Super-Resolution on the Two-Dimensional Unit Sphere

TL;DR

This paper addresses the problem of reconstructing atomic measures on the 2-sphere from spherical harmonic moments, proposing an optimization-based approach with explicit dual certificates and numerical validation.

Contribution

It introduces a novel method for super-resolution on the 2-sphere using explicit kernel-based dual certificates and analyzes the interpolation problem in detail.

Findings

Successful construction of dual certificates using explicit kernels

Effective numerical examples demonstrating the method's performance

Theoretical analysis of the interpolation problem on the sphere

Abstract

We study the problem of recovering an atomic measure on the unit 2-sphere $\mathbb{S}^2$ given finitely many moments with respect to spherical harmonics. The analysis relies on the formulation of this problem as an optimization problem on the space of bounded Borel measures on $\mathbb{S}^2$ as it was considered by Y. de Castro & F. Gamboa and E. Cand\'es & C. Fernandez-Granda. We construct a dual certificate using a kernel given in an explicit form and make a concrete analysis of the interpolation problem. Numerical examples are provided and analyzed.