Invariant Geometric Evolutions of Surfaces and Volumetric Smoothing

The study of geometric flows for smoothing, multiscale representation, and analysis of two and three dimensional objects has received much attention in the past few years. In this paper, we first present results mainly related to Euclidean invariant geometric smoothing of three dimensional surfaces. We describe results concerning the smoothing of graphs (images) via level sets of geometric heat-type flows. Then we deal with proper three dimensional flows. These flows are governed by functions of the principal curvatures of the surface, such as the mean and Gaussian curvatures. Then, given a transformation group G acting on Rn, we write down a general expression for any G-invariant hypersurface geometric evolution in R n . As an application, we derive the simplest affine invariant flow for surfaces. *This work was supported in part by grants from the National Science Foundation DMS-9116672, DMS9204192, DMS-8811084, and ECS-9122106, by the Air Force Office of Scientific Research AFOSR-90-0024 and F49620-94-1-OOS8DEF, by the Army Research Office DAAL03-92-G-0115, DAAL03-91-G-0019, DAAH04-93G-0332, DAAH04-94-G-0054, by the Rothschild Foundation-Yad Hanadiv, and by Image Evolutions, Ltd.