A mathematical theory of super-resolution and two-point resolution

TL;DR

This paper develops a mathematical framework for super-resolution, introducing identities for source recovery, proving a resolution limit based on SNR, and demonstrating that resolution can surpass the Rayleigh limit under certain conditions.

Contribution

It introduces location-amplitude identities for super-resolution, establishes a theoretical resolution limit in multi-dimensional spaces, and provides an algorithm that achieves this resolution.

Findings

Super-resolution capabilities are characterized by new identities linking true and recovered sources.

A resolution limit formula is derived, depending on SNR and cutoff frequency.

Resolution can surpass the Rayleigh limit when SNR exceeds 2.

Abstract

This paper focuses on the fundamental aspects of super-resolution, particularly addressing the stability of super-resolution and the estimation of two-point resolution. Our first major contribution is the introduction of two location-amplitude identities that characterize the relationships between locations and amplitudes of true and recovered sources in the one-dimensional super-resolution problem. These identities facilitate direct derivations of the super-resolution capabilities for recovering the number, location, and amplitude of sources, significantly advancing existing estimations to levels of practical relevance. As a natural extension, we establish the stability of a specific $l_0$ minimization algorithm in the super-resolution problem. The second crucial contribution of this paper is the theoretical proof of a two-point resolution limit in multi-dimensional spaces. The resolution limit is expressed as: \[ R = \frac{4\arcsin \left(\left(\frac{\sigma}{m_{\min}}\right)^{\frac{1}{2}} \right)}{\Omega} \] for $\frac{\sigma}{m_{\min}}\leq\frac{1}{2}$, where $\frac{\sigma}{m_{\min}}$ represents the inverse of the signal-to-noise ratio ($\mathrm{SNR}$) and $\Omega$ is the cutoff frequency. It also demonstrates that for resolving two point sources, the resolution can exceed the Rayleigh limit $\frac{\pi}{\Omega}$ when the signal-to-noise ratio (SNR) exceeds $2$. Moreover, we find a tractable algorithm that achieves the resolution $R$ when distinguishing two sources.