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.

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 [

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 [

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 [

This paper is organized as follows. Section

In this section, we survey a two-dimensional direct scattering problem and introduce an asymptotic expansion formula. For a more detailed description we recommend [

Let us denote

In this work, we assume that every inhomogeneity is characterized by its dielectric permittivity and magnetic permeability at a given positive angular frequency

For a given fixed frequency

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

Based on several works [

Henceforth, we analyze the mathematical structure of

Assume that

Now, we introduce the main result.

For sufficiently large

Based on the asymptotic expansion formula (

Next, based on the orthonormal property of singular vectors, relations (

For evaluating

Finally, for

Since

Theoretically, if the size, permittivity, and permeability of the random scatterers are smaller than those of the inhomogeneities, then

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

Distribution of inhomogeneities (red-colored dots) and random scatterers (blue-colored “×” mark).

The far-field elements of MSR matrix

Figure

Distribution of normalized singular values (a, c) and maps of

Now, let us examine the effect of total number of directions

Distribution of normalized singular values (a) and map of

Opposite to the previous result, Figure

Distribution of normalized singular values (a, c) and maps of

On the basis of recent works [

Distribution of normalized singular values (a, c) and maps of

From the above results, we can examine that, by having small perturbations of random scatterers

Distribution of normalized singular values (a) and map of

It is well-known that using multifrequency improves the imaging performance; refer to [

Figure

Maps of

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 [

In comparison with the MUSIC, other closely related reconstruction algorithms such as linear sampling method [

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

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.