STAR - Laplacian Spectral Kernels and Distances for Geometry Processing and Shape Analysis

• Published in: Computer graphics forum Vol. 35; no. 2; pp. 599 - 624
• Main Author: Patane, Giuseppe
• Format: Journal Article
• Language: English
• Published: Oxford Blackwell Publishing Ltd 01.05.2016
• Subjects: Analysis; and object representations; Categories and Subject Descriptors (according to ACM CCS); Computation; Computer graphics; Differential equations; Diffusion; diffusion geometry; Eigenvalues; harmonic equation; heat equation; I.3.3 [Numerical Analysis]: Approximation/Image Generation-Special function approximations; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling/Curve; I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling/Curve, surface, solid, and object representations; I.3.6 [Computer Graphics]: Methodology and Techniques; Kernels; Laplace transforms; Laplace-Beltrami operator; Laplacian eigenmproblem; Laplacian spectral distance and kernels; Laplacian spectrum; Mathematical analysis; Mathematical models; numerical analysis; Readers; shape analysis; solid; Spectra; spectral geometry processing; Studies; surface; Topological manifolds
• ISSN: 0167-7055; 1467-8659
• DOI: 10.1111/cgf.12866

In geometry processing and shape analysis, several applications have been addressed through the properties of the spectral kernels and distances, such as commute‐time, biharmonic, diffusion, and wave distances. Our survey is intended to provide a background on the properties, discretization, computation, and main applications of the Laplace‐Beltrami operator, the associated differential equations (e.g., harmonic equation, Laplacian eigenproblem, diffusion and wave equations), Laplacian spectral kernels and distances (e.g., commute‐time, biharmonic, wave, diffusion distances). While previous work has been focused mainly on specific applications of the aforementioned topics on surface meshes, we propose a general approach that allows us to review Laplacian kernels and distances on surfaces and volumes, and for any choice of the Laplacian weights. All the reviewed numerical schemes for the computation of the Laplacian spectral kernels and distances are discussed in terms of robustness, approximation accuracy, and computational cost, thus supporting the reader in the selection of the most appropriate method with respect to shape representation, computational resources, and target application.