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: \[ R = 4 ( ( m_ )^12 ) \] for m_12, where m_ represents the inverse of the signal-to-noise ratio (SNR) and is the cutoff frequency. It also demonstrates that for resolving two point sources, the resolution can exceed the Rayleigh limit when the signal-to-noise ratio (SNR) exceeds 2. Moreover, we find a tractable algorithm that achieves the resolution R when distinguishing two sources.

Tasks

Reproductions