Méthodes de reconstruction en tomographie de diffraction 3D. (Reconstruction methods for 3D diffraction tomography)

Corentin Friedrich

Abstract: This thesis is focused on microwave tomography. This imaging technique consists in estimating a three-dimensional mapping of the dielectric properties of an unknown volume from measurements of the electromagnetic field from a known incident wave and scattered by this volume. This is a promising technique that is used in various applications (medical imaging, geophysics, non- destructive testing, ...) but suffers from high computational costs. This is a reason why microwave imaging is not widely used in industry. In this thesis, microwave imaging is considered as an inverse problem, where the error between the measurements and a forward model that describes the scattered field is minimized as a function of the properties of the volume. The physical model is discretized using the method of moments. This inverse problem is ill-posed because the number of unknowns is higher than the number of measurements. It is tackled through the minimization of a regularized leastsquares cost function, which is addresed by local iterative optimization algorithms. Moreover, the forward model is non-linear. Thus, reconstruction is a difficult and costful procedure. The computation of the objective function and of its gradient requires the resolution of a high number of linear systems, which are performed at each iteration of the optimization algorithm and represent most of the computational cost. In this thesis, we propose to reduce the computational costs of the reconstruction algorithms by focusing on the resolution of these linear systems. Two contributions are presented. The first one is a procedure in order to reduce the number of linear systems depending on the configuration of the measurement setup. The second contribution offers an efficient way to speed up the resolutions of the systems. We adapt block resolution algorithms, in order to jointly solve multiple linear systems involving a common operator matrix. These methods are validated on simulated, realistic, 3D problems, and applied to the reconstruction of real objects from experimental measurements of scattered fields. satisfactory results are obtained, where the computation time can be reduced by a factor of two, in particular for the most difficult reconstruction problems.