Resolution analysis of imaging with $\ell_1$ optimization

TL;DR

This paper analyzes the resolution limits of array imaging of sparse scenes using $$ optimization, considering both narrow-band and broad-band cases, and introduces deterministic bounds based on mutual coherence and interaction coefficients.

Contribution

It provides new deterministic resolution limits for $$ imaging that account for worst-case scenarios and arbitrary unknown locations, extending compressed sensing theory.

Findings

Resolution limits depend on scene sparsity and array configuration.

Derived bounds are valid for worst-case unknowns locations.

Numerical simulations confirm theoretical predictions.

Abstract

We study array imaging of a sparse scene of point-like sources or scatterers in a homogeneous medium. For source imaging the sensors in the array are receivers that collect measurements of the wave field. For imaging scatterers the array probes the medium with waves and records the echoes. In either case the image formation is stated as a sparsity promoting $\ell_1$ optimization problem, and the goal of the paper is to quantify the resolution. We consider both narrow-band and broad-band imaging, and a geometric setup with a small array. We take first the case of the unknowns lying on the imaging grid, and derive resolution limits that depend on the sparsity of the scene. Then we consider the general case with the unknowns at arbitrary locations. The analysis is based on estimates of the cumulative mutual coherence and a related concept, which we call interaction coefficient. It complements recent results in compressed sensing by deriving deterministic resolution limits that account for worse case scenarios in terms of locations of the unknowns in the imaging region, and also by interpreting the results in some cases where uniqueness of the solution does not hold. We demonstrate the theoretical predictions with numerical simulations.