Towards a Mathematical Theory of Super-Resolution

TL;DR

This paper establishes a mathematical framework for super-resolution, demonstrating that exact recovery of point sources from low-frequency Fourier samples is possible via convex optimization, with robustness to noise and extensions to higher dimensions.

Contribution

It introduces a convex optimization approach for super-resolution that guarantees exact recovery under certain conditions, extending previous results and analyzing noise robustness.

Findings

Exact super-resolution is achievable with infinite precision when sources are sufficiently separated.

Recovery can be formulated as a semidefinite program, enabling practical computation.

The method is robust to noise and applicable in higher dimensions.

Abstract

This paper develops a mathematical theory of super-resolution. Broadly speaking, super-resolution is the problem of recovering the fine details of an object---the high end of its spectrum---from coarse scale information only---from samples at the low end of the spectrum. Suppose we have many point sources at unknown locations in $[0,1]$ and with unknown complex-valued amplitudes. We only observe Fourier samples of this object up until a frequency cut-off $f_c$. We show that one can super-resolve these point sources with infinite precision---i.e. recover the exact locations and amplitudes---by solving a simple convex optimization problem, which can essentially be reformulated as a semidefinite program. This holds provided that the distance between sources is at least $2/f_c$. This result extends to higher dimensions and other models. In one dimension for instance, it is possible to recover a piecewise smooth function by resolving the discontinuity points with infinite precision as well. We also show that the theory and methods are robust to noise. In particular, in the discrete setting we develop some theoretical results explaining how the accuracy of the super-resolved signal is expected to degrade when both the noise level and the {\em super-resolution factor} vary.