A Renewal Theorem for Strongly Ergodic Markov Chains in Dimension d≥3d\geq3 and Centered Case

In dimension $d\geq3$, we present a general assumption under which the renewal theorem established by Spitzer for i.i.d. sequences of centered nonlattice r.v. holds true. Next we appeal to an operator-type procedure to investigate the Markov case. Such a spectral approach has been already developed by Babillot, but the weak perturbation theorem of Keller and Liverani enables us to greatly weaken thehypotheses in terms of moment conditions. Our applications concern the v$-geometrically ergodic Markov chains, the$\rho$-mixing Markov chains, and the iterative Lipschitz models, for which the renewal theorem of the i.i.d. case extends under the (almost) expected moment condition

Multidimensional renewal theory in the non-centered case. Application to strongly ergodic Markov chains.

International audienceLet $(S_n)_n$ be a $R^d$-valued random walk ($d\geq2$). Using Babillot's method [2], we give general conditions on the characteristic function of $S_n$ under which $(S_n)_n$ satisfies the same renewal theorem as the classical one obtained for random walks with i.i.d. non-centered increments. This statement is applied to additive functionals of strongly ergodic Markov chains under the non-lattice condition and (almost) optimal moment conditions

Restoration of hyperspectral astronomical data with spectrally varying blur

International audienceIn this paper we present a method for hyper-spectral image restoration for integral field spectrographs (IFS) data. We specifically address two topics: (i) the design of a fast approximation of spectrally varying operators and (ii) the comparison between two kind of regularization functions: quadratic and spatial sparsity functions. We illustrate this method with simulations coming from the Multi Unit Spectroscopic Explorer (MUSE) instrument. It shows the clear increase of the spatial resolution provided by our method as well as its denoising capability

Fast model of space-variant blurring and its application to deconvolution in astronomy

International audienceImage deblurring is essential to high resolution imaging and is therefore widely used in astronomy, microscopy or com- putational photography. While shift-invariant blur is modeled by convolution and leads to fast FFT-based algorithms, shift- variant blurring requires models both accurate and fast. When the point spread function (PSF) varies smoothly across the field, these two opposite objectives can be reached by inter- polating from a grid of PSF samples. Several models for smoothly varying PSF co-exist in the literature. We advocate that one of them is both physically- grounded and fast. Moreover, we show that the approximation can be largely improved by tuning the PSF samples and inter- polation weights with respect to a given continuous model. This improvement comes without increasing the computa- tional cost of the blurring operator. We illustrate the developed blurring model on a deconvo- lution application in astronomy. Regularized reconstruction with our model leads to large improvements over existing re- sults

Exact discrete minimization for TV+L0 image decomposition models

International audiencePenalized maximum likelihood denoising approaches seek a solution that fulfills a compromise between data fidelity and agreement with a prior model. Penalization terms are generally chosen to enforce smoothness of the solution and to reject noise. The design of a proper penalization term is a difficult task as it has to capture image variability. Image decomposition into two components of different nature, each given a different penalty, is a way to enrich the modeling. We consider the decomposition of an image into a component with bounded variations and a sparse component. The corresponding penalization is the sum of the total variation of the first component and the L0 pseudo-norm of the second component. The minimization problem is highly non-convex, but can still be globally minimized by a minimum s-t-cut computation on a graph. The decomposition model is applied to synthetic aperture radar image denoising

Sparse + smooth decomposition models for multi-temporal SAR images

International audienceSAR images have distinctive characteristics compared to optical images: speckle phenomenon produces strong fluctuations, and strong scatterers have radar signatures several orders of magnitude larger than others. We propose to use an image decomposition approach to account for these peculiarities. Several methods have been proposed in the field of image processing to decompose an image into components of different nature, such as a geometrical part and a textural part. They are generally stated as an energy minimization problem where specific penalty terms are applied to each component of the sought decomposition. We decompose temporal series of SAR images into three components: speckle, strong scatterers and background. Our decomposition method is based on a discrete optimization technique by graph-cut. We apply it to change detection tasks

Un modèle rapide de flou variable dans le champ de flou variable dans le champ et son application a la déconvolution en astronomie

National audienceLa déconvolution d'images est essentielle pour l'imagerie haute résolution et est par conséquent largement utilisée en astronomie et en microscopie. Alors qu'un flou invariant dans le champ est modélisé par une convolution conduisant à des algorithmes rapides à base de FFT, les flous variant dans le champ nécessitent des modèles à la fois précis et suffisamment rapides. Lorsque la réponse impulsionnelle (RI) varie continument dans le champ, un compromis entre ces deux objectifs contradictoires peut être atteint en interpolant une grille de RI. Plusieurs modèles pour les RI variant continûment dans le champ co-existent dans la littérature. Nous montrons que l'un d'entre eux est à la fois bien fondé physiquement et rapide. De plus, nous montrons que la qualité d'approximation peut être améliorée en ajustant les RI et les poids d'interpolation par rapport à un modèle continu choisi. Cette amélioration ne modifie pas la complexité de l'application de l'opérateur de flou. Nous illustrons le modèle développé sur une application de déconvolution en astronomie et montrons qu'une reconstruction régularisée avec le modèle proposé améliore largement les résultats existants

Quasi-compactness of Markov kernels on weighted-supremum spaces and geometrical ergodicity

Let $P$ be a Markov kernel on a measurable space \X and let V:\X\r[1,+\infty). We provide various assumptions, based on drift conditions, under which $P$ is quasi-compact on the weighted-supremum Banach space (\cB_V,\|\cdot\|_V) of all the measurable functions f : \X\r\C such that \|f\|_V := \sup_{x\in \X} |f(x)|/V(x) < \infty. Furthermore we give bounds for the essential spectral radius of $P$. Under additional assumptions, these results allow us to derive the convergence rate of $P$ on \cB_V, that is the geometric rate of convergence of the iterates $P^n$ to the stationary distribution in operator norm. Applications to discrete Markov kernels and to iterated function systems are presented.Comment: 45 page