Two-Dimensional Super-Resolution via Convex Relaxation

TL;DR

This paper presents a convex optimization approach for super-resolution in two dimensions, enabling exact recovery of point sources from band-limited Fourier measurements under certain separation conditions.

Contribution

It introduces a convex relaxation method using TV norm minimization for 2D super-resolution, providing theoretical guarantees for exact source recovery.

Findings

Exact recovery when sources are separated by at least 1.68/fc

Convex optimization via TV norm effectively solves the super-resolution problem

The dual certificate condition ensures successful reconstruction

Abstract

In this paper, we address the problem of recovering point sources from two dimensional low-pass measurements, which is known as super-resolution problem. This is the fundamental concern of many applications such as electronic imaging, optics, microscopy, and line spectral estimation. We assume that the point sources are located in the square $[0,1]^2$ with unknown locations and complex amplitudes. The only available information is low-pass Fourier measurements band-limited to integer square $[-f_c,f_c]^2$. The signal is estimated by minimizing Total Variation $(\mathrm{TV})$ norm, which leads to a convex optimization problem. It is shown that if the sources are separated by at least $1.68/f_c$, there exist a dual certificate that is sufficient for exact recovery.