On thermo-acoustic and photo-acoustic Imaging of small absorbers ; Sur l’imagerie thermoacoustique et photoacoustique de petits absorbeurs

This thesis is divided into two parts. The first part is dedicated to the study of inverse problems for wave equations and their application to medical imaging. More precisely, we focus our work on the study of the photo-acoustic and thermo-acoustic tomography techniques. They are multi-wave imaging techniques based on the photo-acoustic effect that was discovered in 1880 by Alexander Graham Bell. The inverse problem we are concerned in throughout this thesis is the problem of recovering small absorbers in a bounded domain Ω R3. We provide a direct reconstruction method based on the algebraic algorithm that was developed first in without following the quantitative photo-acoustic tomography approach (qPAT). This algorithm allows us to reconstruct the number of the absorbers and their locations from a single Cauchy data, in addition to some information on optical parameters such as the conductivity and the absorption coefficient that can serve as an important diagnostic information in detecting tumors. The main difference between PAT and TAT is in the type of optical pulse used. In PAT, a high frequency radiation is delivered into the biological tissue to be imaged, while in TAT low frequency radiations are used, which makes some differences in the physical and mathematical setting of the problem. In this dissertation we study the both mathematical models, and propose reconstruction algorithms for the two inverse problems. The second part of this thesis is devoted to the study of non-autonomous semilinear elliptic equations. We study the existence of radial solution in Rn with non zero limiting behavior. ; Cette thèse est divisée en deux parties. La première partie porte sur l’étude de problèmes inverses pour les équations des ondes et leur application à l’imagerie photoacoustique (PAT) et thermoacoustique (TAT). Ces sont des techniques d’imagerie multi-ondes basées sur l’effetphotoacoustique découvert en 1880 par Alexander Graham Bell . Le problème inverse qui nous intéresse consiste à récupérer des petits ...