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In this short paper, we formulate parameter estimation for finite mixture models in the context of discrete optimal transportation with convex regularization. The proposed framework unifies hard and soft clustering methods for general mixture models. It also generalizes the celebrated $k$\nobreakdash-means and expectation-maximization algorithms in relation to associated Bregman divergences when applied to exponential family mixture models.

International audience; We consider the problem of estimating a low-rank signal matrix from noisy measurements under the assumption that the distribution of the data matrix belongs to an exponential family. In this setting, we derive generalized Stein's unbiased risk estimation (SURE) formulas that hold for any spectral estimators which shrink or threshold the singular values of the data matrix. This leads to new data-driven spectral estimators, whose optimality is discussed using tools from random matrix theory and through numerical experiments. Under the spiked population model and in the asymptotic setting where the dimensions of the data matrix are let going to infinity, some theoretical properties of our approach are compared to recent results on asymptotically optimal shrinking rules for Gaussian noise. It also leads to new procedures for singular values shrinkage in finite-dimensional matrix denoising for Gamma-distributed and Poisson-distributed measurements.

International audience; We focus on the maximum regularization parameter for anisotropic total-variation denoising. It corresponds to the minimum value of the regularization parameter above which the solution remains constant. While this value is well know for the Lasso, such a critical value has not been investigated in details for the total-variation. Though, it is of importance when tuning the regularization parameter as it allows fixing an upper-bound on the grid for which the optimal parameter is sought. We establish a closed form expression for the one-dimensional case, as well as an upper-bound for the two-dimensional case, that appears reasonably tight in practice. This problem is directly linked to the computation of the pseudo-inverse of the divergence, which can be quickly obtained by performing convolutions in the Fourier domain.

International audience; This chapter is dedicated to very high resolution (VHR) SAR imagery, including interferometric applications. First, the principles of SAR data acquisition are presented as well as the different types of configurations. The widely adopted Gaussian complex model of fully developed speckle is described as well as more advanced statistical models for VHR SAR data that account for textures. The following two parts are devoted to SAR image estimation and to image denoising within two different frameworks. First, Markovian modeling is introduced and the associated optimization approaches are presented, including graph-cut based optimization. The second framework is the patch-based non-local modeling of SAR complex data. Both frameworks are adapted to SAR images through the use of statistical models specific to SAR imagery. Their applications to amplitude data, interferometry, and fusion with optical data are illustrated. A special focus is given to phase unwrapping applied to single and multi- channel interferometry, showing the usefulness of local and global contextual models.

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International audience; Restoration of a piece-wise constant signal can be performed using anisotropic Total-Variation (TV) regularization. Anisotropic TV may capture well discontinuities but suffers from a systematic loss of contrast. This contrast can be re-enhanced in a post-processing step known as least-square refitting. We propose here to jointly estimate the refitting during the Douglas-Rachford iterations used to produce the original TV result. Numerical simulations show that our technique is more robust than the naive post-processing one.

International audience; The classical super-resolution (SR) setting starts with a set of low-resolution (LR) images related by subpixel shifts and tries to reconstruct a single high-resolution (HR) image. In some cases, partial observations about the HR image are also available. Trying to complete the missing HR data without any reference to LR ones is an inpainting (or completion) problem. In this paper, we consider the problem of recov- ering a single HR image from a pair consisting of a complete LR and incomplete HR image pair. This setting arises in par- ticular when one wants to fuse image data captured at two different resolutions. We propose an efficient algorithm that allows to take advantage of both image data by first learning nonlocal interactions from an interpolated version of the LR image using patches. Those interactions are then used by a convex energy function whose minimization yields a super- resolved complete image.

International audience; In this paper, an extension of the framework proposed by Deledalleet al.[1] for Non Local Means (NLM) method is proposed. This extensionis a general adaptive method to denoise images containingmultiple noises. It takes into account a segmentation stage that indicatesthe noise type of a given pixel in order to select the similaritymeasure and suitable parameters to perform the denoising task, consideringa certain patch on the image. For instance, it has been experimentallyobserved that fetal 3D ultrasound images are corruptedby different types of noise, depending on the tissue. Finally, the proposedmethod is applied to denoise these images, showing very goodresults.

Noise in images often limits visual and automatic interpretation of the scene. Speckle in synthetic aperture radar (SAR) imagery and shot noise in photon-limited imagery are two examples of strong corruptions that require the use of denoising techniques. Patches are small image parts that capture both textures and local structures. Though being crude low-level features (compared to higher level descriptors), they have led to very powerful image processing approaches by exploiting the natural redundancy of images. Patch-based methods achieve state-of-the-art denoising performance. The classical patch-based denoising technique non-local (NL) means is designed for images corrupted by an additive Gaussian noise (i.e., fluctuations being symmetrical, signal-independent without outliers). NL means cannot be applied directly on images corrupted by a non-Gaussian process especially with non-symmetrical distribution, signal-dependence and heavy-tail such as speckle and shot noise. The goal of this thesis is to bridge the gap between patch-based denoising methods restricted to Gaussian noise and techniques dedicated to SAR despeckling. After reviewing image denoising techniques for Gaussian noise and for non-Gaussian noise, we propose an extension of the NL means that adapts to a given noise distribution. Besides the problem of image denoising, we study the problem of patch comparison under non-Gaussian conditions. Many tasks in computer vision require matching image parts. We introduce a similarity criterion grounded on the generalized likelihood ratio test and illustrate its effectiveness on different applications including detection, stereo-vision and motion-tracking. This criterion is at the heart of the proposed patch-based estimator. An iterative scheme is proposed to deal with strong noise corruptions and we develop an unsupervised method for parameter setting. Our approach leads to state-of-the-art denoising results in SAR imagery for amplitude images, as well as interferometric or polarimetric data. The proposed technique is applied successfully to one of the latest aerial SAR sensor: F-SAR from the German Aerospace Center (DLR). Images with strong contrasts suffer from denoising artefacts known as noise halo due to the absence of similar patches in the vicinity of some structures. This residual noise can be reduced by considering patches with shapes of various scales and orientations. Local selection of relevant shapes leads to an improved denoising quality, especially close to edges., Le bruit dans les images limite souvent l'interprétation visuelle ou automatique de la scène. Le chatoiement ou "speckle" en imagerie radar à synthèse d'ouverture (RSO) et le bruit de grenaille ou "shot noise" en imagerie à faible luminosité sont deux exemples de fortes corruptions qui nécessitent l'utilisation de techniques de débruitage. Les vignettes ou "patchs" sont de petites imagettes qui capturent à la fois les textures et les structures locales. Bien qu'étant assez rudimentaires (comparées à des descripteurs de plus haut niveau), elles ont mené à de puissantes approches de traitement d'images tirant parti de la redondance naturelle des images. Les méthodes à patchs représentent l'état-de-l'art des méthodes de débruitage. La technique classique de débruitage à patchs, les moyennes non-locales (NL), est conçue pour les images corrompues par du bruit additif gaussien (c-à-d., pour des fluctuations symétriques, indépendantes du signal et sans valeurs extrêmes). Les moyennes NL ne peuvent pas être appliquées directement sur des images corrompues par un bruit non-gaussien surtout pour des distributions asymétriques, dépendantes du signal et à queues lourdes telles que le bruit de chatoiement et le bruit de grenaille. Le but de cette thèse est de combler le fossé entre les méthodes de débruitage à patchs, restreintes au bruit gaussien, et les techniques dédiées aux images RSO. Après avoir examiné les techniques de débruitage d'image pour le bruit gaussien puis non-gaussien, nous proposons une extension des moyennes NL qui s'adapte à la distribution d'un bruit donné. Au-delà du problème du débruitage d'image, nous étudions le problème de la comparaison de patchs sous conditions non-gaussiennes. La plupart des tâches de vision par ordinateur requièrent de mettre en correspondance des parties d'images. Nous introduisons un critère de similarité fondé sur le rapport de vraisemblance généralisé et nous illustrons son efficacité sur différentes applications dont la détection, la vision stéréoscopique et le suivi de mouvement. Ce critère est au coeur de l'estimateur à patchs proposé. Un schéma itératif est élaboré pour faire face aux fortes corruptions de bruit et nous développons une méthode non-supervisée pour le réglage des paramètres. Notre approche mène à des résultats de débruitage état-de-l'art en imagerie RSO pour les images d'amplitude, ainsi que les données interférométriques ou polarimétriques. La technique proposée est appliquée avec succès sur l'un des derniers capteurs aérien RSO: F-SAR de l'agence aérospatiale allemande (DLR). Les images avec de forts contrastes souffrent d'un artéfact de débruitage de type "halo de bruit" dû à l'absence de patchs similaires dans les environs de certaines structures. Ce bruit résiduel peut être réduit en considérant des patchs avec des formes d'échelle et d'orientation variées. La sélection locale des formes pertinentes permet d'améliorer la qualité du débruitage, surtout à proximité des contours.

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This paper presents a likelihood ratio test based method of change detection and classification for synthetic aperture radar (SAR) time series, namely NORmalized Cut on chAnge criterion MAtrix (NORCAMA). This method involves three steps: 1) multi-temporal pre-denoising step over the whole image series to reduce the effect of the speckle noise; 2) likelihood ratio test based change criteria between two images using both the original noisy images and the denoised images; 3) change classification by a normalized cut based clustering-and-recognizing method on change criterion matrix (CCM). The experiments on both synthetic and real SAR image series show the effective performance of the proposed framework.

This document provides additional information and results of the method NL-SAR described in our paper: "NL-SAR: a unified Non-Local framework for resolution-preserving (Pol)(In)SAR denoising" submitted to IEEE Trans. on Geoscience and Remote Sensing [Deledalle et al., 2013]. NL-SAR is a fully automatic method for speckle reduction that handles amplitude, polarimetric and/or interferometric SAR data. It can process single look and multi-look images. The source code of the method is freely available at: http://www.math.u-bordeaux1.fr/cdeledal/nlsar.php.

International audience; Cet article traite du problème de débruitage des images via une nouvelle formulation variationnelle exploitant localement le caractère redondant ou bien régulier des images. Nous effectuons une minimisation de la variation totale avec une attache aux données non locale. En tirant parti à la fois des approches non locales et variationnelles, nous montrons que l'on peut pallier deux défauts respectifs de ces deux approches : l'effet de crénelage et de bruit résiduel. Par ailleurs, le modèle proposé s'étend à différentes statistiques de bruit et nous montrons qu'une solution peut être obtenue efficacement dans le cas de lois issues de la famille exponentielle. Les résultats obtenus, en associant l'approche non locale aux méthodes variationnelles, montrent une réduction des effets de crénelage et de bruit résiduel. Nos résultats offrent à la fois un gain un terme de qualité visuelle et de rapport signal à bruit.

National audience; Cet article traite des propriétés structurelles des solutions de problèmes inverses avec régularisation favorisant des modèles de faible complexité. Plus exactement, la régularisation appartient à la classe générique de semi-normes définies comme des normes décomposables composées par un opérateur linéaire, d'où l'a priori décomposable type analyse. Nous proposons une analyse théorique unifiée des propriétés structurelles des solutions de problèmes inverses. Nous prouvons de nouveaux résultats d'unicité et des bornes de stabilité. Notre cadre inclut de nombreux cas particuliers comme la variation totale discrète, le Lasso par blocs analyse ou alors la norme nucléaire. Nos résultats principaux établissent des conditions suffisantes garantissant l'unicité de la solution régularisée et sa stabilité à un bruit arbitraire borné. En chemin, nous montrons une condition suffisante fine d'unicité dont la portée va bien au delà des normes décomposables.

In this paper, we investigate in a unified way the structural properties of solutions to inverse problems. These solutions are regularized by the generic class of semi-norms defined as a decomposable norm composed with a linear operator, the so-called analysis type decomposable prior. This encompasses several well-known analysis-type regularizations such as the discrete total variation (in any dimension), analysis group-Lasso or the nuclear norm. Our main results establish sufficient conditions under which uniqueness and stability to a bounded noise of the regularized solution are guaranteed. Along the way, we also provide a strong sufficient uniqueness result that is of independent interest and goes beyond the case of decomposable norms., Comment: 4 pages, to appear in SAMPTA 2013

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