Contemporary Mathematics
Use of random matrix theory for target detection,
localization, and reconstruction
Josselin Garnier
Abstract. The detection, localization, and characterization of a target em-
bedded in a medium is an important problem in wave sensor imaging. The
responses between each pair of source and receiver are collected so that the
available information takes the form of a response matrix between the source
array and the receiver array. When the data are corrupted by additive noise
we show how the target can be efficiently detected, localized and characterized
using recent tools of random matrix theory.
1. Introduction
The detection, localization, and characterization of a target embedded in a
medium is an important problem in wave sensor imaging [8, 22]. Sensor array
imaging involves two steps. The first step is experimental, it consists in recording
the waves generated by the sources and received by the sensors. The data set
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 some relevant features of the medium (reflector
locations,. . . ). The main applications of sensor array imaging are medical imaging,
geophysical exploration, and non-destructive 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 [9, 10, 11, 1, 2]. This paper extends the results contained
in [1, 2] 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 present a detailed analysis of the critical case
when the singular value associated to the target is at the edge of the deformed
2000 Mathematics Subject Classification. 78A46, 15B52.
Key words and phrases. Imaging, random matrix theory.
This work was supported by National Institute for Mathematical Sciences (2010 Thematic
Program, TP1003). We thank Habib Ammari, Hyeonbae Kang, and Knut Sølna for useful and
stimulating discussions during the summer 2010 that we spent at Inha University.
c 0000 (copyright holder)
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Contemporary Mathematics
Volume 548, 2011
c 2011 American Mathematical Society
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