International Journal of Antennas and Propagation

Volume 2015, Article ID 129823, 18 pages

http://dx.doi.org/10.1155/2015/129823

## A Simple Quantitative Inversion Approach for Microwave Imaging in Embedded Systems

^{1}University of Naples “Parthenope”, 80143 Naples, Italy^{2}IREA, National Research Council of Italy, 80124 Naples, Italy

Received 20 February 2015; Accepted 31 May 2015

Academic Editor: Felipe Cátedra

Copyright © 2015 M. Ambrosanio et al. 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.

#### Abstract

In many applications of microwave imaging, there is the need of confining the device in order to shield it from environmental noise as well as to host the targets and the medium used for impedance matching purposes. For instance, in MWI for biomedical diagnostics a coupling medium is typically adopted to improve the penetration of the probing wave into the tissues. From the point of view of quantitative imaging procedures, that is aimed at retrieving the values of the complex permittivity in the domain under test, the presence of a confining structure entails an increase of complexity of the underlying modelling. This entails a further difficulty in achieving real-time imaging results, which are obviously of interest in practice. To address this challenge, we propose the application of a recently proposed inversion method that, making use of a suitable preprocessing of the data and a scenario-oriented field approximation, allows obtaining quantitative imaging results by means of quasi-real-time linear inversion, in a range of cases which is much broader than usual linearized approximations. The assessment of the method is carried out in the scalar 2D configuration and taking into account enclosures of different shapes and, to show the method’s flexibility different shapes, embedding nonweak targets.

#### 1. Introduction

Microwave imaging (MWI) is an emerging and appealing technique to retrieve morphological and quantitative maps of the electromagnetic properties of notaccessible domains. The physical phenomenon which MWI is based on is the electromagnetic scattering, which is due to differences in the electromagnetic properties of the targets with respect to those of the background. Based on this principle, there is a number of applications in which MWI is finding place, ranging from civil engineering to cultural heritage and archaeology, from security to medical diagnostics [1].

Owing to such a remarkably broad range of applications, several efforts have been carried out and are ongoing in the design of accurate imaging devices and in the development of reliable strategies for the processing of the data. In particular, these latter represent a challenging task, as they have to cope with the nonlinearity and ill-posedness of the inverse problem underlying MWI.

Among the various applications, one that has gained a growing interest is that of biomedical diagnostics, wherein the interest in MWI is mainly motivated by the nonionizing nature of microwave radiations and their potential of providing quantitative images through relatively low cost devices.

In such a framework, microwave imaging systems are typically confined, that is, delimited by a metallic or dielectric casing. The role of such a casing is twofold. On the one hand, it is meant to isolate the apparatus from external interference. On the other, it has to contain the coupling medium (usually liquid) in which the target of the diagnostic survey is embedded and which is adopted to facilitate the penetration of the probing wave inside the inspected tissues and maximize their interaction. Some examples of such systems are the system for breast cancer imaging developed at Dartmouth College [2], the scanner developed at the Institute Fresnel [3], and the imaging system of Manitoba University [4].

On the other hand, the presence of a confining structure implies a considerable increase in the complexity of the interaction between the field and the target. This entails that, in the development of any inversion strategy of the data, an adequate electromagnetic modeling of the actual system is necessary.

While canonical cases (such as a cylindrical circular cross section embedding) can be accounted for by means of analytic tools [5, 6], in more complex scenarios the use of appropriate numerical techniques is the obvious solution. For instance, several authors have pursed the hybridization of iterative inversion techniques based on gradient optimization with forward solver and modeling based on the Finite Element Method (FEM) [7]. This is, for instance, the case of the works by Lencrerot and coworkers [8], that have exploited a FEM solver as the forward engine of a Netwton Kantorovich approach, by Zakaria and coworkers [9], that have proposed a hybrid FEM contrast source inversion scheme, and by Attardo et al. who have introduced a FEM based contrast source extended Born inversion scheme [10].

While the abovementioned approaches are certainly effective ways to tackle the inverse scattering problem in a noncanonical configuration, such as the one in which the system is enclosed into a shielding, they have a drawback related to their nonnegligible computational cost deriving from their iterative nature. This circumstance prevents them from providing real-time or quasi-real-time imaging results. In addition, as they tackle the nonlinear problem through a local optimization scheme, they are prone to the occurrence of false solutions, which is an obviously detrimental outcome in any imaging application, but it is even worse in medical applications.

On the downside, approaches based on the linearization arising from the Born approximation (BA) [1, 11, 12], as well as shape reconstruction methods [13–15], could met the computational efficiency required to provide real-time results, but they are not able to provide quantitative images, which represent the true added value of MWI. Moreover, methods based on the BA have a limited range of applicability, as they can be indeed confidently applied only for weakly scattering targets [16].

With respect to such a scenario, this paper presents the application of a recently proposed inversion method [17] to the case of embedded imaging systems. The considered method has the unique capability of handling the quantitative imaging of nonweak targets within a linear framework. This is accomplished by exploiting an original field approximation that stems from an original exploitation of the well-known Linear Sampling Method [13]. Thanks to such a preprocessing of the scattered field data, the problem is recast into a linearized one, wherein the linearization has a much broader validity as compared to BA, since it implicitly takes the unknown targets into account.

As a consequence of the above, the adopted method allows tackling the inversion task in a computationally inexpensive fashion (in quasi-real-time) and free from false solutions, thus overcoming the aforementioned drawbacks. On the other side, the proposed method still relies on an approximation, so that its validity is in any case limited as compared to the full-wave iterative inversion schemes recalled above [8–10]. Nevertheless, results achieved in the canonical free space, as well as those presented in the following, confirm that the method is capable of succeeding also for scatterer far outside from the validity range of BA and that it can handle target usually handled via full-wave schemes [17, 18].

The paper is structured as follows. Section 2 recalls the electromagnetic scattering problem in presence of an embedding device. In Section 3 the Quantitative Linear Sampling Method is recalled and how the presence of the casing affects its implementation is described. Section 4 is devoted to a broad numerical analysis concerning different shaped cavities and different measurement strategies. Finally, in Section 5 some conclusions are given.

#### 2. Statement of the Problem in a PEC Cavity

Let us consider the canonical 2D scalar inverse scattering problem in a PEC enclosure.

Let denote the domain enclosed in the casing, in which is embedded the cross section of the unknown scatterers, whose properties, at the frequency , are related to those of the homogeneous host medium through the contrast function:wherein indicates the position of a generic point in and and denote the complex permittivity of the scatterer and of the background medium, respectively.

The scatterers are probed by a set of incident fields radiated by the transmitting antennas located at on a closed curve inside , which can be a circle inscribed into the PEC enclosure, as well as a generic curve embedded into the casing. Receivers are located at the same positions of transmitters ().

By assuming the TM polarization for the electric field, the equations governing the scattering phenomenon in the considered geometry can be expressed in an integral form aswherein is the wavenumber in the host medium, , , and are the total, incident, and scattered field, respectively, and is Green’s function pertaining to the empty system, possibly filled with a homogeneous medium. The Green function is the kernel of the radiation operators and that relate the contrast source in to the field it radiates on and in , respectively.

In free space, that is, in absence of any enclosure, such Green’s function is , being the zero-order second-kind Hankel function. In the cylindrical perfect electric conductor (PEC) embedding, it has been shown that such Green’s function can be expressed as a single summation of Bessel functions [3, 19].

For noncanonical casing, instead, Green’s function is not known in closed form, so its numerical evaluation is needed.

The inverse scattering problem at hand is then cast as retrieving the unknown contrast from the measured scattered field for known incident fields .

#### 3. A Simple Inversion Strategy: The Quantitative Linear Sampling Method

##### 3.1. First Linear Step: The Linear Sampling Method

The LSM belongs to the class of qualitative reconstruction methods, as it provides an estimate of the targets support, but not its electric properties. The LSM requires solving an auxiliary linear inverse problem rather than the nonlinear one formulated through (2)-(3) [13]. With respect to the scenario described above, the auxiliary problem is cast aswhere denotes a point of an arbitrary grid that samples the region under test , is the unknown to be determined, and is the far field operator [13].

Due to compactness of , (4) corresponds to a linear ill-posed inverse problem [13]. Hence, a stable solution of (4) in the generic sampling point requires a regularization. Usually, this is done considering the Tikhonov regularization [20].

The estimated support is achieved by evaluating the energy (i.e., the norm) of , as this assumes its lowest values in the points of the investigated region belonging to the target, while it diverges in points external to it [13]. Therefore, the support is simply determined by plotting the LSM indicator over and associating the sampling points where the indicator is low to the unknown objects.

The explicit expression of the LSM indicator readswhose evaluation is computationally straightforward as it requires a single evaluation of the SVD of . Moreover, the Tikhonov parameter can be determined only once for all sampling points [21–23]. In the examples of Section 4 this parameter is computed according to the physics based criterion introduced in [22].

##### 3.2. Field Approximation

In order to better understand how the quantitative LSM (QLSM) works, let us note that(1)by construction, represents a scattered field, as it is obtained through a linear combination of the measured data through the function ;(2)given the linearity of the relationship between incident fields and scattered ones, is the scattered field which is obtained on when is probed by means of the incident field:(3)in those points wherein the two sides of (4) match, the probing wave forces the targets to scatter on the field of a point source located in (obviously in presence of the PEC).

These simple observations have an interesting implication. As a matter of fact, as long as the LSM equation (4) can be solved in the sampling point , can be used to define an incident field whose corresponding scattered field is like the one radiated by a source centered on . Conversely, the incident wave (6) required to enforce this field depends on the scatterer under test through . Clearly, this is the* opposite* of what happens in usual scattering experiments, where the scattered field changes with the scatterer, while the incident field is independent of it.

This concept can be exploited to cast a new approximation of the relationship between the contrast and the scattered fields. In particular, one can assume that when using as incident wave, the scattered field corresponds, in the* whole* space into the cavity and external to the sampling point at hand (rather than only on as enforced by the LSM equation, this corresponds to analytically continuing the scattered field from to and then exploiting the continuity of the field’s tangential component), to the field radiated by an elementary source located in . Accordingly, the total field will be

By relying on the above concepts, the data-to-unknown relationship can be recast aswhich provides a linear formulation of the inverse scattering problem, since the total field is assigned through (7).

With respect to the nature of the introduced approximation, it is worth remarking that the total field given in (7) is not the outcome of a straightforward linearization of the scattering equation (2). As a matter of fact, the linearization is achieved by replacing the unknown total field with a known (approximated) one, which takes the scatterer under test into account via the LSM preprocessing. As such, the exploited approximation is completely different from the Born one, in which the effect of the target on the total field is simply neglected.

##### 3.3. Quantitative Linear Inversion

In this subsection, we recall the inversion method which relies on the LSM based approximation introduced in the previous subsection. To this end, let us first note that(1)to achieve the new data equation (8) no additional measurements are necessary, as everything is done “virtually” via a rearrangement of the scattered fields ruled by LSM solution;(2)by considering different sampling points for which the LSM equation is properly solved, it is possible to devise several “virtual” experiments, so as to rearrange the available multiview data into* multiple* virtual experiments.

Let us now assume that the original multiview multistatic configuration consists of transmitters and receivers.

The first step of the proposed method consists in the application of the LSM to the data matrix , to estimate the targets’ shape through indicator (5).

Then, we select a subset of sampling points belonging to the targets, . These points identify the virtual experiments. For each of them, we compute the approximated total field , , via (7) and then use these fields to build the data-to-unknown matrix equation that corresponds to the considered multiple virtual experiments:

In (9), assuming that the domain under test has been discretized into cells,is the matrix containing the values of the unknown contrast in the discretized investigation domain, with denoting the matrix transposition,is the matrix in which each element is a matrix containing the samples of the scattered field for the th virtual experiment, , andis matrix that represents the discretization of the approximated data equations for the considered multiple virtual experiments, with .

By denoting with the SVD of , (9) can be solved via truncated SVD (TSVD) [20], thus achieving the expression for the estimated unknown contrast:wherein is the th column of , is the th element of the th column of , and is the th element on the diagonal of .

The truncation index , which is the regularization parameter of this inversion scheme, is determined to get a tradeoff between the accuracy and the stability of the reconstruction.

#### 4. Numerical Analysis

In this section we present some numerical examples in order to assess the performance of the QLSM, recalled in previous section, in noncanonical embedding systems. The case is supposed to be a perfectly electric conductor (PEC), filled by a matching medium whose electromagnetic features are and mS/m. Three different kinds of metallic enclosures have been considered: a circular cavity, a square, and a triangular one. In these systems the imaging domain is represented by a square whose size is equal to 0.38 m (about ). The working frequency has been fixed at 1 GHz and the probes have been located in two different configurations.(i)*Measurement configuration A:* the probes are equally spaced on a circle surrounding the imaging domain, whose radius is equal to 0.38 m.(ii)*Measurement configuration B:* the probes are equally spaced along the edges of the cavity at a distance equal to (about 4.75 cm), as suggested by the analysis in [5].

The transmitter/receiver antennas are infinite-length filamentary sources located inside the cavity. When an antenna is transmitting, all the others are in receiving mode.

Figure 1 shows the two configurations for the considered kinds of enclosures, as well as the size of the metallic cases.