Total Variation as a local filter

International audienceIn the Rudin-Osher-Fatemi (ROF) image denoising model, Total Variation (TV) is used as a global regularization term. However, as we observe, the local interactions induced by Total Variation do not propagate much at long distances in practice, so that the ROF model is not far from being a local filter. In this paper, we propose to build a purely local filter by considering the ROF model in a given neighborhood of each pixel. We show that appropriate weights are required to avoid aliasing-like effects, and we provide an explicit convergence criterion for an associated dual minimization algorithm based on Chambolle's work. We study theoretical properties of the obtained local filter, and show that this localization of the ROF model brings an interesting optimization of the bias-variance trade-off, and a strong reduction a ROF drawback called "staircasing effect". We finally present a new denoising algorithm, TV-means, that efficiently combines the idea of local TV-filtering with the non-local means patch-based method

Total Variation Restoration of Images Corrupted by Poisson Noise with Iterated Conditional Expectations

International audienceInterpreting the celebrated Rudin-Osher-Fatemi (ROF) model in a Bayesian framework has led to interesting new variants for Total Variation image denoising in the last decade. The Posterior Mean variant avoids the so-called staircasing artifact of the ROF model but is computationally very expensive. Another recent variant, called TV-ICE (for Iterated Conditional Expectation), delivers very similar images but uses a much faster fixed-point algorithm. In the present work, we consider the TV-ICE approach in the case of a Poisson noise model. We derive an explicit form of the recursion operator, and show linear convergence of the algorithm, as well as the absence of staircasing effect. We also provide a numerical algorithm that carefully handles precision and numerical overflow issues, and show experiments that illustrate the interest of this Poisson TV-ICE variant

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

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).Le modèle ROF (Rudin, Osher, Fatemi), introduit en 1992 en utilisant la variation totale comme terme de régularisation pour la restauration d'images, a fait l'objet de nombreuses recherches théoriques et numériques depuis. Dans cette thèse, nous présentons de nouveaux modèles inspirés de la variation totale mais construits par analogie avec une méthode de débruitage beaucoup plus récente et radicalement différente : les moyennes non locales (NL-means). Dans une première partie, nous transposons le modèle ROF dans un cadre bayésien, et montrons que l'estimateur associé à un risque quadratique (moyenne a posteriori) peut être calculé numériquement à l'aide d'un algorithme de type MCMC (Monte Carlo Markov Chain), dont la convergence est soigneusement contrôlée compte tenu de la dimension élevée de l'espace des images. Nous montrons que le débruiteur associé permet notamment d'éviter le phénomène de "staircasing", défaut bien connu du modèle ROF. Dans la deuxième partie, nous proposons tout d'abord une version localisée du modèle ROF et en analysons certains aspects : compromis biais-variance, EDP limite, pondération du voisinage, etc. Enfin, nous discutons le choix de la variation totale en tant que modèle a priori, en confrontant le point de vue géométrique (modèle ROF) au cadre statistique (modélisation bayésienne)

Posterior Expectation of the Total Variation model: Properties and Experiments

International audienceThe Total Variation image (or signal) denoising model is a variational approach that can be interpreted, in a Bayesian framework, as a search for the maximum point of the posterior density (Maximum A Posteriori estimator). This maximization aspect is partly responsible for a restoration bias called ''staircasing effect'', that is, the outbreak of quasi-constant regions separated by sharp edges in the intensity map. In this paper we study a variant of this model that considers the expectation of the posterior distribution instead of its maximum point. Apart from the least square error optimality, this variant seems to better account for the global properties of the posterior distribution. Theoretical and numerical results are presented, that demonstrate in particular that images denoised with this model do not suffer from the staircasing effect

Transport optimal régularisé semi-déséquilibré pour la restauration d'images

In this paper we consider the use of a penalty based on optimal transport in order to regularize inverse problems in imaging. The proposed approach is formulated in a variational setting and aims at promoting images whose patch distribution is close either to the one learned by a generative model, or to available uncorrupted patches. With the aid of numerical illustrations, we argue in favor of adopting an asymmetric form of unbalanced transport. We then provide details concerning the computation and the differentiation of the proposed penalty. Finally, we detail the application of our approach to a particular super-resolution setting: the image zoom completion problem.Nous étudions dans cet article l'utilisation d'une pénalité basée sur le transport optimal afin de régulariser des problèmes inverses en traitement d'images. L'approche proposée est formulée dans un cadre variationnel et vise à favoriser des images dont les patchs ont une distribution proche de celle apprise par un modèle génératif, ou d'exemples non dégradés disponibles. À l'aide d'illustrations numériques, nous montrons la nécessité d'adopter une forme non symétrique de déséquilibre dans la formulation du transport optimal. Nous donnons ensuite les détails permettant de calculer et de différentier cette formulation. Enfin, nous détaillons son application à un problème particulier de super-résolution : la complétion de zoom

Digital Image Processing- DEA MVA

As digital images become more widely used, digital image analysis must find more tools t