International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 9304371, 8 pages

http://dx.doi.org/10.1155/2016/9304371

## A Banach Space Regularization Approach for Multifrequency Microwave Imaging

^{1}Department of Mathematics, University of Genoa, Via Dodecaneso 35, 16146 Genova, Italy^{2}Department of Electrical, Electronic, Telecommunications Engineering, and Naval Architecture, University of Genoa, Via Opera Pia 11A, 16145 Genova, Italy

Received 15 October 2015; Accepted 24 February 2016

Academic Editor: Ahmed T. Mobashsher

Copyright © 2016 Claudio Estatico 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

A method for microwave imaging of dielectric targets is proposed. It is based on a tomographic approach in which the field scattered by an unknown target (and collected in a proper observation domain) is inverted by using an inexact-Newton method developed in Banach spaces. In particular, the extension of the approach to multifrequency data processing is reported. The mathematical formulation of the new method is described and the results of numerical simulations are reported and discussed, analyzing the behavior of the multifrequency processing technique combined with the Banach spaces reconstruction method.

#### 1. Introduction

The regularization of ill-posed problems in Banach spaces exhibits several advantages over the corresponding classical regularization in Hilbert spaces [1–3]. A significant example is microwave imaging (MI). MI is a well-known technique in which an unknown target is inspected by using an electromagnetic radiation [4]. When illuminated by an incident field, the target scatters the incident radiation depending on its physical properties, geometrical shape, and dimensions. For dielectric targets with dimensions comparable with the wavelength of the incident waves, the scattering phenomena are quite complicated [5]. However, the scattered electric field can be collected around the target and the properties of the object can be retrieved by inverting the equation governing the electromagnetic scattering problem. Several approaches to solve this inverse problem with qualitative [6–8] or quantitative techniques [9–11] have been reported in the scientific literature, with reference to both two- and three-dimensional configurations [12–14]. Among these methods, it has been proved in [15] that, exploiting the properties of the norms in the Banach spaces, it is possible to obtain better reconstructions than developing the corresponding approach in Hilbert spaces. In particular, in [16] an inexact-Newton method has been applied to a tomographic configuration, in which the target is assumed to be a cylindrical one and the imaging process is performed in free space and at a fixed frequency. In this paper, the above formulation is extended to multifrequency imaging, in which the use of more than one incident radiation results in an improved processing, due to the additional information included in the input data (samples of the scattered electric field at different frequencies).

The mathematical formulation of the proposed approach is described in the following with reference to transverse-magnetic illumination conditions. The capabilities and limitations of the proposed extended method are evaluated by means of numerical simulations concerning targets in noisy environments for which the forward problem is solved by using the method of moments. The inverse solution is obtained, as mentioned, by using a two-loop inexact-Newton method, which has been developed in the framework of Banach spaces. In particular, after the discretization of the continuous model (based on the electric field integral equation (EFIE)), the outer loop performs a linearization of the resulting nonlinear algebraic equation, whereas the inner loop solves the obtained (ill-posed) linear equation in a regularized sense by using a truncated Landweber scheme in Banach spaces.

The paper is organized as follows. The mathematical formulation of the developed approach is discussed in Section 2. Section 3 reports some numerical results aimed at validating the inversion procedure. Finally, conclusions are drawn in Section 4.

#### 2. Mathematical Formulation

Let us consider the configuration shown in Figure 1. A cylindrical target, whose cross section is enclosed in an investigation domain , is illuminated by a set of known time-harmonic, TM- electromagnetic fields characterized by angular frequencies . As it is well known, under such hypotheses, the scattering problem turns out to be two-dimensional and scalar [4]. The resulting scattered electric field is collected in a measurement domain .