Applications of random matrix theory for sensor array imaging with measurement noise

Summary

The imaging of a small target embedded in a medium is a central problem in sensor array imaging. The goal is to find a target embedded in a medium. The medium is probed by an array of sources, and the signals backscattered by the target are recorded by an array of receivers. The responses between all pairs of source and receiver are collected so that the available information takes the form of a response matrix. When the data are corrupted by additive measurement noise we show how tools of random matrix theory can help to detect, localize, and characterize the target.

1. Introduction

The imaging of a small target embedded in a medium is a central problem in wave sensor imaging [Angelsen 2000; Stergiopoulos 2001]. Sensor array imaging involves two steps. The first step is experimental, it consists in emitting waves from an array of sources and recording the backscattered signals by an array of receivers. The data set then consists of a matrix of recorded signals whose indices are the index of the source and the index of the receiver. The second step is numerical, it consists in processing the recorded data in order to estimate the quantities of interest in the medium, such as reflector locations. The main applications of sensor array imaging are medical imaging, geophysical exploration, and nondestructive testing.

Recently it has been shown that random matrix theory could be used in order to build a detection test based on the statistical properties of the singular values of the response matrix [Aubry and Derode 2009a; 2009b; 2010; Ammari et al. 2011; 2012]. This paper summarizes the results contained in [Ammari et al. 2011; 2012] and extends them into several important directions. First we address in this paper the case in which the source array and the receiver array are not coincident, and more generally the case in which the number of sources is different from the number of receivers. As a result the noise singular value distribution has the form of a deformed quarter circle and the statistics of the singular value associated to the target is also affected. Second we study carefully the estimation of the noise variance of the response matrix. Different estimators are studied and an estimator that achieves an efficient trade-off between bias and variance is proposed.