Organisée par SMAI-SIGMA, groupe thématique de la
Société de Mathématiques
Appliquées et Industrielles (SMAI), cette rencontre annuelle a
pour objectif de rassembler la communauté française autour des thémes Signal, Image, Géométrie Algorithmique, Modélisation Géométrique, et Approximation. Cette année, la journée a été préparée en collaboration avec les GDR MOA et MSPC ainsi que le laboratoire JLL. Parmi les sept exposés de niveau accessible pour non-experts, trois exposés de l'après-midi seront axés sur le sujet parcimonie en signal et image.
Programme
La journée comportera sept communications orales d'une durée de 40 minutes plus 5 minutes de discussion. Cliquer sur un titre pour voir le résumé de l'exposé.
Les exposés se déroulent le matin en salle 309, couloir 15-16, et l'après-midi
en salle 326, couloir 15-25, toutes les deux
au troisième étage. Les pauses se déroulent dans une salle café en face de la salle 309.
Liste de participants
Stephen Becker Laboratoire JLL, Paris VI
Laurent Condat GREYC, Image team, Caen
Rémi Gribonval INRIA Rennes
Daniel Kressner ANCHP, EPF Lausanne
Steve Oudot INRIA Saclay
Nelly Pustelnik IMS, Bordeaux
François-Xavier Vialard CEREMADE, Université Paris-Dauphine
Marie-Laurence Mazure Laboratoire Jean Kuntzmann, Université Joseph Fourier, Grenoble
Bang Cong Vu Laboratoire Jacques-Louis Lions, Paris 6
Frederic Nataf LJLL
Jean-Marie Mirebeau Ceremade, Université Paris-Dauphine
Patrice Brault LSS Laboratoire des Signaux et Systemes, CNRS UMR8506/ Supelec/ U-Psud
Jerôme Bobin CEA
Valérie Perrier Laboratoire Jean Kuntzmann, Grenoble INP
Cécile Louchet Laboratoire MAPMO, Université d'Orléans
Eric Nyiri Arts et Métiers ParisTech
Yannick Kergosien LIM&BIO, Université Paris13
Frédéric Chazal INRIA Saclay
Christian Gout LMI, INSA de Rouen
Alexandre Bos INRIA Saclay
Christophe Rabut Université de Toulouse (INSA, IMT, IREM)
Anna Jezierska Université Paris-Est, Institut Gaspard Monge
Emilie Chouzenoux LIGM, Université Paris Est Marne-La-Vallée
Caroline Chaux LIGM UMR CNRS 8049
Ana Matos Université de Lille 1
Patrick Chenin Laboratoire LJK Université de Grenoble (UJF)
Louie John VALLEJO Laboratoire Jacques-Louis Lions, Paris VI / Universite des Philippines
Giap Nguyen Laboratoire L3i, Université de La Rochelle
Laurent Duval IFP Energies nouvelles
Andrés ALMANSA CNRS LTCI - Telecom ParisTech
Laurent Sifre Ecole Polytechnique
Gérard Chollet CNRS-LTCI, TELECOM-ParisTech
Thomas Oberlin Laboratoire Jean Kuntzmann
El Hadji Diop CMM, Mines Paris Tech
Samuel Vaiter CEREMADE
Dominique Pastor Lab-STICC Télcom Bretagne
Georgios Tzagkarakis CEA, IRFU/SEDI
Hugo Raguet CEREMADE
Eva Wesfreid CMLA, ENS Cachan
Moulay Abdellah Chkifa UPMC
Rachel ABABOU Ecoles de Saint-Cyr Coëtquidan
Alain PERRONNET LJLL UPMC
Giovanni Chierchia TSI, Telecom Paris-Tech
Jean-Francois Cardoso LTCI CNRS & IAP
Laurent Gajny LSIS - Arts et Métiers ParisTech
Nicolas Schmidt CEREMADE
Fernand Meyer Mines-ParisTech
Charles-Alban Deledalle CEREMADE, Paris Dauphine
Bernhard Beckermann Labo Painlevé, Université de Lille 1
Albert Cohen Laboratoire JLL, Paris VI
Patrick Louis Combettes Laboratoire JLL, Paris VI
Gabriel Peyré Ceremade, Université Paris-Dauphine
...
Les organisateurs
Bernhard Beckermann, Laboratoire Paul Painlevé, Université des Sciences et Technologies de Lille
Albert Cohen,
Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Patrick Louis Combettes, Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie, Paris
Stephen Becker (Laboratoire JLL, Paris VI) : TFOCS: A General First-Order Framework for Solving Constrained
Optimization
There are many specialized solvers that solve very specific convex programs efficiently, but there are
few algorithms that are general and can deal with complicated constraints. To address this and other
problems, we introduce a framework and software package called TFOCS. The method relies on two
tricks: dualization and smoothing. This talk describes the framework and also discusses recent splitting
methods such as the method by Chambolle and Pock.
Laurent Condat (Caen) : Modèles et méthodes pour l'acquisition des images couleurs
matricées
Les images couleurs sont acquises dans les appareils photo au moyen d’un capteur unique sur lequel une
matrice de filtres couleurs (CFA) est superposée. Le problème de dématriçage/débruitage conjoint
consiste à reconstruire une image couleurs à partir des données brutes délivrées par le capteur.
Nous étudions les problématiques liées au choix du CFA et à la reconstruction des images couleurs et
proposons quelques solutions, dont les résultats représentent l'état de l'art.
Rémi Gribonval (INRIA Rennes) :
Sparsity & Co.: Analysis vs Synthesis in Low-Dimensional Signal Models
In the past decade there has been a great interest in a synthesis-based model for signals, based on sparse and redundant representations.
Such a model, which assumes that the signal of interest can be composed as a linear combination of few columns from a given matrix (the dictionary), has been extensively exploited in signal and image processing.
Its applications range from compression, denoising, deblurring & deconvolution, to blind signal separation, and even more recently to new approaches to acquire and measure data with the emerging paradigm of compressive sensing.
The talk will begin with a brief review of the main existing algorithmic and theoretical results dedicated to the recovery of sparse vectors from low-dimensional projections, which form the basis of a number of signal reconstruction approaches for such generic linear inverse problems (e.g., compressed sensing, inpainting, source separation, etc.).
An alternative analysis-based model can be envisioned, where an analysis operator multiplies the signal, leading to a so-called cosparse outcome.
How similar are the two signal models ? Can one derive cosparse regularization algorithms with performance guarantees when the data to be reconstructed is cosparse rather than sparse ?
Existing empirical evidence in the litterature suggests that a positive answer is likely.
In recent work we propose a uniqueness result for the solution of linear inverse problems under a cosparse hypothesis, based on properties of the analysis operator and the measurement matrix. Unlike with the synthesis model, where recovery guarantees usually require the linear independence of sets of few columns from the dictionary, our results suggest that linear dependencies between rows of the analysis operators may be desirable. The nature and potential of these new results will be discussed and illustrated with toy image processing and acoustic imaging experiments (joint work with S. Nam, M. Davies and M. Elad).
Daniel Kressner (EPF Lausanne) : Low-rank matrix and tensor techniques in scientific computing
Matrices with (approximate) low rank structure have
played a pivotal role in the development of fast
solvers in numerical linear algebra. Recently,
significant progress has been made in extending
these ideas to tensors. Nowadays, low-rank
tensor techniques have been applied to a wide range
of problems, including the solution of high-dimensional,
parameter-dependent PDEs and the simulation of
quantum many-body systems. This talk aims at
providing a survey of these developments, focussing
on the use of multivariate approximations for analysing
the accuracy and convergence of algorithms based
on low-rank tensors.
Steve Oudot (Saclay) : Unsupervised Learning using Topological Persistence
In this talk I will present a clustering scheme that
combines a mode-seeking phase with a cluster merging phase in the
corresponding density map. While mode detection is done by a standard
graph-based hill-climbing scheme, the novelty of the proposed approach
resides in its use of topological persistence to guide the merging of
clusters. The algorithm provides additional feedback in the form of a
set of points in the plane, called a persistence diagram, which
provably reflects the prominences of the modes of the density. In
practice, this feedback enables the user to choose relevant parameter
values, so that under mild sampling conditions the algorithm will
output the correct number of clusters, a notion that can be made
formally sound within persistence theory. After presenting the
algorithm and showing some experimental data, I will move on to the
theory and introduce topological persistence together with its most
fundamental result: the stability of persistence diagrams.
Nelly Pustelnik (Bordeaux) : Proximal methods for constrained cosparse modelling
The concept of cosparsity has been recently introduced in the arena
of compressed sensing. It consists of minimizing the l0 norm (or the l1 norm)
of an analysis-based representation of the target signal under a data fidelity
constraint. The main contribution of this work is the introduction of a
new projection technique, which allows us to consider more
flexible data fidelity constraints than the standard quadratic one. The
validity of our approach is illustrated through an application to
image restoration in the presence of Poisson noise.
(work in collaboration with G. Chierchia, J.-C. Pesquet, and B. Pesquet-Popescu)
François-Xavier Viallard (Ceremade, Université Dauphine) : Geodesic regression and cubic splines on shape spaces
After a brief introduction to our target applications in biomedical imaging, we present the generalisation of two
standard mathematical tools to the space of shapes in a diffeomorphic framework, namely
linear regression and cubic splines.
These two generalizations have very different goals that are both motivated by the quantitative and
statistical study of time sequences of shapes (for which the time sampling is relatively sparse) which
is a subject of growing interest in the biomedical imaging community. The mathematical tools
involved are riemannian geometry infinite dimension and simple optimal control tools.