Slepian Scale-Discretised Wavelets on the Sphere

TL;DR

This paper introduces a new spherical wavelet basis using Slepian functions for analyzing incomplete spherical data, enabling localized convolution and multiscale analysis.

Contribution

The work develops Slepian scale-discretised wavelets on the sphere, extending wavelet analysis to incomplete datasets with a novel basis and convolution method.

Findings

Constructed a new spherical wavelet basis for incomplete data

Enabled convolution on incomplete spheres using Slepian functions

Demonstrated denoising application on Earth topography data

Abstract

This work presents the construction of a novel spherical wavelet basis designed for incomplete spherical datasets, i.e. datasets which are missing in a particular region of the sphere. The eigenfunctions of the Slepian spatial-spectral concentration problem (the Slepian functions) are a set of orthogonal basis functions which are more concentrated within a defined region. Slepian functions allow one to compute a convolution on the incomplete sphere by leveraging the recently proposed sifting convolution and extending it to any set of basis functions. Through a tiling of the Slepian harmonic line, one may construct scale-discretised wavelets. An illustration is presented based on an example region on the sphere defined by the topographic map of the Earth. The Slepian wavelets and corresponding wavelet coefficients are constructed from this region and are used in a straightforward denoising example.