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)

1

Contemporary Mathematics

Volume 548, 2011

c 2011 American Mathematical Society

1

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)

1

Contemporary Mathematics

Volume 548, 2011

c 2011 American Mathematical Society

1