Slepian Scale-Discretised Wavelets on Manifolds

Abstract: Inspired by recent interest in geometric deep learning, this work generalises the recently developed Slepian scale-discretised wavelets on the sphere to Riemannian manifolds. Through the sifting convolution, one may define translations and, thus, convolutions on manifolds - which are otherwise not well-defined in general. Slepian wavelets are constructed on a region of a manifold and are therefore suited to problems where data only exists in a particular region. The Slepian functions, on which Slepian wavelets are built, are the basis functions of the Slepian spatial-spectral concentration problem on the manifold. A tiling of the Slepian harmonic line with smoothly decreasing generating functions defines the scale-discretised wavelets; allowing one to probe spatially localised, scale-dependent features of a signal. By discretising manifolds as graphs, the Slepian functions and wavelets of a triangular mesh are presented. Through a wavelet transform, the wavelet coefficients of a field defined on the mesh are found and used in a straightforward thresholding denoising scheme.