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The proximity operator of a convex function is a natural extension of the notion of a projection operator onto a convex set. This tool, which plays a central role in the analysis and the numerical solution of convex optimization problems, has recently been introduced in the arena of inverse problems and, especially, in signal processing, where it has become… (More)

- Jean-Christophe Pesquet, Hamid Krim, Hervé Carfantan
- IEEE Trans. Signal Processing
- 1996

A simple construction of an orthonormal basis starting with a so called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or… (More)

We propose a primal-dual splitting algorithm for solving monotone inclusions involving a mixture of sums, linear compositions, and parallel sums of set-valued and Lipschitzian operators. An important feature of the algorithm is that the Lipschitzian operators present in the formulation can be processed individually via explicit steps, while the set-valued… (More)

A broad range of inverse problems can be abstracted into the problem of minimizing the sum of several convex functions in a Hilbert space. We propose a proximal decomposition algorithm for solving this problem with an arbitrary number of nonsmooth functions and establish its convergence. The algorithm fully decomposes the problem in that it involves each… (More)

- Caroline Chaux, Laurent Duval, Jean-Christophe Pesquet
- IEEE Transactions on Image Processing
- 2006

We propose a two-dimensional generalization to the M-band case of the dual-tree decomposition structure (initially proposed by Kingsbury and further investigated by Selesnick) based on a Hilbert pair of wavelets. We particularly address: 1) the construction of the dual basis and 2) the resulting directional analysis. We also revisit the necessary… (More)

- Marc Castella, Saloua Rhioui, Eric Moreau, Jean-Christophe Pesquet
- IEEE Transactions on Signal Processing
- 2007

This paper deals with the problem of source separation in the case when the observations result from a multiple-input multiple-output convolutive mixing system. In a blind framework, higher order contrast functions have been proved to be efficient for extracting sources. Inspired by a semiblind approach, we propose new contrast functions for blind signal… (More)

- Patrick L. Combettes, Jean-Christophe Pesquet
- IEEE Transactions on Image Processing
- 2004

Total variation has proven to be a valuable concept in connection with the recovery of images featuring piecewise smooth components. So far, however, it has been used exclusively as an objective to be minimized under constraints. In this paper, we propose an alternative formulation in which total variation is used as a constraint in a general convex… (More)

- Patrick L. Combettes, Jean-Christophe Pesquet
- SIAM Journal on Optimization
- 2007

The notion of soft thresholding plays a central role in problems from various areas of applied mathematics, in which the ideal solution is known to possess a sparse decomposition in some orthonormal basis. Using convex-analytical tools, we extend this notion to that of proximal thresholding and investigate its properties, providing in particular several… (More)

- Emilie Chouzenoux, Jean-Christophe Pesquet, Audrey Repetti
- J. Optimization Theory and Applications
- 2014

We consider the minimization of a function G defined on R , which is the sum of a (non necessarily convex) differentiable function and a (non necessarily differentiable) convex function. Moreover, we assume that G satisfies the KurdykaLojasiewicz property. Such a problem can be solved with the Forward-Backward algorithm. However, the latter algorithm may… (More)

- Jean-Christophe Pesquet, Hamid Krim, David Leporini, E. Hamman
- ICASSP
- 1996