Sampling Sparse Signals on the Sphere: Algorithms and Applications
This addresses sampling challenges for sparse signals on the sphere, with applications in diffusion processes, shot noise removal, and sound source localization, representing an incremental advance in finite-rate-of-innovation sampling.
The paper tackles the problem of reconstructing sparse signals (spikes) on the sphere from lowpass-filtered samples, achieving perfect reconstruction of K spikes using (K+√K)² spatial samples, which improves over previous methods by a factor of four for large K.
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.