An Introduction to Laplacian Spectral Distances and Kernels: Theory, Computation, and Applications

Abstract

In geometry processing and shape analysis, several applications have been addressed through the properties of the Laplacian spectral kernels and distances, such as commute- time, biharmonic, di↵usion, and wave distances. Spectral distances are easily defined through a filtering of the Laplacian eigenpairs and include random walks [FPS05, RS13], heat di↵usion [BBK+10, BBOG11, CL06, GBAL09, LKC06, LSW09], biharmonic [LRF10, Rus11b], and wave kernel [BB11a, ASC11] distances. Biharmonic [LRF10, Rus11b] and di↵usion [BBK+10, BBOG11, CL06, GBAL09, LKC06, LSW09, PS13b] distances pro- vide a trade-o↵ between a nearly geodesic behavior for small distances and the encod- ing of global surface properties for large distances, thus guaranteeing an intrinsic and multi-scale characterisation of the input shape. The heat kernel [BBG94] is also central in di↵usion geometry [BN03, CL06, GK06, Sin06], dimensionality reduction with spectral embeddings [BN03, XHW10], and data classification [SK03].