1. Introduction
The main purpose of microwave imaging is to retrieve the parameter (permittivity or conductivity) distribution in a domain from scattering parameter data. This is an old and difficult problem due to its intrinsic ill-posedness and nonlinearity [
1]. Nevertheless, it is a very important problem to current scientists and engineers in developing reliable tools and techniques that can be applied to real-world problems such as medical imaging [
2,
3,
4], damage detection in civil structure [
5,
6,
7], radar imaging [
8,
9,
10], etc. In order to solve this problem, various techniques have proposed various reconstruction algorithms. For example, Newton method [
11] for reconstructing crack shape, Gauss–Newton method for 3D imaging [
12] and biomedical imaging [
13], Levenberg–Marquadt algorithm for detection and monitoring of leukemia [
14] and reconstructing permittivity distribution [
15], level-set technique [
16] for inverse scattering problem, and optimal control approach [
17] for reconstructing extended targets.
For a successful application of iterative-based techniques, one must begin the iteration procedure with a good initial guess. Otherwise, they will encounter the nonconvergence issue or the occurrence of a local minimizer. Moreover, a huge amount of computational cost will be required to perform the large number of iteration procedures; refer to [
18]. Hence, it is natural to consider the generation of a good initial guess before the iteration procedure. Due to this reason, various noniterative techniques have been investigated and applied to inverse scattering problems and microwave imaging, for example, MUltiple SIgnal Classification (MUSIC) algorithm [
19,
20], Kirchhoff and subspace migrations [
21,
22], Factorization method [
23,
24], topological sensitivity [
25,
26], and linear sampling method [
27,
28].
Direct sampling method (DSM) is a fast, effective, and stable noniterative technique for identifying the location or the outline shape of small or extended targets related to various inverse problems. As such, DSM is currently widely applied in many fields, including the identification of two- and three-dimensional small scatterers [
29,
30,
31,
32,
33,
34] or perfectly conducting cracks [
35], diffusive optical tomography [
36], electrical impedance tomography [
37], detecting sources in a stratified ocean waveguide [
38], phaseless inverse source scattering problem [
39], and mono-static imaging [
40].
Recently, the application of DSM has been extended to microwave imaging that uses synthetic [
41] and real [
42] data for a noniterative identification of small anomalies from the collected scattering parameter data. The rapid identification of unknown anomalies is crucial in microwave imaging and DSM can be regarded as a suitably fast microwave imaging technique because it has certain advantages, such as the requirement of only a few (e.g., one or two) sources for generating incident fields and the low computational cost for performing the imaging procedure. However, difficulties can emerge in terms of identifying why inaccurate results had been obtained (see [
41,
43]). Heuristically, it has turned out that this phenomenon is due to the coupling effect between the anomaly and antennas, and the distance between the transmitter and the anomaly. However, a related mathematical theory of this phenomenon has yet to be satisfactorily established because previous studies do not consider the coupling effect between teh anomaly and antennas. Motivated by this issue, we attempted to identify the factors that influence the measurement data and correspondingly introduced another indicator function of DSM for a better imaging performance by removing certain scattering parameter data influenced by the coupling effect. This is based on the hypothesis that there exists a coupling effect between anomaly and antennas. We subsequently determined the mathematical structure of both the traditional and the introduced indicator functions by establishing relationships among the Bessel functions of the integer order of the first kind, the antenna setting, and the material properties. The theoretical results revealed the reason behind the appearance of inaccurate results using the traditional DSM and allowed for achieving a better imaging performance with the introduced indicator function.
Various simulation results from synthetic data generated by the CST STUDIO SUITE demonstrate the theoretical result and explored behaviors of traditional and designed DSM. It is worth emphasizing that once the outline shape of the anomaly is recognized, it can be selected as a good initial guess and one can retrieve a complete shape via iteration-based schemes.
The remainder of this paper is organized as follows. In
Section 2, we outline the basic concept of the scattering parameter, introduce the traditional indicator functions of DSM, and design another DSM indicator function for improving the imaging performance. In
Section 3, we investigate the structure of the indicator functions by establishing relationships among the Bessel functions of the integer order of the first kind, the antenna configuration, and the material properties. In
Section 4, a set of simulation results with synthetic data are presented to support the investigation. A short conclusion including an outline of the current and future work is given in
Section 5.
2. Scattering Parameter and Indicator Functions of Direct Sampling Method
Suppose that there exists a circular cylindrical obstacle with infinite length in the vertical direction (parallel to the
z-axis) in a given region of interest (ROI) and it is surrounded by a number of dipole antennas in the vertical direction. Then, based on mathematical treatment of the scattering of time-harmonic electromagnetic waves from thin infinitely long cylindrical obstacles, this can be considered the two-dimensional inverse problem. An illustration of the experimental setup and the two-dimensional cross-section of the obstacle is given in
Figure 1; refer to [
20] for a detailed description.
We denote
as the 2D cross section of a cylindrical obstacle with radius
and location
such that
where
denotes the two-dimensional unit circle centered at the origin (in general,
is a simply connected domain with smooth boundary that describes the shape of
) and
is the dipole antenna located at
,
to transmit or receive signals. Here,
denotes the homogeneous background, which is the intersection between the ROI and
, and
is the exterior of
. Throughout this paper, we consider the single-source case, i.e., an antenna
is used for signal transmission only and the antennas
,
are used for signal reception.
All materials involved are nonmagnetic i.e., they are characterized by their dielectric permittivity and electrical conductivity at a given angular frequency
. Correspondingly, we set the value of magnetic permeability to be constant at every location
such that
. Meanwhile, we denote
and
as the permittivity of
and the
, respectively, where
is the vacuum permittivity. The conductivities
and
could then be defined analogously. Following this, we introduce the piecewise constant permittivity
and conductivity
as follows:
respectively. With this, let
k be the background wave number that satisfies
and assume that
.
Let us denote
as the
S-parameter (or scattering parameter) defined as
where
denotes the input voltage (or incident wave) at
and
is the corresponding output voltage (or reflected wave) at
. We also denote
and
as the total and incident
S-parameters in the presence and absence of
. Throughout this paper, the measurement data are the scattered-field
S-parameter defined as
Note that this subtraction is essential to remove unknown modeling errors and is useful in designing an indicator function because it can be expressed as the following integral equation:
where
denotes the objective function
denote the
z-component of the incident field
in a homogeneous medium due to the point current density at
, and
is the
z-component of the total field
. Notice that, based on the Maxwell equation, the incident field
satisfies
and the corresponding total field
satisfies
with a transmission condition at the boundary
. Here,
and
denote the magnetic fields defined analogously.
At this moment, we cannot use
to design an indicator function because the total field
of (
1) cannot be formulated without a priori information of
. Now, let us assume that the cross section
is a small ball such that
where
denotes the background wavelength. Then, based on [
44], it is possible to apply the Born approximation so that the total field
can be approximated by the incident field
. With this, on the basis of the reciprocity property of the incident field,
can be approximated as follows:
Let
be the set of measurement data
and
be the unit vector, which is the arrangement of measurement datain
:
where the inner product and corresponding norm are defined as
respectively. Then, based on (
2),
can be written by
Based on the above expression, let us define the following unit vector: for
Then, by testing orthonormality relation between
and
, it will be possible to extract
so that the location of
can be identified. To this end, the typical indicator function
has been designed as follows (see [
29,
30,
31,
33]):
Then, it is expected that the map will contain a peak of largest magnitude 1 at and a small magnitude at so that the location can be identified via the map of .
However, judging by the simulation results presented in
Section 4, the imaging performance of
is somehow poor. Note that the imaging performance of
significantly depends on the location of the source
. Moreover, if the antenna
is used for both signal transmission and reception, the measurement data
will be influenced not only by the anomaly but also by the other antennas
,
. In contrast, if
,
is influenced by the anomaly only. For a detailed description, refer to ([
21] Section 1). Hence, it is feasible to design a new indicator function by disregarding the measurement data
, i.e., an antenna
is used for signal transmission only and the antennas
,
and
, are used for signal reception. With this, let us introduce the set of measurement data
and an arrangement of measurement data:
where the inner product and corresponding norm are defined as
respectively. Based on the structure of
, let us introduce the following unit vector: for
and corresponding indicator function
such that
Then, it is expected that the imaging performance of is better than the traditional performance of .
3. Theoretical Results and Related Discussion
To compare the imaging performance of and , we established a mathematical structure of the indicator functions, as outlined below.
Theorem 1 (Structure of the indicator functions with single source).
Let , , , and for all n. If satisfies for and , , then the following relations hold uniformly: Here, denotes the Bessel function of integer order s of the first kind, with is the set of integer numbers, , and for all n.
Proof. Let us recall that, since
is a small anomaly,
is given by (
2):
when
. If
, then because the measurement data are influenced by the antennas
,
and
,
can also be regarded as anomalies with permittivity
and conductivity
that significantly depend on the applied frequency. Generally, all antennas are the same size and made of the same material; it is feasible to assume that
and
for all
n. It should be noted that, because the size of antenna
is small enough, it is possible to apply the Born approximation to (
1) such that
Thus, applying (
2) and (
9) to (
3), we can examine
As demonstrated in ([
1] Theorem 2.5), given that for
,
and the following Jacobi–Anger expansion holds uniformly
we can derive
Here,
denotes the Hankel function of order zero of the first kind. Thus, we can examine
Similarly, since
we can derive
By applying (
12), (
13), and Hölder’s inequality
we can obtain (
6).
To derive (
7), let us apply (
8) and (
10) to (
4). Then since
similar to the derivation of (
13), we have
and by applying Hölder’s inequality
we can obtain (
7), thereby completing the proof of theory. □
Now, let us discuss some properties of and based on the result in Theorem 1.
Remark 1 (Performance of the indicator functions).
Based on (6) and (7), the imaging performance of is significantly affected by
- ①
the material properties of the anomaly and the antennas due to the factors and ;
- ②
the antenna configuration such as total number (factors N and ) and arrangement (factors and );
- ③
the applied frequency (factors k and ω); and
- ④
the location of the transmitter and the distance between the transmitter and the anomaly due to the factors of .
Notice that the factor ④ is due to the coupling effect so that the imaging performance of traditional DSM is significantly influenced by the coupling effect. However, the imaging performance of the is independent from the factor ④, which means that, instead of using , a good result can be obtained via the map of .
Remark 2 (Performance of the indicator functions with multiple frequencies).
Generally, the application of multiple frequencies should guarantee a good imaging result [32,45,46]. However, based on ④ of Remark 1, the application of multiple frequencies to will not guarantee a good imaging performance, while it is expected that such an application of will. This is the theoretical reasoning behind why negative results have surfaced [43]. 4. Simulation Results and Discussion
To demonstrate the theoretical results and to compare the imaging performance of
and
, simulation results with synthetic data are presented in this section. To this end,
dipole antennas with a location of
were selected. Meanwhile, the background was selected as a homogeneous medium with
and
, while the ROI
was set to the interior of a circle with a diameter of
centered at the origin. We then selected an anomaly
as a circle with the following properties: diameter =
, location
, permittivity
, and conductivity
. We refer to
Figure 1 for illustration. The measurement data
and incident field data
for every
were generated using CST STUDIO SUITE.
Example 1 (Simulation Result at
).
Figure 2 shows the maps of and for at . As discussed above, the identified location of via the map of is not exactly in line with the actual location. Moreover, for each m, the identified locations of the anomaly obtained via were different. This supports the observations discussed in Remark 1. In contrast, the identified location of via the map of is very close to the actual location and is independent from the location of the transmitter. Hence, we could assess the imaging performance of the in relation to that of the . Example 2 (Simulation Result at
).
Figure 3 shows the maps of and for at . When comparing these with those presented in Figure 2, it is clear that, even if the location of the transmitter is the same, the identified location of the anomaly is different. Meanwhile, almost the same result can be obtained via the map of with a different frequency. Example 3. Figure 4 shows the maps of and for at . Notice that, although the distance between and is close, the identified location of via the map of is not accurate. Additionally, it is very hard and easy to identify the locations of via the maps of at and , respectively. Thus, as we mentioned in Remark 1, the imaging performance of significantly depends on the applied frequency and distance between the anomaly and the transmitter. Meanwhile, fortunately, the identified location of via the map of is very accurate to the actual location and is independent of the location of the transmitter. Hence, on the basis of the results in Figure 2 and Figure 3, we can conclude that imaging performance of is better in relation to that of the . Remark 3 (Discussion of Examples 1, 2, and 3).
Based on the results in Examples 1, 2, and 3, we can examine that the imaging performance of is better than the one of . Moreover, as we observed ③ in Remark 1, the imaging performance of is significantly dependent on the applied frequency. The imaging performance of is also influenced by the applied frequency; however, recognization of the location of anomaly is very stable.
Example 4. Figure 5 presents the results of the multi-frequency imaging of and for with a frequency band of . As was discussed in Remark 2, the identified location via the multi-frequency is not accurate in the case of certain , which means that there is no improvement to the imaging performance. However, the location of the anomaly can be identified clearly via the multi-frequency because the magnitudes of several artifacts were reduced successfully. Throughout the theoretical and simulation results, we can examine that established structures (
5) and (
6) prove the main research question about the coupling effect and provide answers to some phenomena that cannot be explained via previous research, and various imaging results of
successfully support the theoretical result. Moreover, the established structures (
5) and (
7) illustrate the improvement in imaging performance, and various imaging results of
show not only the verification of theoretical results but also the stability.
5. Concluding Remarks
In this contribution, two different indicator functions of DSM were introduced and designed for the fast identification of small anomalies from collected scattered-field S-parameters. To explain the influence of the coupling effect of traditional DSM and the improvement in the imaging performance of the designed indicator function, mathematical structures of traditional and designed indicator functions were analyzed by establishing a relationship between an infinite series of Bessel functions of integer order of the first kind and the antenna configuration.
We presented various simulation results from synthetic data computed by CST-STUDIO SUITE, and we examined that the imaging performance of traditional DSM is significantly influenced by the coupling effect, the designed indicator function is independent from the coupling effect and successfully improves the traditional one. Moreover, we confirmed that the designed DSM is very fast, stable, and effective for identifying small anomalies in microwave imaging. It is worth noticing that the identified shape of anomaly does not guarantee the accurate shape. Fortunately, it can be regarded as an initial guess and one can evolve it to retrieve a better shape via the iterative schemes. Therefore, it will be possible to examine that only a few iteration procedures are required, i.e., it will not require tremendous computations.
Here, we considered the two-dimensional problem. Following to [
30], we expect that the designed indicator function can be extended to the more-realistic three-dimensional microwave imaging. It has been confirmed that DSM can be applied to the limited-aperture inverse scattering problem. The application of DSM in real-world limited-aperture microwave imaging and designing an improved DSM will be interesting research topics.