Compressive Inverse Scattering I. High Frequency SIMO Measurements

TL;DR

This paper applies compressed sensing theory to high-frequency inverse scattering problems, demonstrating exact and stable recovery of sparse targets using SIMO, MISO, and MIMO measurements, with theoretical bounds and reciprocity analysis.

Contribution

It provides a high-frequency analysis of recoverability in inverse scattering using compressed sensing, including stability, reciprocity, and coherence bounds for various measurement configurations.

Findings

Exact recovery of sparse targets in noise-free conditions.

Stability results for weak or separated scatterers.

Coherence bounds for diffraction tomography with limited data.

Abstract

Inverse scattering from discrete targets with the single-input-multiple-output (SIMO), multiple-input-single-output (MISO) or multiple-input-multiple-output (MIMO) measurements is analyzed by compressed sensing theory with and without the Born approximation. High frequency analysis of (probabilistic) recoverability by the $L^1$-based minimization/regularization principles is presented. In the absence of noise, it is shown that the $L^1$-based solution can recover exactly the target of sparsity up to the dimension of the data either with the MIMO measurement for the Born scattering or with the SIMO/MISO measurement for the exact scattering. The stability with respect to noisy data is proved for weak or widely separated scatterers. Reciprocity between the SIMO and MISO measurements is analyzed. Finally a coherence bound (and the resulting recoverability) is proved for diffraction tomography with high-frequency, few-view and limited-angle SIMO/MISO measurements.