The main purpose of this paper is to investigate the structure of the weighted multifrequency multiple signal classification (MUSIC) type imaging function in order to improve the traditional MUSIC-type imaging. For this purpose, we devise a weighted multifrequency MUSIC-type imaging function and examine a relationship between weighted multifrequency MUSIC-type function and Bessel functions of integer order of the first kind. Some numerical results are demonstrated to support the survey.

Inverse problem, which deals with the reconstruction of cracks or thin inclusions in homogeneous material (or space) with physical features different from space, is of interest in a wide range of fields such as physics, engineering, and image medical science which are closely related to human life; refer to [

The noniterative algorithms such as multiple signal classification (MUSIC), subspace migrations, topological derivative, and linear sampling method can contribute to yielding the appropriate image as an initial guess. Previous attempts to investigate MUSIC-type algorithm presented various experiments with the use of MUSIC-type algorithm. For instance, the use of MUSIC-type algorithm for eddy-current nondestructive evaluation of three-dimensional defects [

This paper is organized as follows. In Section

In this section, we simplify surveying the two-dimensional direct scattering problem for the existence of perfectly conducting cracks and the single- and multifrequency MUSIC algorithm. For more information, see [

First, we consider the two-dimensional electromagnetic scattering by a perfectly conducting crack located in the homogeneous space

The far-field pattern is defined as function

Second, we present the traditional MUSIC-type algorithm for imaging of perfectly conducting cracks. For the sake of simplicity, we exclude the constant

We design multifrequency MUSIC-type imaging function and try to describe its structure. First, we introduce a multifrequency MUSIC-type algorithm

Assume that

So we can recognize the mathematical structure of multifrequency MUSIC-type algorithm. However, the finite representation of

In order to propose the weighted multifrequency MUSIC-type imaging algorithm, we introduce the following lemma derived from [

For sufficiently large

With this, let us define an alternative projection operator weighted by applied frequency

Assume that

The following equations are satisfied by the definition of

Next, we introduce a weighted multifrequency MUSIC-type imaging function based on MUSIC-type imaging function

Assume that

By Theorem

Hence, we can obtain

Therefore,

Looking at the results of Theorem

In this section, some numerical examples are displayed in order to support our analysis in the previous section. Applied frequencies are of the form

For illustrating arc-like cracks, three

It is worth emphasizing that all the far-field data

Figures

Shape reconstruction of

Map of

Map of

Same as Figure

Map of

Map of

Blue- and red-colored lines are

Map of

Graph of oscillating pattern

Map of

Graph of oscillating pattern

Figure

Same as Figure

Map of

Map of

Figure

Influence of noise. 30 dB ((a) and (b)) and 10 dB ((c) and (d)) white Gaussian random is added.

Map of

Map of

Map of

Map of

Now, we consider the imaging of oscillating crack. For this, we consider the following cracks (see Figure

Shape of oscillating cracks

Crack

Crack

Figure

Shape reconstruction of

Map of

Map of

Figure

Same as Figure

Map of

Map of

Based on the structure of multifrequency MUSIC-type imaging function, we introduced a weighted multifrequency MUSIC-type imaging function. Through a careful analysis, a relationship between imaging function and Bessel function of the first kind of integer order is established, and we have confirmed that the proposed imaging algorithm is an improved version of the traditional one.

Although, the proposed algorithm produces very good results and improves the traditional MUSIC algorithm, it still needs some upgrade, for example, imaging of highly oscillating cracks. Development of this should be an interesting and remarkable research project. In this paper, we considered the MUSIC algorithm in full-view inverse scattering problem. Based on the result in [

The author would like to express his thanks to Won-Kwang Park (Kookmin University) for his valuable discussions and MATLAB simulations to generate the forward data and MUSIC-type imaging function. The author 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.