A Partition Based Method for Spectrum-Preserving Mesh Simplification.

Abstract

The majority of the simplification methods focus on preserving the appearance of the mesh, ignoring the spectral properties of the differential operators derived from the mesh. The spectrum of the Laplace-Beltrami operator is essential for a large subset of applications in geometry processing. Coarsening a mesh without considering its spectral properties might result in incorrect calculations on the simplified mesh. Given a 3D triangular mesh, this article aims to simplify the mesh using edge collapses, while focusing on preserving the spectral properties of the associated cotangent Laplace-Beltrami operator. Unlike the existing spectrum-preserving coarsening methods, we consider solely the eigenvalues of the operator in order to preserve the spectrum. The presented method is partition based, that is the input mesh is divided into smaller patches which are simplified individually. We evaluate our method on a variety of meshes, by using functional maps and quantitative norms, to measure how well the eigenvalues and eigenvectors of the Laplace-Beltrami operator computed on the input mesh are maintained by the output mesh. We demonstrate that the achieved spectrum preservation is at least as effective as the existing spectral coarsening methods.