Geometric Heat Equation and Nonlinear Diffusion of Shapes and Images

Abstract

We propose a geometric smoothing method based on local curvature in shapes and images which is governed by the geometric heat equation and is a special case of the reaction-diffusion framework proposed by Faugeras (1990). For shapes, the approach is analogous to the classical heat equation smoothing, but with a renormalization by arc-length at each infinitesimal step. For images, the smoothing is similar to anisotropic diffusion in that, since the component of diffusion in the direction of the brightness gradient is nil, edge location and sharpness are left intact. We present several properties of curvature deformation smoothing of shape: it preserves inclusion order, annihilates extrema and inflection points without creating new ones, decreases total curvature, satisfies the semi-group property allowing for local iterative computations, etc. Curvature deformation smoothing of an image is based on viewing it as a collection of iso-intensity level sets, each of which is smoothed by curvature and then reassembled. This is shown to be mathematically sound and applicable to medical, aerial and range images.<<ETX>>