Sampling theorems and compressive sensing on the sphere

TL;DR

This paper introduces a new sampling theorem on the sphere that reduces the number of samples needed for band-limited signals, improving efficiency and performance in compressive sensing applications like inpainting.

Contribution

The paper presents a novel sampling theorem on the sphere that requires fewer samples than previous methods, enhancing compressive sensing and signal reconstruction.

Findings

Fewer samples needed for exact representation of band-limited signals.

Superior inpainting reconstruction performance with the new sampling theorem.

Implications for reduced dimensionality and increased sparsity in spherical signals.

Abstract

We discuss a novel sampling theorem on the sphere developed by McEwen & Wiaux recently through an association between the sphere and the torus. To represent a band-limited signal exactly, this new sampling theorem requires less than half the number of samples of other equiangular sampling theorems on the sphere, such as the canonical Driscoll & Healy sampling theorem. A reduction in the number of samples required to represent a band-limited signal on the sphere has important implications for compressive sensing, both in terms of the dimensionality and sparsity of signals. We illustrate the impact of this property with an inpainting problem on the sphere, where we show superior reconstruction performance when adopting the new sampling theorem.