International Journal of Antennas and Propagation

Volume 2018, Article ID 3020417, 15 pages

https://doi.org/10.1155/2018/3020417

## Embedding Approach to Modeling Electromagnetic Fields in a Complex Two-Dimensional Environment

^{1}Department of Electrical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, Netherlands^{2}Department of Information Technology, Ghent University, iGent-Technologiepark-Zwijnaarde 15, 9052 Gent, Belgium^{3}Aix Marseille University, CNRS, Centrale Marseille, Institut Fresnel, Marseille, France

Correspondence should be addressed to Ann Franchois; eb.tnegu@siohcnarf.nna

Received 31 October 2017; Revised 19 February 2018; Accepted 6 March 2018; Published 10 May 2018

Academic Editor: Paolo Burghignoli

Copyright © 2018 Anton Tijhuis 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

An approach is presented to combine the response of a two-dimensionally inhomogeneous dielectric object in a homogeneous environment with that of an empty inhomogeneous environment. This allows an efficient computation of the scattering behavior of the dielectric cylinder with the aid of the CGFFT method and a dedicated extrapolation procedure. Since a circular observation contour is adopted, an angular spectral representation can be employed for the embedding. Implementation details are discussed for the case of a closed 434 MHz microwave scanner, and the accuracy and efficiency of all steps in the numerical procedure are investigated. Guidelines are proposed for choosing computational parameters such as truncation limits and tolerances. We show that the embedding approach does not increase the CPU time with respect to the forward problem solution in a homogeneous environment, if only the fields on the observation contour are computed, and that it leads to a relatively small increase when the fields on the mesh are computed as well.

#### 1. Introduction

In almost any computational approach to solving nonlinear inverse-scattering problems, a discretized configuration is introduced that depends on a fixed number of parameters. Subsequently, a cost functional is defined in terms of simulated and known scattered fields. Here, two different strategies can be distinguished. Conventionally, the corresponding forward problem is treated as an auxiliary problem, which is solved exactly for successive approximate configurations [1–6]. For multidimensional problems, this requires a number of field computations for a varying physical parameter such as frequency or source position. The cost function then refers to the known measured field information and preferably includes a regularizing function of the configuration parameters [7–9]. In the so-called modified gradient method and subsequent generalizations, the configuration and the unknown fields are determined simultaneously [10, 11]. The conventional approach has the advantage that the formulation of the inverse problem directly relates the parameterized configuration to the known field data. From a practical point of view, however, it is sometimes considered as less feasible because of the computational effort required in the repeated field computations. The argument is that it is not needed to compute the field with full accuracy in a configuration that still deviates considerably from the actual one.

For the case of an inhomogeneous, lossy dielectric cylinder in a homogeneous surrounding medium, however, it was demonstrated that a highly efficient implementation is obtained when the fields are computed by solving a contrast-source integral equation with a combination of the conjugate-gradient FFT (CGFFT) method and a special extrapolation procedure [12]. The extrapolation can be performed for almost any physical parameter [13], such as frequency or source position. Thus, the forward scattering problem can be solved for each new value of the physical parameter in a few iterations of the CGFFT procedure. This technique has been demonstrated successfully in the context of Newton-type inverse scattering [5, 6]. It is the authors’ experience that these schemes, with only the parameters in the profile parameterization as fundamental unknowns, are generally more efficient than schemes where the field and the profile are determined simultaneously.

A special feature of our implementation is that its efficiency is based on the circumstance that the dielectric cylinder is embedded in a homogeneous surrounding medium. This means that Green’s function in the integral equation above exhibits convolution symmetry. Preserving that symmetry in the relevant space discretization allows the application of FFT operations in evaluating the operator products in the conjugate gradient method [14–16]. In practical experiments, however, the surrounding medium may be inhomogeneous and the symmetry is broken. In that case, FFT operations are no longer applicable. The same problem also arises in the modified gradient method, where FFT operations are used as well to compute the field updates.

To circumvent this problem, we use the feature that the scattering operator characterizes the complete electromagnetic response of the region inside a closed observation contour. Hence, it must be possible to determine the scattered-field data from a cylinder in an arbitrary environment from the scattering operator for the same object in a homogeneous environment. That data, in turn, can then be obtained with the existing implementation. We introduced this so-called embedding approach for a cylinder inside a circular observation contour in [17, 18]. The choice of this particular configuration was inspired by the experimental research with a 434 MHz scanner of the third author [19]. Several authors have shown interest in quantitative imaging with a circular scanner with metallic enclosure, including [20–22]. It is well-known that employing the 2D Green’s function of the empty casing is computationally expensive [19, 20].

In the present paper, we formulate the embedding approach in the angular spectral representation for a general surrounding 2D medium and subsequently specialize to the case where the dielectric cylinder is surrounded by a perfectly conducting circular container. Besides a more comprehensive theoretical formulation than in [17, 18], we provide details on the numerical implementation and on its performance in accuracy and speed as a function of various parameters. We show that, with well-chosen values for these parameters, the embedding approach does not increase the CPU time as compared to the forward problem solution in a homogeneous environment, if only the fields on the observation contour are computed, and that it leads to a relatively small increase in CPU time, when the fields in the object are needed as well, for example, to compute the Jacobian matrix in a Newton-type inversion scheme.

The embedding approach relies on the identification of the scattering and reflection operators for the dielectric cylinder and the empty microwave scanner, respectively. The idea of using such an operator to characterize scattering properties has a long tradition in the electromagnetic literature [23–26]. The use of a numerically computed scattering operator, however, is new and originates from the availability of the “march in source position” method [13]. A generalization for multiple interacting domains of arbitrary shapes is given in [27, 28].

Finally, it should be remarked that in [29], a procedure is proposed based on reciprocity that is capable of converting the field in the complete configuration into the field in a homogeneous environment, that is, the reverse procedure from what is proposed in the present paper. The suggestion is to perform the profile inversion on the thus corrected data, using inverse-profiling algorithms for objects in a homogeneous background. However, this idea has two possible drawbacks. First, in order to carry out the conversion from one environment to another, complete data on a contour surrounding the scatterer must be available. In an actual experiment, such data may not always be available, while theoretical results for an estimated configuration can always be computed. Second, the conversion renormalizes the experimental data including the measurements, while the present procedure allows a comparison with the actual data. This makes it easier to account for the accuracy of these data, for example, by including appropriate weighting coefficients in the cost functional.

The paper is organized as follows. In Section 2, we describe the scanner configuration and its mathematical idealization. Section 3 summarizes the field computation for an object in a homogeneous environment. The scattering operators are introduced in Section 4 and used to formulate the embedding approach in Section 5. Section 6 presents the computational details for a homogeneous environment, the empty scanner and the complete configuration. In Section 7, the computational complexity of the algorithm is analyzed, and a procedure is given for tuning the computational parameters. The conclusions are drawn in Section 8.

#### 2. Formulation of the Problem

In the present paper, we describe and investigate an efficient procedure to calculate the electric field inside a cylindrical scanner for a given permittivity profile. This field may then be used in inverse-profiling algorithms. For our numerical experiments, we adopted the configuration of the scanner described in [19] and shown in Figure 1, which was developed to conduct biomedical imaging experiments. This scanner comprises a circular array of 64 transmitting/receiving conical dipole antennas, which operate at 434 MHz in a multi-incidence mode, that is, one antenna at a time is transmitting and the others are receiving. The array has a radius of 27.6 cm and is placed inside a water-filled metal casing with a slightly larger radius of 29.0 cm. Measurements of the relative permittivity of the water typically yielded , which corresponds to a wavelength cm in the water. The diameter of the *T*/*R* circle thus is about , the antennas are spaced apart about , and they are at a distance of about from the casing.