Localised and shape-aware functions for spectral geometry processing and shape analysis: A survey & perspectives

Basis functions provide a simple and effective way to interpolate signals on a surface with an arbitrary dimension, topology, and discretisation. However, the definition of local and shape-aware basis functions is still an open problem, which has been addressed through optimisation methods that are time-consuming, over-constrained, or admit more than one solution. In this context, we review the definition, properties, computation, and applications of the class of Laplacian spectral basis functions, which are defined as solutions to the harmonic equation, the diffusion equation, and PDEs involving the Laplace-Beltrami operator. The resulting functions are efficiently computed through the solution of sparse linear systems and satisfy different properties, such as smoothness, locality, and multi-scale shape encoding. Finally, the discussion of the properties of the Laplacian spectral basis functions is integrated with an analysis of their behaviour with respect to different measures (i.e., conformal and area-preserving metrics, transformation distance) and of their applications to spectral geometry processing and shape analysis.