SIGMA 2015

Journée scientifique SMAI-SIGMA 2015

2 novembre 2015
Institut Henri Poincaré
11, rue Pierre et Marie Curie, 75005 Paris

Conférenciers invités

• Markus Bachmayr, LJLL, Université Pierre et Marie Curie
• Claire Boyer, IMT, Université Toulouse III
• Marianne Clausel, LJK, Université de Grenoble
• Carole Le Guyader, LMI, INSA de Rouen
• Jean-Marie Mirebeau, LMO, Université Paris-Sud
• Konstantin Usevich, Gipsa-Lab, Grenoble INP

Inscription

L’inscription est gratuite mais obligatoire (pour tenir compte de la capacité de la salle et dimensionner les pauses café), et doit se faire via le formulaire suivant.

Programme

Les exposés auront lieu à l’Institut Henri Poincaré, amphithéâtre Darboux. Les horaires des exposés sont susceptibles de changer.

HoraireConférencier
9h15 - 9h30Accueil
9h30 - 10h15Jean-Marie Mirebeau
10h15 - 10h45Pause café et discussions
11h30 - 12h15Marianne Clausel
12h15 - 13h45Déjeuner
13h45 - 14h45Assemblée générale SMAI-SIGMA (ouverte à tous)
14h45 - 15h30Claire Boyer
15h30 - 16h00Pause café et discussions
16h00 - 16h45Konstantin Usevich
16h45 - 17h30Markus Bachmayr

Markus Bachmayr: Sparse approximation of elliptic PDEs with lognormal coefficients

We consider elliptic partial differential equations with diffusion coefficients of lognormal form, that is a=exp(b) where b is a Gaussian random field. For such problems, we study the \ell^p-summability properties of the Hermite polynomial expansion of the solution in terms of the countably many scalar parameters appearing in a given representation of $b$. These summability results have direct consequences on the approximation rates of best $n$-term truncated Hermite expansions. Our results significantly improve on the state of the art estimates established for this problem by Hoang and Schwab. In particular, the new estimates take into account the support properties of the basis functions involved in the representation of b, in addition to the size of these functions. Furthermore, we also outline new results in the same spirit for affinely parametrized diffusion coefficients, which in a similar manner improve on known estimates by Cohen, DeVore, and Schwab. One interesting conclusion from our analysis in the lognormal case is that in certain relevant examples, the Karhunen-Loève representation of b is not the best choice in terms of the sparsity and approximability of the resulting Hermite expansion. This is joint work with Albert Cohen, Ronald DeVore, and Giovanni Migliorati.

Claire Boyer: Block-constrained compressed sensing

L’exposé s’articulera autour de 2 thèmes. Premièrement, nous essaierons de justifier théoriquement l’utilisation du compressed sensing dans des applications réelles. Pour ce faire, nous présenterons de nouveaux théorèmes CS compatibles avec des contraintes d’acquisition réalistes et souvent rencontrées en pratique. Ces nouveaux résultats prennent en compte -à la fois- une acquisition structurée mais aussi une parcimonie structurée du signal à reconstruire. Dans un second temps, nous présenterons une nouvelle manière de générer des schémas de (sous-)échantillonnage, implémentables et donnant de bons résultats de reconstruction. Ces travaux s’appuient sur les projections de mesures et seront illustrés dans le cadre de l’imagerie IRM.

Marianne Clausel: Hyperbolic wavelet transform: a new tool for analysis of anisotropic textures

Anisotropic images – that is images having different geometric characteristics along different directions – naturally appear in various areas (biomedical, hydrology, geostatistics and spatial statistics, etc.) ([1], [2], [3], [4]…). The detection and characterization of anisotropy is then an important issue in many applications (see [2] for example).

In our work, we analyze global and local anisotropic regularity of any image, using hyperbolic wavelet basis, well–suited to this framework. Further, we relate local regularity features to global quantities related to hyperbolic wavelet coefficients of the analyzed texture. We then pave the way to the introduction of new multifractal attributes for images, allowing to describe simultaneously possible complex scales invariances properties and anisotropic characteristics.

1. Benson, D., Meerschaert, M.M., Baumer, B., and Sheffler, H.P. (2006). Aquifer Operator-Scaling and the effect on solute mixing and dispersion. Water Resour.Res. 42 W01415,1–18.

2. Bonami, A. and Estrade, A. (2003). Anisotropic analysis of some Gaussian models. The Journal of Fourier Analysis and Applications 9, 215-236.

3. Bierme, H., Meerschaert, M.M. and Scheffler, H.P. (2007). Operator Scaling Stable Random Fields. Stoch. Proc. Appl. 117 (3), 312–332.

4. Davies, S. and Hall, P. (1999). Fractal analysis of surface roughness by using spatial data (with discussion). J. Roy. Statist. Soc. Ser. B 61, 3-37.

Carole Le Guyader: Joint Segmentation/Registration Model by Shape Alignment Via Weighted Total Variation Minimization and Nonlinear Elasticity

This presentation falls within the scope of joint segmentation-registration using nonlinear elasticity principles. Saint Venant-Kirchhoff materials being the simplest hyperelastic materials (hyperelasticity being a suitable framework when dealing with large and nonlinear deformations), we propose viewing the shapes to be matched as such materials. Then we introduce a variational model combining a measure of dissimilarity based on weighted total variation and a regularizer based on the stored energy function of a Saint Venant-Kirchhoff material. Adding a weighted total variation based criterion enables to align the edges of the objects even if the modalities are different. We derive a relaxed problem associated to the initial one for which we are able to provide a result of existence of minimizers. A description and analysis of a numerical method of resolution based on a decoupling principle is then provided including a theoretical result of Γ-convergence. Applications on academic and biological images are provided.

Jean-Marie Mirebeau: Computing shortest paths on manifolds: algorithms and applications.

The Fast Marching algorithm is an efficient numerical method for computing the shortest path between points of a domain in R^d, by solving an eikonal PDE. It has numerous applications, ranging from motion planning to medical image segmentation. The unit of length, for computing the path length, may vary on the domain.

Motivated by applications, we generalize the algorithm to the case where the unit of length also depends on the path direction. A conflict arises between this anisotropic geometry and the rigid structure of the cartesian grid. Its solution involves elegant and uncommon tools in numerical analysis, such as the classification of low dimensional lattices, and Stern-Brocot’s arithmetic tree. As an illustration, we extract the retina irrigation network by identifying vessels with weighted euler elastica curves (in collaboration with Laurent Cohen, Da Chen).

Konstantin Usevich: Structured low-rank matrix approximation and completion

Structured low-rank approximation (SLRA) is the problem of approximating a given structured matrix (for example, Hankel/Toeplitz, block-Hankel, Sylvester, etc.) by a low-rank matrix with the same structure. SLRA occurs in signal/image processing, computer algebra, tensor decompositions, among others. An important case is when some of the elements of the given matrix are not specified (SLRA with missing data), also known as structured low-rank matrix completion (SLRMC). In general, SLRA and SLRMC are difficult non-convex optimization problems, except for few special cases.

In the first part of the talk, I will give a brief overview of applications and algorithms for SLRA/SLRMC. In the second part, I will focus on a convex relaxation for these problems based on the nuclear norm heuristic. I will present some recent results on performance of this convex relaxation for exact SLRMC.

Participants

La liste des participants sera mise à jour régulièrement.

1. Marie-Laurence Mazure, Université de Grenoble
2. Quentin Mérigot, CNRS / Université Paris-Dauphine
3. Albert Cohen, UPMC
4. Gabriel Peyré, CNRS et Paris-Dauphine
5. Bernd Beckermann, Labo Painlevé, Université Lille 1
7. Dario Prandi, Univ. Paris-Dauphine
8. Jonathan Vacher, Ceremade & Unic
9. Jerome Bobin, CEA
10. Frederic Nataf, CNRS-UPMC-INRIA
12. Vincent Duval, INRIA (MOKAPLAN)
13. Roman Andreev, UP7D/LJLL
14. Jean-Luc Starck, CEA, Service d’Astrophysique
16. Arthur Leclaire, ENS Cachan
17. Li-Thiao-Té Sébastien, Université Paris 13
18. Viet Tran, Doctorant
19. Jean-Charles Pinoli, ENS des Mines (Saint-Etienne)
20. Rachel Ababou, Ecoles de Saint-Cyr Coëtquidan
21. Ana Matos, Université de Lille 1
22. Martin Campos Pinto, LJLL, UPMC
23. Carole Le Guyader, INSA Rouen
24. Konstantin Usevich, GIPSA-lab, CNRS
25. Markus Bachmayr, LJLL, UPMC Paris 6
26. Claire Boyer, Institut de Mathématiques de Toulouse
27. Luca Nenna, INRIA
28. Hugo Raguet, Institut de Mathématiques de Marseilles
29. Valérie Perrier, Laboratoire Jean Kuntzmann
30. Clarice Poon, Universite Paris Dauphine
31. Antoine Levitt, Inria & CERMICS
32. Cécile Louchet, Université d’Orléans
33. Laurent Gajny, IBHGC, Arts et Métiers ParisTech
34. Kévin Degraux, UCLouvain - Belgium