Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics

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 Candès and Fernandez-Granda (Commun Pure Appl Math 67:906–956, 2014), where the theory was developed for trigonometric polynomials. We also show that in the special case of nonnegative ensembles, a sparsity condition is sufficient for exact recovery.