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

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.