Super-Resolution by Compressive Sensing Algorithms

TL;DR

This paper compares four compressive sensing algorithms for super-resolution with Fourier measurements, introducing a new metric for spike recovery quality, and demonstrates that BLOOMP and BP-BLOT achieve significant super-resolution capabilities.

Contribution

The paper introduces a new filtered error norm metric and compares four algorithms, showing BLOOMP and BP-BLOT can perform grid-independent super-resolution of spikes.

Findings

BLOOMP and BP-BLOT achieve grid-independent recovery of well-separated spikes.

Both algorithms can localize spike support within a few percent of the Rayleigh length.

BP-BLOT can super-resolve closely spaced spikes separated by less than the Rayleigh length.

Abstract

In this work, super-resolution by 4 compressive sensing methods (OMP, BP, BLOOMP, BP-BLOT) with highly coherent partial Fourier measurements is comparatively studied. An alternative metric more suitable for gauging the quality of spike recovery is introduced and based on the concept of filtration with a parameter representing the level of tolerance for support offset. In terms of the filtered error norm only BLOOMP and BP-BLOT can perform grid-independent recovery of well separated spikes of Rayleigh index 1 for arbitrarily large super-resolution factor. Moreover both BLOOMP and BP-BLOT can localize spike support within a few percent of the Rayleigh length. This is a weak form of super-resolution. Only BP-BLOT can achieve this feat for closely spaced spikes separated by a fraction of the Rayleigh length, a strong form of super-resolution.