Two-Dimensional Super-Resolution via Convex Relaxation

Abstract

In this paper, we address the problem of recovering point sources from two-dimensional low-pass measurements, which is known as the super-resolution problem. This is the fundamental concern of many applications such as electronic imaging, optics, microscopy, and line spectral estimations. We assume that the point sources are located in the square [0,1] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> with unknown locations and complex amplitudes. The only available information is low-pass Fourier measurements band limited to the integer square [-f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> ] <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> . The sources are estimated by minimizing the total variation norm, which leads to a convex optimization problem. We observe in numerical results that if the sources are closer than 1.4/f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , they might not be recovered by the proposed optimization. We theoretically show that there exists a dual certificate which guarantees exact recovery when the sources are separated by at least 1.68/f <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</sub> , reducing the gap between the available theoretical guarantee for the source separations and the observed results in the simulations.