Recovery of Sparse Positive Signals on the Sphere from Low Resolution Measurements

TL;DR

This paper addresses the challenge of reconstructing sparse positive signals on a sphere from low-resolution measurements using convex optimization, demonstrating that recovery accuracy depends on noise and signal density.

Contribution

It introduces a convex optimization approach for recovering positive Diracs on a sphere from low-resolution spherical harmonic projections, with theoretical validation.

Findings

Recovery error scales with noise level

Method successfully reconstructs sparse signals

Validation through numerical experiments

Abstract

This letter considers the problem of recovering a positive stream of Diracs on a sphere from its projection onto the space of low-degree spherical harmonics, namely, from its low-resolution version. We suggest recovering the Diracs via a tractable convex optimization problem. The resulting recovery error is proportional to the noise level and depends on the density of the Diracs. We validate the theory by numerical experiments.