572 IEEE TRANSACTIONS ON IMAGE PROCESSING, VOL. 12, NO. 5, MAY 2003 Statistical Regularization in Linearized Microwave Imaging Through MRF-Based MAP Estimation: Hyperparameter Estimation and Image Computation Vito Pascazio, Member, IEEE, and Giancarlo Ferraiuolo Abstract—The application of a Markov random fields (MRF) based maximum a posteriori (MAP) estimation method for microwave imaging is presented in this paper. The adopted MRF family is the so-called Gaussian-MRF (GMRF), whose energy function is quadratic. In order to implement the MAP estimation, first, the MRF hyperparameters are estimated by means of the expectation-maximization (EM) algorithm, extended in this case to complex and nonhomogeneous images. Then, it is implemented by minimizing a cost function whose gradient is fully analytically evaluated. Thanks to the quadratic nature of the energy function of the MRF, well posedness and efficiency of the proposed method can be simultaneously guaranteed. Numerical results, also per- formed on real data, show the good performance of the method, also when compared with conventional techniques like Tikhonov regularization. Index Terms—Bayesian estimation, image formation, Markov random fields (MRF), microwave tomography. I. INTRODUCTION T OMOGRAPHIC imaging at microwaves (in classical microwave tomograhpy [1], or in subsurface sensing [2]) amounts to form (it would be better to say, to compute) images of internal sections (tomograms) of objects through the use of noninvasive and nondestructive techniques. Such images are usually formed by the samples of the so-called contrast function, which is related to the permittivity of the object. There are a lot of applications where tomographic imaging can be useful, for instance in medical imaging of human body, in the detection of internal defeats of objects used in aircrafts and nuclear plants, in ground penetrating radar imaging for archeology, for underground tunnel detection, and many others. The images are obtained by processing the data collected by illuminating the objects with known incident fields and by mea- suring the fields they scatter outside. Unfortunately, the data (the measured scattered field data samples) are related to the un- knowns (the tomographic image samples) through a non linear mapping [3], which induces a strong filtering between images Manuscript received May 9, 2001; revised November 15, 2002. This work was supported in part by the Agenzia Spaziale Italiana (ASI) under research Contract COD I/R/136/00. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Eric L. Miller. V. Pascazio is with the Istituto di Teoria e Tecnica delle Onde Elettromag- netiche, Università di Napoli Parthenope, 38-80133 Napoli, Italy (e-mail: vito.pascazio@uninav.it). G. Ferraiuolo is with the Dipartimento di Ingegneria Elettronica e delle Tele- comunicazioni, Università di Napoli Federico II, 21-80125 Napoli, Italy (e-mail: gferraiu@unina.it). Digital Object Identifier 10.1109/TIP.2003.811507 and scattered fields. For these reasons, it can be shown that the tomographic imaging problem is ill-posed [4], so that it can be very difficult to develop reliable image formation procedures. Several methods to solve this (inverse) problem have been developed. Some of these are based on the linearization of the above mentioned mapping (hereafter referred as model) [5], while others try to solve it by preserving its intrinsic nonlinearity [3], [6]. Methods based on the Born or the Rytov approximations, and their derivations, belong to the class of linear methods. In the Born approximation the total electric field inside the object is assumed to be approximately equal to the incident electric field, and the contrast function is the unknown. It has to be remembered that the Born approximation cannot manage large objects in terms of wavelength, and/or objects characterized by high contrast values [5]. In order to overcome limitations of linearized models, approaches based on approximation-free models can be adopted [3], [6]. These nonlinear methods, usually solved through iterative optimiza- tion procedures, suffer for the presence of false solutions of the problem (which are the local minima of the nonconvex functional to be optimized) [7]. A feature common to both linearized and non linearized cases is their ill-posedness. They do not allow to reconstruct arbitrary (i.e., belonging to ) images, so that the imaging problem re- quires proper regularization. All is further worsened, if we con- sider the unavoidable presence of the measurement noise. The work presented in this paper is concerned with the reg- ularization of the image formation by means of a technique based on Markov Random Fields (MRF), used to describe from a statistical point of view the unknown images [8]. Their use is very popular to model the contextual information in image processing and formation. As we want to test and verify their usefulness on regularization of solution procedures, in the fol- lowing we choose to adopt a linear approximation of the model, in order to avoid any problem connected to the false solutions occurrence: if the procedure does not work well, we want to be sure that it is not due to the presence of a local minimum. The image formation procedure that we adopt is based on a maximum a posteriori (MAP) estimation [9], where the a priori probability term is given by a Gibbs distribution [8], that is expo- nential distribution whose exponent, the so-called energy func- tion, characterizes the different MRF families. The final imple- mentation of the MAP estimation algorithm requires that the a priori model for the images be tuned on the images to be re- constructed. This can be done by choosing the expression of the 1057-7149/03$17.00 © 2003 IEEE