Méthodes variationnelles en traitement d'image

Abstract : The purpose of this thesis is to investigate the mathematical properties of some models which are currently used in image processing. Generalizing an approach by S. J. Osher, L. Rudin and E. Fameti, we decompose an image f of L² as a sum u+v where u belongs to somme functional Banach space E while v belongs to L². The Banach space is aimed at modeling the objects contained in the given image and the optimal decomposition minimizeq the energy J(u)=||u||_E+\lambda||f-u||^2_2. The main difficulty is to choose an adapted Banach space E. The common choice are E=\dot{B}^{1,1}_1(\R^2) which leads to the well-known Donoho's wavelet thresholding or E=BV(\R^2) the space of functions of bounded variations. The latter choice is the Osher Rudin Fatemi algorithm. These two choices are suffering from severe drawbacks. In the first case, sharp edges are erased. The second choicedoes not lead to a wavelet thresholding. That is why we propose E=\B1inf(\R^2) which yields sharp edges and is given by wavelet thresholding. This is the two first parts of the thesis. In the third part, we investigate the mathematical properties of the Osher-Vese newest algorithm which keeps track if the textured components.
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Ali Haddad. Méthodes variationnelles en traitement d'image. Mathématiques [math]. École normale supérieure de Cachan - ENS Cachan, 2005. Français. ⟨tel-00133405⟩

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