Super-Resolution in Phase Space

TL;DR

This paper introduces a general super-resolution approach in Phase Space, unifying various transformations and demonstrating effective recovery of Dirac distributions from low-pass measurements via Total Variation minimization.

Contribution

It extends super-resolution techniques to Phase Space transformations, providing a unified framework compatible with classical Fourier-based methods.

Findings

Super-resolution in Phase Space generalizes classical Fourier domain results.

Total Variation minimization effectively recovers Dirac distributions.

Minimum separation bounds are comparable to existing methods.

Abstract

This work considers the problem of super-resolution. The goal is to resolve a Dirac distribution from knowledge of its discrete, low-pass, Fourier measurements. Classically, such problems have been dealt with parameter estimation methods. Recently, it has been shown that convex-optimization based formulations facilitate a continuous time solution to the super-resolution problem. Here we treat super-resolution from low-pass measurements in Phase Space. The Phase Space transformation parametrically generalizes a number of well known unitary mappings such as the Fractional Fourier, Fresnel, Laplace and Fourier transforms. Consequently, our work provides a general super- resolution strategy which is backward compatible with the usual Fourier domain result. We consider low-pass measurements of Dirac distributions in Phase Space and show that the super-resolution problem can be cast as Total Variation minimization. Remarkably, even though are setting is quite general, the bounds on the minimum separation distance of Dirac distributions is comparable to existing methods.