Superresolution without Separation

TL;DR

This paper demonstrates that, under ideal conditions, arbitrarily close point sources can be resolved beyond the diffraction limit by leveraging a combination of polynomial interpolation and compressed sensing techniques.

Contribution

It provides a theoretical framework showing exact recovery of source positions and amplitudes from limited observations using a weighted basis pursuit approach.

Findings

Exact resolution of arbitrarily close sources in ideal scenarios.

Recovery of source parameters from 2M + 1 observations.

Integration of polynomial interpolation with compressed sensing methods.

Abstract

This paper provides a theoretical analysis of diffraction-limited superresolution, demonstrating that arbitrarily close point sources can be resolved in ideal situations. Precisely, we assume that the incoming signal is a linear combination of M shifted copies of a known waveform with unknown shifts and amplitudes, and one only observes a finite collection of evaluations of this signal. We characterize properties of the base waveform such that the exact translations and amplitudes can be recovered from 2M + 1 observations. This recovery is achieved by solving a a weighted version of basis pursuit over a continuous dictionary. Our methods combine classical polynomial interpolation techniques with contemporary tools from compressed sensing.