## 1. Introduction

[2] A 2D TM quantitative microwave imaging algorithm for complex permittivity reconstructions of lossy dielectric objects in a 434 MHz scanner is presented. The scanner, which was developed at CNRS/Supélec to conduct biomedical imaging experiments [*Geffrin*, 1995], consists of a water-filled circular cylindrical metal casing containing an array of transmitting/receiving antennas located on a circle with a slightly smaller diameter. The object under test is placed in the center of this circle. This system can be regarded as a complicated environment, since the reflections from the metal casing, and to a lesser extent the antenna interactions, should be accounted for in a reconstruction algorithm. In the past, quantitative microwave reconstruction algorithms for biomedical imaging generally were configured for the object in a homogeneous medium of infinite extent [e.g., *Chew and Wang*, 1990; *Meaney et al.*, 1995; *Franchois et al.*, 1998; *Tijhuis et al.*, 2001]. In more recent years, *Paulsen and Meaney* [1999] and *Meaney et al.* [1999] have incorporated a nonactive antenna compensation model in their reconstruction algorithm, leading to an improved image quality. In the present paper, we take into account the influence of the metal casing in a computationally efficient way by applying an embedding technique to the forward model, as described by *Tijhuis et al.* [2000]. With this technique, the forward problem is first solved for the object in a homogeneous medium of infinite extent by means of the efficient conjugate gradient FFT method [*Peng and Tijhuis*, 1993], and a scattering matrix of the object is computed. Next, the casing is introduced and the scattered field on the measurement circle is expressed in terms of “outgoing” and “source-free” cylindrical wave functions, the coefficients of which are determined by imposing the boundary conditions at the casing and the scattering matrix for the object. Finally, the induced currents on the casing are replaced by equivalent current sources on the measurement circle, leading to an elegant expression for the total field in the object in terms of the homogeneous medium solutions. We have implemented this adapted model into a Broyden–Fletcher–Goldfarb–Shanno (BFGS) quasi-Newton optimization algorithm with line search [*Tijhuis et al.*, 2001; *Franchois and Tijhuis*, 2001]. This method accumulates approximate second derivative information during the iterations. In our case, the optimization operates on a nonlinear least squares cost function, representing the error between the measured field data and the field computed for a parameterized complex permittivity distribution, possibly augmented with a regularization term. Since we use different meshes for the forward modeling and the parameter reconstruction, the ill conditioning also can be alleviated by choosing a sufficiently coarse mesh for the latter.

[3] Since the beginning of the 1990s, several authors have applied Newton-like nonlinear least squares optimization techniques, such as the linearized Gauss–Newton and Levenberg–Marquardt methods, in their 2D microwave reconstruction algorithms [e.g., *Chew and Wang*, 1990; *Joachimowicz et al.*, 1991; *Franchois and Pichot*, 1997]. Others have applied conjugate gradient optimization schemes, either to the forementioned cost function [*Harada et al.*, 1995; *Lobel et al.*, 1997; *Rekanos et al.*, 1999; *Tijhuis et al.*, 2001] or to a modified gradient type of cost function, in which the field is also included as a parameter [*Kleinman and van den Berg*, 1992; *Belkebir et al.*, 1997]. Still others have tried to tackle the problem of local minima by using global optimization techniques such as simulated annealing [*Garnero et al.*, 1991; *Caorsi et al.*, 1994] or genetic algorithms [*Caorsi et al.*, 2001], often at the expense of considerable computational power. All these methods have yielded results with varying success in terms of dynamic range, spatial resolution, sensitivity, and convergence speed, which can be ascribed to the difficulty of the quantitative microwave reconstruction problem at hand, a (very) nonlinear, ill-conditioned large residual optimization problem, and this seems to hold for the BFGS quasi-Newton algorithm as we have implemented it here as well [*Fletcher*, 1990].

[4] In section 2, we introduce the configuration and formulate the inverse problem. In section 3, we describe the BFGS quasi-Newton algorithm and in section 4, we expose, for the sake of completeness, the embedding procedure applied to the 434 MHz circular scanner configuration. In section 5, we show reconstruction results from simulated data for some piecewise homogeneous lossy dielectric cylinders of moderate and high contrast in the 434 MHz scanner configuration.