IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 60, NO. 7, JULY 2012 3373 A Novel Microwave Imaging Approach Based on Regularization in Banach Spaces C. Estatico, M. Pastorino, Fellow, IEEE, and A. Randazzo, Member, IEEE Abstract—Inverse problems arising in microwave imaging suffer from high ill-posedness. As it is well known, it is necessary to employ regularized inversion methods, in order to mitigate such behavior. Usually, such approaches are formulated in standard Hilbert spaces. Recently, a more generic regularization theory, working in Banach spaces, has been investigated, in order to overcome some limitations of the Hilbert-space regularization. In this paper, a novel imaging algorithm, performing a Ba- nach-space regularization, is proposed for 2-D electromagnetic inverse scattering problems. The reconstruction capabilities of the methods are evaluated by using numerical and experimental data. Index Terms—Inverse problems, iterative algorithms, mi- crowave imaging, Newton method. I. INTRODUCTION T HE inverse scattering problem represents the basic formu- lation for a number of imaging and nondestructive testing methods [1]–[20]. Usually, the object under test is illuminated by incident waves and the field it scatters is collected in a set of measurement points located near the object. Depending on the adopted configuration (e.g., tomography, buried object de- tection, etc.) the measurement sensors can be positioned along different probing lines. In any case, by “inverting” the measured samples of the scattered field, one searches for estimates of the distributions of key parameters, such as the dielectric permit- tivity and the electric conductivity (for penetrable materials). In several cases, the electromagnetic inverse problem is formu- lated through integral equations, which are properly discretized by using suitable basis functions. The resulting discrete inverse scattering problem turns out to be ill-conditioned, and conse- quently, regularized methods must be used. Several regularization methods for ill-posed functional equa- tions have been firstly and deeply investigated in the context of Hilbert spaces (see [21] for a general overview). Basically, any linear (or linearized) operator in Hilbert spaces can be decomposed into a set of eigencomponents by using the spec- tral theory, which can be understood as a generalization of the Manuscript received May 03, 2011; revised September 16, 2011; accepted February 15, 2012. Date of publication April 30, 2012; date of current version July 02, 2012. The work of C. Estatico was supported in part by MIUR under Grant 20083KLJEZ, and by the GNCS-INDAM project “Analisi di strutture nella ricostruzione di immagini e monumenti.” C. Estatico is with the Department of Science and High Technology, Univer- sity of Insubria, I-22100 Como, Italy (e-mail: claudio.estatico@uninsubria.it). M. Pastorino and A. Randazzo are with the Dynatech Department, University of Genoa, I-16145 Genoa, Italy (e-mail: matteo@unige.it, pastorino@unige.it; andrea.randazzo@unige.it). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAP.2012.2196925 classical eigenvalues/eigenvectors decomposition of matrices. This way, convergence and regularization properties of any solving method can be analyzed by considering the behavior of any single eigencomponent. Although spectral decomposition is a powerful tool that highly simplifies the mathematical study, regularization methods in Hilbert spaces usually give rise to smooth (and sometimes over-smooth) solutions. This represents a drawback in all the practical applications where nonsmoothness naturally arises, as in most imaging problems. In fact, even if a pixelated discretization is used (piecewise constant basis functions), the reconstruction usually suffers from over-smoothing effects. Actually, in practical applica- tions, objects with smoothly varying dielectric distributions are rarely encountered. On the contrary, in most cases, there are strong boundaries between essentially homogeneous materials. More recently, some regularization methods have been intro- duced and investigated in Banach spaces (i.e., complete vector spaces endowed with a norm that only allow “length” and “dis- tances” between its elements to be measured, without any scalar product) [22]–[25]. Due to the geometrical properties of Ba- nach spaces, these regularization methods allow to obtain so- lutions endowed with lower over-smoothness, which results, as instance, in a better localization and restoration of the discon- tinuities or localized impulsive signals in imaging applications. Another useful property of the regularization in Banach space is that the solutions are sparse, that is, in general they can be represented by few values. Indeed, it has been shown that the resolution of functional equations in Banach spaces, with , leads to solutions which usually have few compo- nents. This is very useful when dealing with large scale prob- lems, where sparsity gives rise to low numerical cost in compu- tation and storage. We point out that sparsity in Banach spaces is a fast emerging research field, with a lot of real applications in learning theory and compressive sensing [26]–[28]. In this paper, a new formulation developed in Banach spaces is applied to a tomographic imaging configuration devoted to the inspection of cylindrical scatterers under transverse magnetic illumination (2-D imaging). To the best of the authors’ knowl- edge, it is the first time that mathematical tools from the theory of regularization in Banach spaces are applied to the inversion of the Lippman-Schwinger nonlinear integral operator arising in electromagnetic inverse scattering problems. The paper is organized as follows. In Section II the math- ematical formulation of the new approach is detailed and discussed in four subsections, whereas in Section III the results of some inversions are provided. They concern both numerical and experimental input data. Moreover, comparisons with reconstructions obtained by using a standard Hilbert-space Newton method under the same imaging conditions are also 0018-926X/$31.00 © 2012 IEEE