Super-Resolution of Positive Sources: the Discrete Setup

TL;DR

This paper demonstrates that super-resolution of positive point sources in single-molecule microscopy can be achieved through linear programming, with performance depending on source support regularity, and provides near-optimal theoretical guarantees supported by numerical experiments.

Contribution

It establishes a stable linear programming approach for super-resolution of positive sources, with performance guarantees based on Rayleigh regularity, and proves near-optimality of the method.

Findings

Super-resolution is achievable via linear programming.

Performance depends on the Rayleigh regularity of source support.

Numerical experiments validate the theoretical results.

Abstract

In single-molecule microscopy it is necessary to locate with high precision point sources from noisy observations of the spectrum of the signal at frequencies capped by $f_c$, which is just about the frequency of natural light. This paper rigorously establishes that this super-resolution problem can be solved via linear programming in a stable manner. We prove that the quality of the reconstruction crucially depends on the Rayleigh regularity of the support of the signal; that is, on the maximum number of sources that can occur within a square of side length about $1/f_c$. The theoretical performance guarantee is complemented with a converse result showing that our simple convex program convex is nearly optimal. Finally, numerical experiments illustrate our methods.