Super-Resolution of Positive Sources: the Discrete Setup

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 fc, 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/fc. 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.