Super-Resolution of Positive Sources: the Discrete Setup
Provides rigorous theoretical guarantees for a practical super-resolution problem in single-molecule microscopy, enabling stable recovery under realistic conditions.
This paper establishes that super-resolution of positive point sources from low-frequency Fourier measurements can be solved stably via linear programming, with reconstruction quality depending on the Rayleigh regularity of the source support. The method is proven to be nearly optimal.
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.