Sampling Sparse Signals on the Sphere: Algorithms and Applications

TL;DR

This paper introduces a novel sampling scheme for perfect reconstruction of sparse signals on the sphere, utilizing a generalized annihilating filter method that improves sampling efficiency and is applicable to various problems like diffusion, noise removal, and sound localization.

Contribution

It presents a new sampling algorithm based on a generalized annihilating filter for sparse signals on the sphere, reducing sampling requirements and demonstrating versatility across multiple applications.

Findings

Reconstructs K spikes from (K+√K)^2 samples, improving efficiency.

Applicable to diffusion, noise removal, and sound localization.

Reduces sampling complexity by a factor of four for large K.

Abstract

We propose a sampling scheme that can perfectly reconstruct a collection of spikes on the sphere from samples of their lowpass-filtered observations. Central to our algorithm is a generalization of the annihilating filter method, a tool widely used in array signal processing and finite-rate-of-innovation (FRI) sampling. The proposed algorithm can reconstruct $K$ spikes from $(K+\sqrt{K})^2$ spatial samples. This sampling requirement improves over previously known FRI sampling schemes on the sphere by a factor of four for large $K$. We showcase the versatility of the proposed algorithm by applying it to three different problems: 1) sampling diffusion processes induced by localized sources on the sphere, 2) shot noise removal, and 3) sound source localization (SSL) by a spherical microphone array. In particular, we show how SSL can be reformulated as a spherical sparse sampling problem.