MUSIC-type imaging of perfectly conducting cracks in limited-view inverse scattering problems

doi:10.1016/j.amc.2014.04.097

Applied Mathematics and Computation

Volume 240, 1 August 2014, Pages 273-280

https://doi.org/10.1016/j.amc.2014.04.097 Get rights and content

Abstract

Although standard MUltiple SIgnal Classification (MUSIC) algorithm has been considered a promising non-iterative imaging technique for cracks, its application is restricted to the full-view inverse scattering problems. Many experimental results revealed that MUSIC can be applied to the imaging of perfectly conducting cracks with small length in limited-view inverse scattering problem. On the other hand, MUSIC is not applicable for imaging of extended cracks in limited-view problems but the reason behind this restricted application has not been theoretically investigated. This gave an impetus for this study to attempt to identify the structure of MUSIC-type algorithm that appears in the limited-view inverse scattering problems by establishing a relationship between MUSIC and Bessel functions of integer order of the first kind. Some numerical experiments are illustrated to support the identified structure of MUSIC.

Introduction

Inverse scattering problem for reconstructing the shape of perfectly conducting cracks hidden in a structure (such as a bridge or a concrete wall) is an old problem which, nevertheless, still poses great challenges to nowaday scientists for developing novel tools and techniques which can be applied to this problem because this is closely related to human life, refer to [2], [3], [6] and references therein. In order to solve this kind of inverse problem, many authors have proposed various reconstruction algorithms. As we can observe in various works [1], [13], [19], [23], [26], [28], [31], Newton-type iteration scheme has been considered in a long period of research. However, it requires a complex calculation of the so called Fréchet derivative, suitable regularization terms for each iteration step, and a priori information of target, e.g., material properties, locations, and so on. Additionally, a good initial guess close to what is intended to be found must be created in order to guarantee a successful reconstruction.

Motivated by this fact, various non-iterative algorithms such as MUltiple SIgnal Classification (MUSIC), topological derivative, linear sampling, and Kirchhoff migrations have been developed. Among them, MUltiple SIgnal Classification (MUSIC) algorithm was developed for locating point-like scatterers [11], and generalized to electromagnetic imaging various kind of targets such as perfectly conducting cracks, internal corrosion, breast cancer, refer to [5], [7], [8], [9], [10], [12], [14], [16], [24], [25], [29], [30], [32]. Although one can obtain a good result or initial guess via MUSIC, its application is highly restricted to the full-view inverse scattering problems. Based on many experimental researches, in the limited-view inverse problems, locations of point-like scatterers can be localized but desirable shape of extended targets cannot be obtained via MUSIC algorithm. In recent works [17], [18], the structure of MUSIC was identified in full-view inverse scattering problem, but as to the limited-view problem, little has been theoretically studied. This gives an impetus for this study to analyze MUSIC-type imaging algorithm in limited-view inverse scattering problem.

The purpose of this paper is to analyze the structure of MUSIC for imaging of perfectly conducting cracks with small length in limited-view inverse scattering problem by constructing a relationship with Bessel functions of integer order of the first kind. This is based on the structure of left-singular vector of so-called Multi-Static Response (MSR) matrix, and the fact that collected far-field pattern can be represented as an asymptotic expansion formula due to the presence of such cracks. From the identified structure, we can examine certain properties of MUSIC-type imaging in limited-view problem.

This paper is organized as follows. In Section 2, we introduce direct scattering problem and MUSIC-type imaging algorithm. Section 3 brings forth the analysis on the structure of MUSIC-type imaging algorithm in the limited-view inverse scattering problem. In Section 4, we present several numerical experiments to support this result and Section 5 presents a short conclusion.

Section snippets

Direct scattering problems and MUSIC algorithm

First, we consider the two-dimensional electromagnetic scattering by $M$ -different linear perfectly conducting cracks with the same length $2 ℓ$ , denoted by $Γ_{m}, m = 1, 2, \dots, M$ , located in the homogeneous space $R^{2}$ . Detailed description can be found in [8], [19]. Throughout this paper, we represent $Γ_{m}$ such that $Γ_{m} = \{y_{m} = {[x_{m}, y_{m}]}^{T} : - ℓ ⩽ x_{m} ⩽ ℓ\},$ for $m = 1, 2, \dots, M$ , and let $Γ$ be the collection of cracks. In this paper, we assume that $Γ_{m}$ are sufficiently separated from each other. Let $u_{inc} (x) = e^{ik θ \cdot x}$ and $u (x)$ be the given

Structure of MUSIC imaging function in the limited-view problem

In order to perform an analysis of MUSIC-type imaging, we must explore the structure of left-singular vectors $U_{m}$ . Note that based on the asymptotic expansion formula (4), we can observe that the $jl$ th element of (5) can be written as $u_{\infty} (ϑ_{j}; θ_{l}) |_{ϑ_{j} = - θ_{j}} = - \frac{2 π}{\ln (ℓ / 2)} \sum_{m = 1}^{M} e^{ik (θ_{j} + θ_{l}) \cdot y_{m}} .$ Hence, we can observe the following result.

Lemma 3.1

[See [4]] For $θ_{n} \in S^{1}$ , define $D (y) \in C^{N \times 1}$ as $D (y) = \frac{1}{\sqrt{N}} {[e^{ik θ_{1} \cdot y}, e^{ik θ_{2} \cdot y}, \dots, e^{ik θ_{N} \cdot y}]}^{T} .$ Then, the left singular vectors $U_{m}$ of the MSR matrix $K$ is of the form $U_{m} \approx D (y_{m}) for m = 1, 2, \dots, M .$

And we

Numerical simulations

In this section, some numerical results are exhibited to support the identified structure of Theorem 3.3. For this, three small cracks are chosen: $\begin{matrix} Γ_{1} & = \{{[x - 0.6, - 0.2]}^{T} : - ℓ ⩽ x ⩽ ℓ\} \\ Γ_{2} & = \{R_{π / 4} {[x + 0.4, x + 0.35]}^{T} : - ℓ ⩽ x ⩽ ℓ\} \\ Γ_{3} & = \{R_{7 π / 6} {[x + 0.25, x - 0.6]}^{T} : - ℓ ⩽ x ⩽ ℓ\} . \end{matrix}$ Here, $R_{θ}$ denotes the rotation by $θ$ and $ℓ = 0.03$ . We apply $N = 12$ to different incident and observation directions $θ_{n} = - {[\cos θ_{n}, \sin θ_{n}]}^{T}, θ_{n} = θ_{1} + (θ_{N} - θ_{1}) \frac{n - 1}{N - 1}$ with $θ_{1} = π / 4$ and $θ_{N} = 3 π / 4$ for the limited-view configuration. For full-view configuration, we set $θ_{1} = 0$ and $θ_{N} = 2 π$ . Throughout

Conclusions

The structure of MUSIC-type imaging function for imaging of small perfectly conducting cracks in the limited-view inverse scattering problems is considered. Based on its relationship with Bessel functions of integer order of the first kind, we have confirmed the reason of effectiveness of MUSIC algorithm in the limited-view inverse scattering problems. The certain properties and limitations have been revealed. In terms of imaging the small cracks, MUSIC is better than subspace migration,

Acknowledgments

We would like to acknowledge two anonymous referees for their precious comments. This work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (No. 2011-0007705), and the research program of Kookmin University in Korea.

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View full text

MUSIC-type imaging of perfectly conducting cracks in limited-view inverse scattering problems

Abstract

Introduction

Section snippets

Direct scattering problems and MUSIC algorithm

Structure of MUSIC imaging function in the limited-view problem

Numerical simulations

Conclusions

Acknowledgments

J. Comput. Phys.

Appl. Math. Lett.

Appl. Numer. Math.

J. Comput. Phys.

Comput. Phys. Commun.

J. Math. Anal. Appl.

J. Comput. Phys.

An Introduction to Mathematics of Emerging Biomedical Imaging

Mathematical Modeling in Biomedical Imaging II: Optical, Ultrasound, and Opto-Acoustic Tomographies

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SIAM J. Appl. Math.

A MUSIC algorithm for locating small inclusions buried in a half-space from the scattering amplitude at a fixed frequency

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