Generalized Spectral Coarsening

Reducing the size of triangle meshes and their higher-dimensional counterparts, called simplicial complexes, while preserving important geometric or topological properties is an important problem in computer graphics and geometry processing. Such salient properties are captured by local shape descriptors via linear differential operators - often variants of Laplacian matrices. The eigenfunctions of Laplacians yield a convenient and useful set of bases that define a spectral domain for geometry processing (akin to the famous Fourier spectrum which uses eigenfunctions of the derivative operator). Existing methods for spectrum-preserving coarsening focus on 0-dimensional Laplacian operators that are defined on vertices (0-dimensional simplices). We propose a generalized spectral coarsening method that considers multiple Laplacian operators of possibly different dimensionalities in tandem. Our simple algorithm greedily decides the order of contractions of simplices based on a quality function per simplex. The quality function quantifies the error due to removal of that simplex on a chosen band within the spectrum of the coarsened geometry. We demonstrate that our method is useful to achieve band-pass filtering on both meshes as well as general simplicial complexes.