Array imaging of localized objects in homogeneous and heterogeneous media

TL;DR

This paper investigates array imaging techniques for localized objects in various media, proposing new optimization methods that improve robustness and stability in complex scattering environments, with comprehensive analysis and comparisons.

Contribution

It introduces a non-iterative, two-step formulation for imaging in strongly scattering homogeneous media and a hybrid-$ extit{ extlangle} extit{ extlangle} ext{l}_1$ method for random media, enhancing robustness and stability.

Findings

Hybrid-$ extlangle extlangle ext{l}_1$ method is stable in large arrays within random media.

Optimal illuminations improve sparse optimization robustness against noise.

The proposed methods outperform Kirchhoff migration and MUSIC in certain scenarios.

Abstract

We present a comprehensive study of the resolution and stability properties of sparse promoting optimization theories applied to narrow band array imaging of localized scatterers. We consider homogeneous and heterogeneous media, and multiple and single scattering situations. When the media is homogeneous with strong multiple scattering between scatterers, we give a non-iterative formulation to find the locations and reflectivities of the scatterers from a nonlinear inverse problem in two steps, using either single or multiple illuminations. We further introduce an approach that uses the top singular vectors of the response matrix as optimal illuminations, which improves the robustness of sparse promoting optimization with respect to additive noise. When multiple scattering is negligible, the optimization problem becomes linear and can be reduced to a hybrid-$\ell_1$ method when optimal illuminations are used. When the media is random, and the interaction with the unknown inhomogeneities can be primarily modeled by wavefront distortions, we address the statistical stability of these methods. We analyze the fluctuations of the images obtained with the hybrid-$\ell_1$ method, and we show that it is stable with respect to different realizations of the random medium provided the imaging array is large enough. We compare the performance of the hybrid-$\ell_1$ method in random media to the widely used Kirchhoff migration and the multiple signal classification methods.