Super-resolution on the Sphere using Convex Optimization

TL;DR

This paper introduces a convex optimization-based method for super-resolution of Dirac ensembles on a sphere, enabling high-precision recovery from low-resolution measurements under separation conditions.

Contribution

It proposes a three-stage algorithm combining semi-definite programming, root finding, and least-squares fitting for super-resolution on the sphere, with efficiency depending only on measurement count.

Findings

High-precision recovery under separation condition

Efficient algorithm with computation time independent of solution accuracy

Recovery error bounds in noisy discrete settings

Abstract

This paper considers the problem of recovering an ensemble of Diracs on a sphere from its low resolution measurements. The Diracs can be located at any location on the sphere, not necessarily on a grid. We show that under a separation condition, one can recover the ensemble with high precision by a three-stage algorithm, which consists of solving a semi-definite program, root finding and least-square fitting. The algorithm's computation time depends solely on the number of measurements, and not on the required solution accuracy. We also show that in the special case of non-negative ensembles, a sparsity condition is sufficient for recovery. Furthermore, in the discrete setting, we estimate the recovery error in the presence of noise as a function of the noise level and the super-resolution factor.