Interpretation of MUSIC for Location Detecting of Small Inhomogeneities Surrounded by Random Scatterers

Abstract

We consider the MUltiple SIgnal Classification (MUSIC) algorithm for identifying the locations of small electromagnetic inhomogeneities surrounded by random scatterers. For this purpose, we rigorously analyze the structure of MUSIC-type imaging function by establishing a relationship with zero-order Bessel function of the first kind. This relationship shows certain properties of the MUSIC algorithm, explains some unexplained phenomena, and provides a method for improvements.

Academic Editor:Eric Florentin

Department of Mathematics, Kookmin University, Seoul 02707, Republic of Korea

Received 27 October 2015; Revised 5 January 2016; Accepted 6 January 2016

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

One of the purposes of the inverse scattering problem is to identify the characteristics (location, shape, material properties, etc.) of small inhomogeneities from the scattered field or far-field pattern. This problem, which arises in fields such as physics, engineering, and biomedical science, is highly relevant to human life; thus, it remains an important research area. Related works can be found in [1-5] and references therein.

Attempts to address the problem described above have led to the development of the MUltiple SIgnal Classification- (MUSIC-) type algorithm to find unknown inhomogeneities and the algorithm has been applied to various problems, for example, detection of small inhomogeneities in homogeneous space [6-9], location identification of small inhomogeneities embedded in a half-space or multilayered medium [10-12], reconstructing perfectly conducting cracks [13, 14], imaging of internal corrosion [15], shape recognition of crack-like thin inhomogeneities [16-18] and volumetric extended targets [19-21], and application to the biomedical imaging [22]. We also refer to [23, 24] for a detailed and concise description of MUSIC. Several research efforts have contributed to confirming that MUSIC is a fast and stable algorithm that can easily be extended to multiple inhomogeneities and that does not require specific regularization terms that are highly dependent on the problem at hand. However, its feasibility is only confirmed when the background medium is homogeneous; that is, the imaging performance of MUSIC when unknown inhomogeneities are surrounded by random scatterers remains unknown. In several works [25-28], an inverse scattering problem in random media has been concerned. Specially, mathematical theory of MUSIC for detecting point-like scatterers embedded in an inhomogeneous medium has been concerned in [29]. Motivated by these remarkable works, a more careful investigation of the mathematical theory is still required.

Motivated by the above, MUSIC algorithm has been applied for detecting the locations of small electromagnetic inhomogeneities when they are surrounded by electromagnetic random scatterers and confirmed that it can be applied satisfactorily. However, this only relied on the results of numerical simulations, that is, a heuristic approach to some extent, which is the motivation for the current work. In this contribution, we carefully analyze the mathematical structure of MUSIC-type imaging function and discover some properties. This work is based on the relationship between the singular vectors associated with nonzero singular values of a multistatic response (MSR) matrix and asymptotic expansion formula due to the existence of small inhomogeneities; refer to [23].

This paper is organized as follows. Section 2 introduces the two-dimensional direct scattering problem and an asymptotic expansion formula in the presence of small inhomogeneities. In Section 3, MUSIC-type imaging function is introduced. In Section 4, we analyze the mathematical structure of the MUSIC-type imaging function and discuss its properties. In Section 5, we present the results of numerical simulations to support the analyzed structure of MUSIC and Section 6 presents a short conclusion.

2. Two-Dimensional Direct Scattering Problem

In this section, we survey a two-dimensional direct scattering problem and introduce an asymptotic expansion formula. For a more detailed description we recommend [18, 23, 30]. Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , be an electromagnetic inhomogeneity with a small diameter [figure omitted; refer to PDF] in two-dimensional space [figure omitted; refer to PDF] . Throughout this paper, we assume that every [figure omitted; refer to PDF] is expressed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the location of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is a simple connected smooth domain containing the origin. For the sake of simplicity, we let [figure omitted; refer to PDF] be the collection of [figure omitted; refer to PDF] . Throughout this paper, we assume that inhomogeneities are well separated from each other such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Let us denote [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , as the random scatterer with small radius [figure omitted; refer to PDF] and let [figure omitted; refer to PDF] be the collection of [figure omitted; refer to PDF] . Similarly, we assume that [figure omitted; refer to PDF] is of the form [figure omitted; refer to PDF] As before, suppose that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] and the positions of [figure omitted; refer to PDF] are random but they are fixed for all frequencies discussed later.

In this work, we assume that every inhomogeneity is characterized by its dielectric permittivity and magnetic permeability at a given positive angular frequency [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the wavelength. Let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] be the electric permittivities of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , respectively. Then, we can introduce the piecewise-constant electric permittivity [figure omitted; refer to PDF] and magnetic permeability [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] respectively. For the sake of simplicity, we let [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence, we can set the wavenumber [figure omitted; refer to PDF] .

For a given fixed frequency [figure omitted; refer to PDF] , we denote [figure omitted; refer to PDF] to be a plane-wave incident field with the incident direction [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] denotes the two-dimensional unit circle. Let [figure omitted; refer to PDF] denote the time-harmonic total field that satisfies the following Helmholtz equation [figure omitted; refer to PDF] with transmission conditions on the boundaries of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . This configuration is associated with a scalar scattering problem for [figure omitted; refer to PDF] -polarized (Transverse Magnetic (TM) polarization, corresponding to dielectric contrasts) field; the [figure omitted; refer to PDF] -polarized (Transverse Electric (TE) polarization, corresponding to magnetic contrasts) case could be dealt with per duality. It is well known that [figure omitted; refer to PDF] can be decomposed as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the unknown scattered field that satisfies the Sommerfeld radiation condition [figure omitted; refer to PDF] uniformly in all directions [figure omitted; refer to PDF] . The far-field pattern [figure omitted; refer to PDF] of the scattered field [figure omitted; refer to PDF] is defined on [figure omitted; refer to PDF] . It can be expressed as [figure omitted; refer to PDF] Then by virtue of [31], the far-field pattern [figure omitted; refer to PDF] can be written as the following asymptotic expansion formula, which plays a key role in the MUSIC-type algorithm that will be designed in the next section: [figure omitted; refer to PDF]

3. MUSIC-Type Imaging Algorithm

In this section, we introduce the MUSIC-type algorithm for detecting the locations of small inhomogeneities. For the sake of simplicity, we exclude the constant term [figure omitted; refer to PDF] from (10). For this, let us consider the eigenvalue structure of the MSR matrix [figure omitted; refer to PDF] Suppose that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ; then [figure omitted; refer to PDF] is a complex symmetric matrix but not a Hermitian. Thus, instead of eigenvalue decomposition, we perform singular value decomposition (SVD) of [figure omitted; refer to PDF] (see [24], for instance) [figure omitted; refer to PDF] where superscript [figure omitted; refer to PDF] is the mark of a Hermitian. Then, [figure omitted; refer to PDF] is the orthogonal basis for the signal space of [figure omitted; refer to PDF] . Therefore, one can define the projection operator onto the null (or noise) subspace, [figure omitted; refer to PDF] . This projection is given explicitly by [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes the [figure omitted; refer to PDF] identity matrix. For any point [figure omitted; refer to PDF] and suitable vectors [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , define a test vector [figure omitted; refer to PDF] as [figure omitted; refer to PDF] Then, by virtue of [23], there exists [figure omitted; refer to PDF] such that, for any [figure omitted; refer to PDF] , the following statement holds: [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . This means that if [figure omitted; refer to PDF] or [figure omitted; refer to PDF] then [figure omitted; refer to PDF] . Thus, the locations of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] follow from computing the MUSIC-type imaging function [figure omitted; refer to PDF] The resulting plot of [figure omitted; refer to PDF] will have peaks of large magnitudes at [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Remark 1.

Based on several works [17, 18, 20], selection of [figure omitted; refer to PDF] in (14) is highly depending on the shape of [figure omitted; refer to PDF] . Unfortunately, the shape of [figure omitted; refer to PDF] is unknown; it is impossible to find proper vectors [figure omitted; refer to PDF] . Due to this fact, following [20], we assume that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ; that is, we consider the following test vector instead of (14): [figure omitted; refer to PDF] and we analyze the mathematical structure of [figure omitted; refer to PDF] .

4. Structure of Imaging Function

Henceforth, we analyze the mathematical structure of [figure omitted; refer to PDF] and examine certain of its properties. Before starting, we recall a useful result derived in [32].

Lemma 2.

Assume that [figure omitted; refer to PDF] spans [figure omitted; refer to PDF] . Then, for sufficiently large [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , the following relation holds: [figure omitted; refer to PDF] where [figure omitted; refer to PDF] denotes Bessel function of order [figure omitted; refer to PDF] of the first kind.

Now, we introduce the main result.

Theorem 3.

For sufficiently large [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] can be represented as follows: for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

Based on the asymptotic expansion formula (10) and results in [13], [figure omitted; refer to PDF] can be represented as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] With this, applying (18) and performing a tedious calculation, we arrive at [figure omitted; refer to PDF] where [figure omitted; refer to PDF] for [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . By implementing elementary calculus, we can show that [figure omitted; refer to PDF] where [figure omitted; refer to PDF] First, applying (18), we can obtain [figure omitted; refer to PDF] This leads us to [figure omitted; refer to PDF] and similarly to [figure omitted; refer to PDF]

Next, based on the orthonormal property of singular vectors, relations (2) and (18), and the following asymptotic form [figure omitted; refer to PDF] we can derive [figure omitted; refer to PDF] and similarly [figure omitted; refer to PDF]

For evaluating [figure omitted; refer to PDF] , let us perform an elementary calculus [figure omitted; refer to PDF] Then, we can conclude that [figure omitted; refer to PDF]

Finally, for [figure omitted; refer to PDF] , by applying following integral, for [figure omitted; refer to PDF] , [figure omitted; refer to PDF] we can derive the following: [figure omitted; refer to PDF] Correspondingly, [figure omitted; refer to PDF] Hence, by combining (27)-(36), we can obtain the following mathematical structure: [figure omitted; refer to PDF] This enables us to obtain the desired result. This completes the proof.

Remark 4 (applicability of MUSIC).

Since [figure omitted; refer to PDF] , the value of [figure omitted; refer to PDF] will be sufficiently large when [figure omitted; refer to PDF] or [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence, based on the result in Theorem 3, the locations of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] can be identified via the map of [figure omitted; refer to PDF] . This is the reason why it is possible to detect the locations of small inhomogeneities as well as random scatterers. Note that, for a successful detection, based on the hypothesis in Theorem 3, the value of [figure omitted; refer to PDF] (at least, greater than [figure omitted; refer to PDF] ) and [figure omitted; refer to PDF] must be sufficiently large enough. If applied frequency is low or total number of [figure omitted; refer to PDF] is small, poor result would appear in the map of [figure omitted; refer to PDF] .

Remark 5 (discrimination of singular values).

Theoretically, if the size, permittivity, and permeability of the random scatterers are smaller than those of the inhomogeneities, then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . This means that if it were possible to discriminate singular values associated with small inhomogeneities, then the structure of [figure omitted; refer to PDF] would become [figure omitted; refer to PDF] Hence, it is expected that more good results can be obtained. Our approach presents an improvement. However, if the relation [figure omitted; refer to PDF] were no longer valid, the locations of random scatterers would have to be identified via MUSIC such that poor results would appear in the map of [figure omitted; refer to PDF] .

5. Results of Numerical Simulations

Selected results of numerical simulations are presented here to support the identified structure of the MUSIC-type imaging function. In this section, we only consider the dielectric permittivity contrast case; that is, we set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . The radius of all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is set to [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. The applied angular frequency is [figure omitted; refer to PDF] and a total of [figure omitted; refer to PDF] number of incident directions is applied such that [figure omitted; refer to PDF]

[figure omitted; refer to PDF] small inhomogeneities are selected with locations [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] . We set [figure omitted; refer to PDF] number of small scatterers as being randomly distributed in [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] and also select the permittivities randomly as [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] , is an arbitrary real value within [figure omitted; refer to PDF] . Refer to Figure 1 for a sketch of the distribution of the three inhomogeneities and random scatterers.

Figure 1: Distribution of inhomogeneities (red-colored dots) and random scatterers (blue-colored "×" mark).

[figure omitted; refer to PDF]

The far-field elements of MSR matrix [figure omitted; refer to PDF] are generated by means of the Foldy-Lax framework to avoid an inverse crime . After the generation, a singular value decomposition of [figure omitted; refer to PDF] is performed via the MATLAB command svd. The nonzero singular values of [figure omitted; refer to PDF] are discriminated as follows: first, a [figure omitted; refer to PDF] -threshold scheme (by first choosing the [figure omitted; refer to PDF] singular values [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] ) is applied based on [18], and second, the first [figure omitted; refer to PDF] -singular values are selected.

Figure 2 exhibits the distribution of the normalized singular values of [figure omitted; refer to PDF] and maps of [figure omitted; refer to PDF] with the [figure omitted; refer to PDF] -threshold scheme and with selection of the first [figure omitted; refer to PDF] -singular values when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Note that due to the huge number of artifacts it is very hard to identify the locations of [figure omitted; refer to PDF] with the [figure omitted; refer to PDF] -threshold scheme but, fortunately in this example, one can discriminate three nonzero singular values such that, based on Remark 5, the locations of [figure omitted; refer to PDF] can be identified more clearly. This result supports the derived mathematical structure in Theorem 3.

Figure 2: Distribution of normalized singular values (a, c) and maps of [figure omitted; refer to PDF] with first [figure omitted; refer to PDF] -singular values (b) and with [figure omitted; refer to PDF] -threshold scheme (d).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

Now, let us examine the effect of total number of directions [figure omitted; refer to PDF] in the extreme cases. Figure 3 exhibits normalized singular values and map of [figure omitted; refer to PDF] with small number of [figure omitted; refer to PDF] when [figure omitted; refer to PDF] . Based on Remark 4, the value of [figure omitted; refer to PDF] must be sufficiently large so, as we expected, locations of [figure omitted; refer to PDF] cannot be identified via the map of [figure omitted; refer to PDF] with small [figure omitted; refer to PDF] .

Figure 3: Distribution of normalized singular values (a) and map of [figure omitted; refer to PDF] with first [figure omitted; refer to PDF] -singular values (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

Opposite to the previous result, Figure 4 displays normalized singular values and maps of [figure omitted; refer to PDF] with large number of [figure omitted; refer to PDF] when [figure omitted; refer to PDF] . Similar to the results in Figure 2, locations of [figure omitted; refer to PDF] can be examined clearly via the selection of first [figure omitted; refer to PDF] -singular values. Applying [figure omitted; refer to PDF] -threshold, it is very hard to identify locations of [figure omitted; refer to PDF] but, opposite to the result in Figure 2, their locations can be recognized even though some artifacts still exist.

Figure 4: Distribution of normalized singular values (a, c) and maps of [figure omitted; refer to PDF] with first [figure omitted; refer to PDF] -singular values (b) and with [figure omitted; refer to PDF] -threshold scheme (d).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

On the basis of recent works [13, 20], it has been confirmed that MUSIC is robust with respect to the random noise. In order to examine the robustness, assume that [figure omitted; refer to PDF] Gaussian random noise is added to the unperturbed data [figure omitted; refer to PDF] . Throughout results in Figure 5 when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , although some blurring appears in the map of [figure omitted; refer to PDF] , we can easily find proper singular values and obtain an accurate image. It is interesting to observe that, opposite to the results in Figure 2, locations of [figure omitted; refer to PDF] can be detected despite existence of some artifacts.

Figure 5: Distribution of normalized singular values (a, c) and maps of [figure omitted; refer to PDF] with first [figure omitted; refer to PDF] -singular values (b) and with [figure omitted; refer to PDF] -threshold scheme (d) when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , and collected far-field data is perturbed by a white Gaussian random noise.

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

(c) [figure omitted; refer to PDF]

(d) [figure omitted; refer to PDF]

From the above results, we can examine that, by having small perturbations of random scatterers [figure omitted; refer to PDF] , their effects to the scattered fields are quite small so that [figure omitted; refer to PDF] can be discriminated very accurately. Opposite to the this examination, let us consider the effect of [figure omitted; refer to PDF] when their size and permittivities satisfy [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively (remember that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] ). In this example, it is very hard to discriminate nonzero singular values associated with [figure omitted; refer to PDF] so that it is impossible to detect their exact locations; refer to Figure 6 when [figure omitted; refer to PDF] and [figure omitted; refer to PDF] .

Figure 6: Distribution of normalized singular values (a) and map of [figure omitted; refer to PDF] with [figure omitted; refer to PDF] -threshold scheme (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

It is well-known that using multifrequency improves the imaging performance; refer to [13, 32-34]. At this moment, we consider multifrequency MUSIC-type imaging in order to compare the imaging performance against the traditional single-frequency one. For given [figure omitted; refer to PDF] -different frequencies [figure omitted; refer to PDF] , SVD of MSR matrix [figure omitted; refer to PDF] is [figure omitted; refer to PDF] Then, by choosing test vector [figure omitted; refer to PDF] we can survey the projection operator onto the null (or noise) subspace such that [figure omitted; refer to PDF] and correspondingly multifrequency MUSIC-type imaging function [figure omitted; refer to PDF] can be introduced as [figure omitted; refer to PDF]

Figure 7 shows maps of [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Here, [figure omitted; refer to PDF] directions are applied and [figure omitted; refer to PDF] are equidistributed in the interval [figure omitted; refer to PDF] with [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . By comparing results in Figure 2, we can observe that unexpected artifacts have been eliminated so that applying multiple frequencies yields a more accurate result and then single frequency.

Figure 7: Maps of [figure omitted; refer to PDF] with first [figure omitted; refer to PDF] -singular values (a) and with [figure omitted; refer to PDF] -threshold scheme (b).

(a) [figure omitted; refer to PDF]

(b) [figure omitted; refer to PDF]

6. Concluding Remarks

The mathematical structure of MUSIC-type imaging function is carefully identified by establishing a relationship with integer ordered Bessel functions. This is based on the fact that the elements of the MSR matrix can be expressed by an asymptotic expansion formula. The identified structure explains some unexplained phenomena and provides a method for improvements.

Based on recent work [7], the electric field [figure omitted; refer to PDF] in the existence of small inhomogeneity with radius [figure omitted; refer to PDF] can be expressed as follows: [figure omitted; refer to PDF] where electromagnetic fields [figure omitted; refer to PDF] are the solutions of the Maxwell equations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is Green's function [figure omitted; refer to PDF] Thus, by applying above asymptotic expansion formula and through the similar process in Theorem 3, the result in this paper can be extended to the three-dimensional problem so that MUSIC will be applicable for detecting three-dimensional inhomogeneities surrounded by random scatterers.

In comparison with the MUSIC, other closely related reconstruction algorithms such as linear sampling method [35-37], subspace migration [32, 33, 38], and direct sampling method [39-41] will be applicable for detecting inhomogeneities in random medium. Analysis of imaging functions and exploring their certain properties will be the forthcoming work.

Acknowledgments

This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (no. NRF-2014R1A1A2055225) and the research program of Kookmin University in Korea.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

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