IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 49, NO. 9, SEPTEMBER 2001 1333 Shape Reconstruction From PO Multifrequency Scattered Fields via the Singular Value Decomposition Approach Rocco Pierri, Angelo Liseno, and Franceso Soldovieri Abstract—This paper deals with the problem of determining the shape of unknown perfectly conducting infinitely long cylinders; starting from the knowledge of the scattered electric far field under the incidence of plane waves with a fixed angle of incidence and varying frequency. The problem is formulated as a nonlinear inverse one by searching for a compact support distribution accounting for the objects contour. The nonlinear unknown to data mapping is then linearized by means of the Kirchhoff approximation, which reduces it into a Fourier transform relationship. Then, the Fourier transform inversion from incomplete data is dealt with by means of the singular value decomposition (SVD) approach and the features of the reconstructable unknowns are investigated. Finally, numerical results confirm the performed analysis. Index Terms—Electromagnetic scattering inverse problems, shape, singular value decomposition (SVD). I. INTRODUCTION T HE PROBLEM of reconstructing the shape of unknown impenetrable objects starting from the knowledge of the measured scattered field data is of interest in many applica- tions. For example, ground penetrating radar [1], nondestructive testing or evaluation of materials [2], and geophysics [3], just to mention a few examples. Several approaches have been employed in order to tackle this inverse problem and they can be mainly classified into two groups according to whether the nonlinearity of the formula- tions is preserved [4]–[8] or not [9]–[13]. In addition, since a crucial point concerns the choice of the unknown representing the objects shape, a further classifica- tion can be done into “surface”-reconstruction based [4]–[7] and “volume”-reconstruction based formulations [8], [9]. Surface-type formulation assuming as unknown of the problem the boundary of the scattering objects parametrized in terms of polar coordinates requires the a priori knowledge of the overall number of objects and their rough locations. The equivalent source technique [4] and the Newton–Kan- torovich iterative procedure [5], [6] have been employed as solution strategies for this formulation. However, both share the common drawback that a sufficient a priori information is needed in order to obtain reliable reconstructions. Indeed, Manuscript received August 17, 2000; revised January 11, 2001. This work was supported by the Programma Operativo Plurifondo of Regione Campania. The authors are with the Seconda Università di Napoli, Dipartimento di Ingegneria dell’Informazione, via Roma 29, I-81031, Aversa, Italy (e-mail: pierri@unina.it). Publisher Item Identifier S 0018-926X(01)07640-2. the equivalent source method requires an a priori information about the geometry of the unknown objects in order to suitably locate the equivalent sources and to choose a good initial guess for the iterative minimization procedure of a proper cost function. On the other side the reliability of the Newton–Kan- torovich iterative algorithm is ensured only if the initial guess of the minimization procedure is “close” to the exact shape, which can be achieved only by resorting to a sufficient a priori information about the scatterers shape. Moreover, a thorough understanding of the convergence properties of the regularized Newton–Kantorovich iterative procedure is up to now lacking [5]. It must be noted, however, that in [7] can be found a surface-reconstruction based approach that does not need the knowledge of the center of the scatterer under investigation. Volume-type formulation seeks for the scatterers cross sec- tion by assuming as unknown of the problem a quantity re- lated to the characteristic function. This is performed by using domain integral equations. The local shape function (LSF) ap- proach [8] has been adopted within this volumetric formulation framework. However, the Newton–Kantorovich iterative proce- dure, employed for minimizing the cost function leads to the above mentioned problems about convergence. The common weak point of all the above mentioned iterative solution algorithms actually arises from the nonquadraticity of the cost functions to be minimized, which is due to the “high nonlinearity” of the formulations. The risk of the occurrence of local minima makes it mandatory a proper choice of the initial guess, which traces back to the knowledge of a sufficient a priori information. This drawback can be overcome by resorting to an high-fre- quency (Kirchhoff) approximation in formulation, that leads to a linearization of the data-unknown mapping [9]–[13]. Recently, a formulation of the problem has been setup by adopting volume integrals through the use of compact support distributions [13]. The problem has, thus, been cast into the solu- tion of a couple of operator equations that have the same formal structure of the interior and exterior Lippmann–Schwinger op- erator equations of the dielectric case [14]. The use of the Kirch- hoff approximation has then made it possible to linearize the un- known-data mapping at the cost of redefining the unknown to be searched for and being able to retrieve only the illuminated side of the scatterers. In particular, the Kirchhoff approximation has led to recast the problem into the inversion of a linear oper- ator acting on a distribution space; the results in [13] allow to justify the employ of the singular value decomposition (SVD) approach [15] to invert . This method allows to weaken some 0018–926X/01$10.00 © 2001 IEEE