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Analyse d'images par des méthodes variationnelles et géométriques

Abstract : In this work, we study both theoretical and numerical aspects of an anisotropic Mumford-Shah problem for image restoration and segmentation. The Mumford-Shah functional allows to both reconstruct a degraded image and extract the contours of the region of interest. Numerically, we use the Amborsio-Tortorelli approximation to approach a minimizer of the Mumford-Shah functional. It Gamma-converges to the Mumford-Shah functional and allows also to extract the contours. However, the minimization of the Ambrosio-Tortorelli functional using standard discretization schemes such as finite differences or finite elements leads to difficulties. We thus present two new discrete formulations of the Ambrosio-Tortorelli functional using the framework of discrete calculus. We use these approaches for image restoration and for the reconstruction of normal vector field and feature extraction on digital data. We finally study another similar shape optimization problem with Robin boundary conditions. We first prove existence and partial regularity of solutions and then construct and demonstrate the Gamma-convergence of two approximations. Numerical analysis shows once again the difficulties dealing with Gamma-convergent approximations.
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Marion Foare. Analyse d'images par des méthodes variationnelles et géométriques. Mathématiques générales [math.GM]. Université Grenoble Alpes, 2017. Français. ⟨NNT : 2017GREAM043⟩. ⟨tel-01682981⟩

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