Exact recovery of Dirac ensembles from the projection onto spaces of spherical harmonics

TL;DR

This paper demonstrates that an ensemble of Dirac delta functions on a sphere can be exactly recovered from spherical harmonic projections using TV norm minimization, under certain separation and sparsity conditions.

Contribution

It extends the theory of super-resolution to spherical domains, providing conditions for exact recovery of Dirac ensembles from harmonic projections.

Findings

Exact recovery under separation condition

Sparsity condition suffices for non-negative ensembles

Method based on dual interpolating polynomials

Abstract

In this work we consider the problem of recovering an ensemble of Diracs on the sphere from its projection onto spaces of spherical harmonics. We show that under an appropriate separation condition on the unknown locations of the Diracs, the ensemble can be recovered through Total Variation norm minimization. The proof of the uniqueness of the solution uses the method of `dual' interpolating polynomials and is based on [8], where the theory was developed for trigonometric polynomials. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for exact recovery.