Slepian Scale-Discretised Wavelets on Manifolds

TL;DR

This paper extends Slepian scale-discretised wavelets from the sphere to general Riemannian manifolds, enabling localized, scale-dependent analysis of data on complex geometries, with applications demonstrated on mesh discretizations.

Contribution

It generalizes Slepian wavelets to Riemannian manifolds and introduces a discretization on meshes, broadening their applicability in geometric deep learning.

Findings

Wavelet transform effectively captures localized features.

Denoising via thresholding improves data quality.

Method applicable to mesh discretizations of 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.