Discrete Combinatorial Laplacian Operators for Digital Geometry Processing
Hao Ran Zhang · 2004
Abstract. Digital Geometry Processing (DGP) is concerned with the construction of signal processing style algorithms that operate on surface geometry, typically specified by an unstructured triangle mesh. An active subfield of study involves the utilization of discrete mesh Laplacian operators for eigenvalue decomposition, mimicking the effect of discrete Fourier analysis on mesh geometry. In this paper, we investigate matrix-theoretic properties, e.g., symmetry, stochasticity, and energy-compaction, of well-known combinatorial mesh Laplacians and examine how they would influence our choice of an appropriate operator or numerical method for DGP. We also propose two new symmetric combinatorial Laplacian operators for eigenanalysis of meshes and demonstrate their advantages over existing ones in several practical applications. Frequency-domain characterization and processing of irregular triangle