The Geometric Heat Equation and Surface Fairing

This paper concentrates on analysis and discussion of the heat equation as it pertains to smoothing of geometric shapes and its relationship to the problem of surface fairing. The geometric heat equation distorts a given shape in order to obtain scale-space representation of a shape. This scale-space provides a complete description of the original shape in terms of small to large scale structures. These structures may then be interrogated by recognition algorithms in an attempt to classify the shape. In computer science, researchers have been working on methods for eliminating noise from mesh data obtained via 3D measurements. In this case one wishes to eliminate the noise present in the surface measurements in order to obtain an improved approximation of the true surface. This process is called surface fairing. These two seemingly different problems actually have many common goals. This report will focus on a generic implementation of the geometric heat equation and a particular implementation of surface fairing suggested by Gabriel Taubin [8]. Each method will be explained and their inter-relationships will be made clear. These relationships will be supported via experimental results obtained from 3D measurements on a set of human faces. The input data consisted of 3D mesh data sets obtained via a 3D laser range scanner, specifically the ShapeGrabber product of Vitana Corporation [1].