elliptic points move out; 2) hyperbolic points should not move at all; and 3) parabolic points should also be stationary. In the process, we also establish monotonicity conditions for the magnitude of the curvature-dependent function. Finally, based on these conditions and previously studied literature on the ow of strictly convex surfaces, we select a magnitude proportional to square root of Gaussian curvature leading to: @ @t = sign(H)qG+ jGj ~ N . Informally, the direction of deformation depends on mean curvature and the magnitude of deformation on Gaussian curvature. Our numerical simulations show that for a large class of non-convex surfaces, but not for all, this deformation takes the initial surface to a round point without developing self-intersections. The necessary conditions and the existence of surfaces that self-intersect for this ow indicate that a ow solely based on curvatures would lead to self-intersections for some surface. Figure 1: When the boundary of the above shape moves along the normal with a speed proportional to curvature, does it evolve to a circular point without developing self-intersections? Grayson showed that the high curvature end points will move in much faster that the low curvature points in such a way that self-intersections are avoided. Eventually, the shape evolves to a round point, keeping the boundaries apart in the process!