Abstract — Under consideration is the large body of signal recovery problems that can be formulated as the problem of minimizing the sum of two (not necessarily smooth) lower semicontinuous convex functions in a real Hilbert space. This generic problem is analyzed and a decomposition method is proposed to solve it. The convergence of the method, which is based on the Douglas-Rachford algorithm for monotone operator-splitting, is obtained under general conditions. Applications to non-Gaussian image denoising in a tight frame are also demonstrated. Index Terms — Convex optimization, denoising, Douglas-Rachford, frame, nondifferentiable optimization, Poisson noise,

This paper focuses on optimization problems containing an l 1 kind of regularity criterion and a smooth data fidelity term. A general theorem is applied in this context; it gives an estimate of the distribution law of the “rank ” of the solution to optimization problems, when the initial datum follows a uniform (in a convex compact set) distribution law. It says that, asymptotically, solutions with a large rank are more and more likely. The main goal of this paper is to understand the meaning of this notion of rank for some energies which are commonly used in image processing. We study in detail the energy whose level sets are defined as the convex hull of a finite subset of R N (c.f. Basis Pursuit) and the total variation. For these energies, the notion of rank relates respectively to sparse representation and staircasing. In all cases but the 2D total variation, we are able to adapt the general theorem mentioned above to the energies under consideration. Key words: basis pursuit, sparse representation, total variation, regularization, polytopes

We examine the general regularization model which is based on total-variation for the structural part and a Hilbert-space norm for the oscillatory part. This framework generalizes the Rudin-Osher-Fatemi and the Osher-Sole-Vese models and opens way for new denoising or decomposition methods with tunable norms, which are adapted to the nature of the noise or textures of the image. We give sufficient conditions and prove the convergence of an iterative numerical implementation, following Chambolle’s projection algorithm. In this paper we focus on the denoising problem. In order to provide an automatic solution, a systematic method for choosing the weight between the energies is imperative. The classical method for selecting the weight parameter according to the noise variance is reformulated in a Hilbert space sense. Moreover, we generalize a recent study of Gilboa-Sochen-Zeevi where the weight parameter is selected such that the denoised result is close to optimal, in the SNR sense. A broader definition of SNR, which is frequency weighted, is formulated in the context of inner products. A necessary condition for maximal SNR is provided. Lower and upper bounds on the SNR performance of the classical and optimal strategies are established, under quite general assumptions.

In this paper a total variation (TV) based approach is proposed for pixel level fusion to fuse images acquired using multiple sensors. In this approach, fusion is posed as an inverse problem and a locally affine model is used as the forward model. A total variation norm based approach in conjunction with principal component analysis is used iteratively to estimate the fused image. The feasibility of the proposed algorithm is demonstrated on images from computed tomography (CT) and magnetic resonance imaging (MRI) as well as visible-band and infrared sensors. The results clearly indicate the feasibility of the proposed approach.

Joint regularization of phase and amplitude of InSAR data: application to 3D reconstruction Régularisation conjointe de la phase et de l’amplitude en interférométrie radar: application à la reconstruction 3D

My research project is to design variational models and fast optimization algorithms to solve efficiently problems arising in image processing, machine learning and other applications such as medical imaging and physics. An important part of my research project is to design convex variational models for basic problems in image processing such as image segmentation and image registration. Indeed, most models published in the last twenty years have used non-convex energy minimization models that capture non-optimal solutions. However, in the last few years, I have developed with my collaborators new convex minimization models along with fast algorithms that, I believe, will provide a new paradigm to solve more effectively basic problems in image processing, computer vision, medical imaging, machine learning and physics. Another part of my research project is to develop a unified framework for image processing. Since several image processing problems such as image denoising, image segmentation and image registration have been defined as variational models, I have developed with my collaborators a new variational model, based on the Polyakov energy, to unify these models. The Polyakov model from the physics of high-energy seems a very promising method to unify image processing models. Unlike standard models, ours can define denoising, segmentation, registration on any smooth and parameterized surface s.a. the sphere. The model is also purely geometric,

Joint regularization of phase and amplitude of InSAR data: application to 3D reconstruction Régularisation conjointe de la phase et de l’amplitude en interférométrie radar: application à la reconstruction 3D

Abstract—Statistical and machine learning is a fundamental task in sensor networks. Real world data almost always exhibit dependence among different features. Copulas are full measures of statistical dependence among random variables. Estimating the underlying copula density function from distributed data is an important aspect of statistical learning in sensor networks. With limited communication capacities or privacy concerns, centralization of the data is often impossible. By only collecting the ranks of the data observed by different sensors, we estimate and evaluate the copula density on an equally spaced grid after binning the standardized ranks at the fusion center. Without assuming any parametric forms of copula densities, we estimate them nonparametrically by maximum penalized likelihood estimation (MPLE) method with a Total Variation (TV) penalty. Linear equality and positivity constraints arise naturally as a consequence of marginal uniform densities of any copulas. Through local quadratic approximation to the likelihood function, the constrained TV-MPLE problem is cast as a sequence of corresponding quadratic optimization problems. A fast gradient based algorithm solves the constrained TV penalized quadratic optimization problem. Numerical experiments show that our algorithm can estimate the underlying copula density accurately. Index Terms—sensor network; dependence; copula; copula density estimation; I.