Three-dimensional shape representation from curvature dependent surface evolution

This paper presents a novel approach to surface representation based on its differential deformations. The evolution of an arbitrary curve by curvature deforms it to a round point while in the process simplifying it. Similarly, we seek a process that deforms an arbitrary surface into a sphere without developing self-intersections, in the process creating a sequence of increasingly simpler surfaces. No previously studied curvature dependent flow satisfies this requirement: mean curvature flow leads to a splitting of the surface, while Gaussian curvature flow leads to instabilities. Thus, in search for such a process, we impose constraints (motivated by visual representation) to narrow down the space of candidate flows. 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; and (2) hyperbolic and parabolic points should not move at all. Accordingly, we propose ∂ψ/∂t=sign(H)√(G+|G|)N&oarr;; 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