Geometric Smoothing of 3d Surfaces and Non-linear Diiusion of 3d Images

In this paper we present a geometric smoothing technique for three-dimensional surfaces and images. The technique relies on curvature-dependent deformations of surfaces and the intuition that highly curved regions should move into their convexity by an amount proportional to their curvature. While this intuition in 2D has lead to a well-de ned process and formal theorems about its smoothing properties, the development of a similar process in 3D has been confounded by the existence of two directions of bending: when both directions of bending agree (elliptic points) the surface moving into the convexity attens; however, when directions of bending disagree (hyperbolic points) the situation is ambiguous. In this paper we show that an appropriate smoothing deformation should leave hyperbolic and parabolic points stationary, but move elliptic points by square root of their Gaussian curvature. The resulting deformation in conjunction with a shock-rarefaction smoothing method results in the Entropy Scale Space for surfaces. Furthermore, this technique can be applied to 3D images by viewing them as a set of ordered iso-intensity surfaces each undergoing this process. We illustrate the e ectiveness of these techniques on several 3D surfaces and 3D images. Since both smoothing surfaces and images are important to a number of computer vision applications, we expect this techniques to be a useful component for them.