This work concerns microwave tomography for application to biomedical imaging. The aim is to retreive both permittivity and conductivity of an unknown object from measurements of the scattered field that results from its interaction with a known interrogating wave. Such a problem is denoted as ``the inverse problem'' as opposed to the associated forward problem that consists of calculating the scattered field while the interrogating wave and the object are known.

The resolution of the inverse problem requires the prior construction of the associated forward model. The latter is based upon an integral representation of the electric field resulting in two coupled integral equations whose discrete counterparts are obtained by means of the method of moments.

Regarding the inverse problem, in addition to the fact that the physical equations involved in the forward modeling make it nonlinear, it is also mathematically ill-posed in the sense of Hadamard, which means that the conditions of existence, uniqueness and stability of its solution are not simultaneously guaranteed. Hence, solving this problem requires its prior regularization which usually involves the introduction of a priori information on the sought solution. This resolution is done here in a Bayesian probabilistic framework where we introduce prior knowledge appropriate to the sought object by considering that the latter is composed of a finite number of homogeneous materials distributed into compact and homogeneous regions. This information is introduced by means of a "Gauss-Markov-Potts" model. In addition, Bayesian computations yield the posterior distribution of all the unknowns from which we can define the point estimators. We proceed then to identify the posterior estimators via variational approximation methods and thereby to reconstruct images of the thought object.

The main contributions of this work are methodological and algorithmic. They are illustrated by an application of microwave tomography to breast cancer imaging. The latter is in itself a very important and original aspect of the thesis. Indeed, imaging of breast cancer using microwaves is a very interesting alternative to X-ray mammography, but it is still at an exploratory stage.

__Members:__

Directeur de thèse Mr Duchêne Bernard Chargé de recherche, CNRS

Co-directeur de thèse Mr Mohammad-Djafari Ali Directeur de recherche, CNRS

Encadrant Mr Ayasso Hacheme Maître de conférences à l'Université de Grenoble

Rapporteurs Mme Litman Amélie Maître de conférences à l'Université d'Aix-Marseille

Mr Massa Andréa Professeur à l'Université de Trento, Italie

Examinateurs Mme Blanc-Feraud Laure Directrice de recherche, CNRS

Mr Pichot du Mezeray Christian Directeur de recherche, CNRS