Three - Dimensional Shape Representation fromCurvature Dependent Surface

This paper presents a novel approach to surface representation based on its di erential deformations, and is an extension to surfaces of the \shape from deformation" framework for curves [31]. A representation of 3D shape for recognition must capture the multidimensional nature of shape, as well as the notion of a hierarchy of scale. A range of deformations can capture both: while constant deformation of shape emphasizes its description as a composition of simpler parts, curvature deformation stresses a view of shape as the result of modi cation of a simpler shape. It is the latter process which is the focus of this paper: we seek a process that deforms arbitrary surfaces to spheres without developing self-intersections, in the process creating a sequence of increasingly simpler shapes. No previously studied curvature-dependent ow satis es this requirement: mean curvature ow leads to a splitting of the shape, 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 points should not move at all; and 3) parabolic points should also be stationary. In the process, we also establish a monotonicity condition for the magnitude of the curvature-dependent function. Finally, we select a candidate magnitude function based on known results of strictly convex surfaces evolving by square root of Gaussian curvature leading to our proposal: @ @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 this deformation takes the initial surface to a round point without developing self-intersections. We illustrate an application of this deformation to the smoothing of two-dimensional range images, as well as three-dimensional CT and MRI data.