Super-Resolution of Point Sources via Convex Programming

TL;DR

This paper introduces a convex optimization method for super-resolving point sources from low-frequency data, achieving exact recovery when sources are sufficiently separated, and extends to related problems.

Contribution

It demonstrates that minimizing a continuous l1 norm can exactly recover point sources separated by at least 1.26/f, with stability to noise, and extends to demixing and shared support estimation.

Findings

Exact recovery when sources are separated by at least 1.26/f.

Method is stable to noise due to dual certificate construction.

Framework extends to demixing sines and spikes, and shared support estimation.

Abstract

We consider the problem of recovering a signal consisting of a superposition of point sources from low-resolution data with a cut-off frequency f. If the distance between the sources is under 1/f, this problem is not well posed in the sense that the low-pass data corresponding to two different signals may be practically the same. We show that minimizing a continuous version of the l1 norm achieves exact recovery as long as the sources are separated by at least 1.26/f. The proof is based on the construction of a dual certificate for the optimization problem, which can be used to establish that the procedure is stable to noise. Finally, we illustrate the flexibility of our optimization-based framework by describing extensions to the demixing of sines and spikes and to the estimation of point sources that share a common support.