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Optimisation de la forme des zones d'observation pour l'équation des ondes. Applications à la tomographie photoacoustique

Abstract : In the first part of this thesis, we consider the wave equation on a regular bounded domain Ω. We investigate the problem of optimizing, in some appropriate sense, the shape and location of sensors spread on an arbitrary measurable subdomain Ω of the boundary of Ω. We introduce a spectral quantity called randomized observability constant, corresponding to the best constant in an average of the classical observability inequality, over random initial data. The pupose of the first chapter is to investigate optimal domains, maximizing the new objective function. The second part consists in applying the previous results to medical imaging, and more precisely to photoacoustic tomography. This imaging technique, constitutes a cutting-edge technology that has drawn considerable attention in the medical imaging area. Firstly because it is non-ionizing and non-invasive, and also because it constitutes a precise and cheap alternative to X imaging. In this framework, we propose here to model the influence of the shape and position of sensors in the inverse problem consisting in the reconstruction of the imaged body. In a nutshell, we build a functional of the shape of the sensors, providing an account for the reconstructed image quality.
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Pierre Jounieaux. Optimisation de la forme des zones d'observation pour l'équation des ondes. Applications à la tomographie photoacoustique. Modélisation et simulation. Université Pierre et Marie Curie - Paris VI, 2016. Français. ⟨NNT : 2016PA066131⟩. ⟨tel-01397189⟩

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