Three-Dimensional Shape Representation from Curvature Dependent Surface Evolution

This paper presents a novel approach to surface representation based on its diierential deformations. The evolution of an arbitrary curve by curvature deforms it to a round point while in the process simplifying it. Similarly, in this paper we seek a process that deforms an arbitrary surface into sphere without developing self-intersections, in the process creating a sequence of increasingly simpler surfaces. No previously studied curvature dependent ow satisses this requirement: mean curvature ow leads to a splitting of the surface, while Gaussian curvature ow leads to instabilities. Thus, in search for such a process, we impose constraints (motivated by visual representation) to narrow down the space of candidate ows. Our main result is to establish a direction for the movement of points to avoid self-intersections: 1) convex elliptic points should move in, while concave elliptic points move out; 2) hyperbolic and parabolic points should not move at all. Accordingly , we propose @ @t = sign(H) p G + jGj ~ N; informally, the direction 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 this deformation has desired properties, leading to a geometric smoothing scheme for 2D and 3D images.