Exact Localization and Superresolution with Noisy Data and Random Illumination

TL;DR

This paper demonstrates that using random illumination and specific recovery algorithms enables exact localization of sparse objects with noisy data, surpassing traditional resolution limits and providing strong theoretical and numerical support.

Contribution

It introduces a novel approach combining random illumination with Lasso and BPDN methods for superresolution and exact localization in noisy environments.

Findings

Lasso and BPDN can localize up to O(m) objects with m data points.

Lasso with random illumination achieves superresolution beyond Rayleigh limit.

Numerical results show Lasso outperforms OST in the proposed setup.

Abstract

This paper studies the problem of exact localization of sparse (point or extended) objects with noisy data. The crux of the proposed approach consists of random illumination. Several recovery methods are analyzed: the Lasso, BPDN and the One-Step Thresholding (OST). For independent random probes, it is shown that both recovery methods can localize exactly $s=\cO(m)$, up to a logarithmic factor, objects where $m$ is the number of data. Moreover, when the number of random probes is large the Lasso with random illumination has a performance guarantee for superresolution, beating the Rayleigh resolution limit. Numerical evidence confirms the predictions and indicates that the performance of the Lasso is superior to that of the OST for the proposed set-up with random illumination.