Modèles variationnels et bayésiens pour le débruitage d'images : de la variation totale vers les moyennes non-locales

Abstract : The ROF (Rudin, Osher, Fatemi, 1992) model, introducing the total variation as regularizing term for image restoration, has since been dealt with intense numerical and theoretical research. In this talk we present new models inspired by the total variation but built by analogy with a much more recent method and diametrically opposed to it: the non-local means. A first model is obtained by transposing the ROF model into a Bayesian framework. We show that the estimator associated to a quadratic risk (posterior expectation) can be numerically computed thanks to a MCMC (Monte Carlo Markov Chain) algorithm, whose convergence is carefully controlled, considering the high dimensionality of the image space. We notably prove that the associated denoiser avoids the staircasing effect, a well-known artifact that frequently occurs in ROF denoising. In a second part of the thesis we propose a neighborhood filter based on the ROF model, and analyze several aspects: stability, limiting PDE, neighborhood weighting... We show that this filter allows to remove noise while maintaining a local control over the noise. Last but not least, we reconsider the choice of total variation as prior image model, by setting the geometrical point of view (ROF model) against the statistical framework (Bayesian modeling).
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Cécile Louchet. Modèles variationnels et bayésiens pour le débruitage d'images : de la variation totale vers les moyennes non-locales. Mathématiques [math]. Université René Descartes - Paris V, 2008. Français. ⟨tel-00371438⟩

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