Recherche de pas par Majoration-Minoration. Application à la résolution de problèmes inverses.

Abstract : The solution to inverse problems encountered in signal and image processing is often defined as the minimizer of a penalized criterion that accounts simultaneously for the data and the prior. This thesis deals with the minimization of differentiable criteria. We focus more specifically on algorithmic strategies for stepsize determination in iterative descent optimization methods. In Christian Labat's thesis [Labat06], a simple and efficient quadratic majorize-minimize line search method has been proposed (MMQ 1D). This method benefits from strong convergence results when the nonlinear conjugate gradient (NLCG) direction is used. In this thesis, we extend those properties to gradient related algorithms. Moreover, we prove that MMQ 1D generalizes into a multi-dimensional search MMQ rD that ensures the convergence of several subspace algorithms. We experimentally show that the super memory gradient algorithm SMG + MMQ 2D performs better than NLCG + MMQ 1D algorithms, for an image deconvolution test problem. MMQ strategies exclude the case of barrier criteria i.e., functions with unbounded gradient. In this thesis, we propose an original MM line search algorithm for barrier functions, based on a quadratic majorizing function augmented with a logarithmic term. This leads to a simple line search MMLQ 1D that ensures the convergence of several classical descent optimization strategies. We experimentally show that the proposal scheme improves the performances of primal interior point algorithm, for a set of quadratic programming test problems. Finally, we apply the MMLQ 1D approach to maximum entropy reconstruction of bi-dimensional NMR spectra.
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Traitement du signal et de l'image. Ecole Centrale de Nantes (ECN), 2010. Français
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Emilie Chouzenoux. Recherche de pas par Majoration-Minoration. Application à la résolution de problèmes inverses.. Traitement du signal et de l'image. Ecole Centrale de Nantes (ECN), 2010. Français. 〈tel-00555643〉

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