Reconstruction of piecewise constant objects in microwave imaging in a 2D-TE configuration O. F´eron, B. Duchˆene and A. Mohammad-Djafari Laboratoire des signaux et syst`emes (CNRS-Sup´elec-UPS), Plateau de Moulon, 3 rue Joliot Curie, 91192 Gif-sur-Yvette, France. Abstract. In this paper we propose a stochastic algorithm applied to an electromagnetic inverse scattering problem. The objective is to characterize an unknown object from measurements of the scattered fields at different frequencies and for several illuminations. This inverse problem is known to be nonlinear and ill-posed. It then needs to be regularized by introducing prior information. The particular prior information we account for is that the object is composed of a finite known number of different materials distributed in compact regions. The algorithm is applied to the inversion of experimental data collected at the Institut Fresnel (Marseille) and has already provided satisfactory results in a 2D-TM configuration. Herein, the goal is to test the same kind of method in a 2D-TE configuration.

Key Words: Microwave imaging, Bayesian estimation, Markov Random Fields.

I. Direct model We deal herein with an electromagnetic inverse obstacle scattering problem where the goal is to characterize unknown targets from measurements of the scattered fields that result from their interactions with a known interrogating (or incident) wave in the microwave frequency range. Modelling is based upon a domain integral representation obtained by applying Green′ s theorem to the Helmholtz wave equations satisfied by the fields and accounting for continuity and radiation conditions. This leads to two coupled contrast-source integral equations that express the ~ inside the domain Ω occupied by the object and the scattered field E ~ dif observed on the total electric field E measurement domain S: Z Z 1 ~~ ′ ~ ′ ′ inc ~ ~ ~ ′ ) dr ′ G(r, r )J(r ) dr + 2 ∇∇ E(r) = E (r) + G(r, r ′ )J(r r∈Ω (1) k0 Ω Ω Z Z ~ dif (r) = ~ ′ ) dr ′ + 1 ∇ ~∇ ~ ~ ′ ) dr ′ E G(r, r ′ )J(r G(r, r ′ )J(r r ∈ S, (2) k02 Ω Ω ~ are the induced currents, χ = k 2 − k 2 (with k0 and k the propagation constants in vacuum and where J~ = χE 0 in the object, respectively), G(r, r ′ ) = 4i H01 (k0 R) is the free space Green′ s function, and R = kr − r ′ k. The contrast function χ characterizes the unknown object and the inverse problem then consists in retrieving this function. Considering a 2D configuration where the object and the sources are cylindrical and infinite along one direction, these coupled equations become simpler and scalar in a TM configuration ([1]). In a TE configuration the electric field is a two-component vector perpendicular to the cylinder axis and the problem is then more involved. Discrete versions of the above equations are obtained by applying the method of moments with pulse basis and point matching: X ~ ~ inc (r) + ~ m) E(r) = E GΩ (r, r m )E(r r∈Ω (3) rm

~ dif

E

(r) =

X

~ m) G (r, r m )E(r S

r ∈ S,

(4)

rm

where r m refers to a pixel in the lattice that partitions Ω. These two equations are of the same kind as in the TM configuration, but obviously the expression of the operators GΩ and GS are a little bit more intricate and 4 times bigger. The operator GΩ can be written as follows:  xx  G Gxy GΩ = , Gyx Gyy OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)

where all the submatrices are NΩ ×NΩ , with NΩ the number of pixels partitioning Ω. The imposition of the continuity ~ = εE ~ (with ε the complex permittivity ε = ε0 εr + i σ ) at condition of the normal component of the electric flux D ω the frontier between two neighbouring pixels leads to the fact that in all the submatrices the contrast function χ, or more precisely the permittivity, appears in two different forms: (i) the value of (ε(r) − ε0 ) for each pixel, and (ii) the difference (1/ε(r) − 1/ε(r ′ )) for two neighbouring pixels. On the one hand, this model introduces an intrinsic correlation between neighbouring pixels, which brings a lot of difficulties into the implementation, but, on the other hand, the prior information that we want to introduce is precisely based upon such a correlation.

II. Prior information and Bayesian approach ~ within The inverse problem in microwave imaging is known to be very ill-posed, and also nonlinear, the total field E Ω being unknown. We then need to introduce additional information in order to regularize the problem. There exists a lot of work that introduce prior information such as the positivity of the real and imaginary parts of the contrast function or the smoothness of the latter, or impose an edge-preserving constraint. Herein, we introduce a particular additional information: although the materials that compose the object are not known, their finite number K is supposed to be known. We introduce this prior information by modelling the contrast image by a hidden Markov random field. This consists in considering a hidden variable z(r) which represents a classification of the contrast image. This hidden variable takes discrete values between 1 and K, each value being associated to a given material. Each material will thus be identified by a label k in the image z and characterized by a mean complex value mk (associated to its estimated complex permittivity) and a variance ρk (which allows some fluctuations around the mean value in the contrast image χ). Therefore we define a prior conditional probability distribution of the contrast as a Gaussian law: for each pixel r of the contrast image, p(χ(r)|z(r) = k) = N (mk , ρk ). Obviously we have also to estimate the classification z and all the parameters mk and ρk , as the materials themselves are not known. Prior probability distributions have then to be assigned to these variables. As the materials are supposed to be distributed in compact regions, a local spatial correlation on the pixels of the classification is introduced by modelling z with a Potts Markov random field:    X X δ(z(r) − z(r ′ )) , (5) p(z) ∝ exp   ′ r∈S r ∈V(r)

where δ is the kronecker function and V(r) is the set made of the four neerest neighbors of the pixel r. The prior probability distributions of the parameters mk and ρk are chosen to be in the so-called ”conjugate priors” family in order to render their estimation easier. ~ χ, z, {mk , ρk }k ) of all the unknowns we All the previous definitions allow us to define a joint prior distribution p(E, have to estimate that takes into account the state equation (3) of the modelling and all the information introduced earlier. By deriving the likelihood term from the observation equation (4) and using the Bayes formula, this allows ~ χ, z, {mk , ρk }k |E ~ dif ) of all the unknowns, given the scattered field us to express the a posteriori distribution p(E, data. This posterior distribution represents all the information we have on the unknowns which comes from our a priori knowledge and from the data. The Bayesian approach consists then in using this posterior distribution to define an estimator for the unknowns.

III. Results The proposed method is applied to experimental data collected at the Institut Fresnel (Marseille, France). Four targets have been considered; they are composed of different dielectric and metallic cylinders. We refer the reader to [2] for a description of their basic features. The data have been measured both in TM and TE configurations, at several frequencies lying in between 2 and 17 GHz, and for various illuminations all around the object. The proposed method, which introduces the a priori knowledge of the number of materials, has already been applied to the TM configuration and has provided good results which have been the subject of paper [1]. In the TE configuration, the direct model has provided good results as compared to the data and work is going on concerning the inversion of the latter.

References [1] O. F´eron, B. Duchˆene, A. Mohammad-Djafari, ”Microwave imaging of inhomogeneous objects made of a finite number of dielectric and conductive materials from experimental data”, Inverse Problems, vol. 21, no. 6, December 2005, pp. S95–S115. [2] J.-M. Geffrin, P. Sabouroux, C. Eyraud, ”Free space experimental scattering database continuation: experimental set-up and measurement precision”, Inverse Problems, vol. 21, no. 6, December 2005, pp. S117–S130.

OIPE 2006 ”The 9th Workshop on Optimization and Inverse Problems in Electromagnetics” - September 13th – 15th, Sorrento (Italy)