Surface wavelets: a multiresolution signal processing tool for 3D computational modelling

摘要

In this paper, we provide an introduction to wavelet representations for complex surfaces (surface wavelets), with the goal of demonstrating their potential for 3D scientific and engineering computing applications. Surface wavelets were originally developed for representing geometric objects in a multiresolution format in computer graphics. These wavelets share all of the major advantages of conventional wavelets, in that they provide an analysis tool for studying data, functions and operators at different scales. However, unlike conventional wavelets, which are restricted to uniform grids, surface wavelets have the power to perform signal processing operations on complex meshes, such as those encountered in finite element modelling. This motivates the study of surface wavelets as an efficient representation for the modelling and simulation of physical processes. We show how surface wavelets can be applied to partial differential equations, stated either in integral form or in differential form. We analyse and implement the wavelet approach for a model 3D potential problem using a surface wavelet basis with linear interpolating properties. We show both theoretically and experimentally that an O(h_n~2) convergence rate, h_n being the mesh size, can be obtained by retaining only O((logN)~(7/2)N) entries in the discrete operator matrix, where N is the number of unknowns. The principles described here may also be extended to volumetric discretizations.