Learning the Morphological Diversity

International audienceThis article proposes a new method for image separation into a linear combination of morphological components. Sparsity in global dictionaries is used to extract the cartoon and oscillating content of the image. Complicated texture patterns are extracted by learning adapted local dictionaries that sparsify patches in the image. These global and local sparsity priors together with the data fidelity define a non-convex energy and the separation is obtained as a stationary point of this energy. This variational optimization is extended to solve more general inverse problems such as inpainting. A new adaptive morphological component analysis algorithm is derived to find a stationary point of the energy. Using adapted dictionaries learned from data allows to circumvent some difficulties faced by fixed dictionaries. Numerical results demonstrate that this adaptivity is indeed crucial to capture complex texture patterns

MPGD's spatial and energy resolution studies with an adjustable point-like electron source

11th Vienna Conference on Instrumentation (February 2007) , to appear in the Proceedings (NIM A)International audienceMicropattern Gaseous Detectors (MPGD), like Micromegas or GEM, are used or foreseen in particle physics experiments for which a very good spatial resolution is required. We have developed an experimental method to separate the contribution of the transverse diffusion and the multiplication process by varying the number of primary electrons generated by a point-like source. A pulsed nitrogen laser is focused by an optical set-up on the drift electrode which is made of a thin metal layer deposited on a quartz lamina. The number of primary electrons can be adjusted from a few to several thousands on a spot which transverse size is less than $100 \mu m$ RMS. The detector can be positioned with an accuracy of $1\mu m$ by a motorized three dimensional system. This method was applied to a small Micromegas detector with a gain set between $10^3$ and $2.10^4$ and an injection of 60 to 2000 photoelectrons. Spatial resolutions as small as $5\mu m$ were measured with 2000 primary electrons. An estimation of the upper limit of the relative gain variance can be obtained from the measurements

A Γ\Gamma-Convergence Result for the Upper Bound Limit Analysis of Plates

Upper bound limit analysis allows one to evaluate directly the ultimate load of structures without performing a cumbersome incremental analysis. In order to numerically apply this method to thin plates in bending, several authors have proposed to use various finite elements discretizations. We provide in this paper a mathematical analysis which ensures the convergence of the finite element method, even with finite elements with discontinuous derivatives such as the quadratic 6 node Lagrange triangles and the cubic Hermite triangles. More precisely, we prove the $\Gamma$-convergence of the discretized problems towards the continuous limit analysis problem. Numerical results illustrate the relevance of this analysis for the yield design of both homogeneous and non-homogeneous materials

Iterative Bregman Projections for Regularized Transportation Problems

International audienceThis article details a general numerical framework to approximate so-lutions to linear programs related to optimal transport. The general idea is to introduce an entropic regularization of the initial linear program. This regularized problem corresponds to a Kullback-Leibler Bregman di-vergence projection of a vector (representing some initial joint distribu-tion) on the polytope of constraints. We show that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form. This allows us to make use of iterative Bregman projections (when there are only equality constraints) or more generally Bregman-Dykstra iterations (when inequality constraints are in-volved). We illustrate the usefulness of this approach to several variational problems related to optimal transport: barycenters for the optimal trans-port metric, tomographic reconstruction, multi-marginal optimal trans-port and in particular its application to Brenier's relaxed solutions of in-compressible Euler equations, partial un-balanced optimal transport and optimal transport with capacity constraints

Sinkhorn Divergences for Unbalanced Optimal Transport

Optimal transport induces the Earth Mover's (Wasserstein) distance between probability distributions, a geometric divergence that is relevant to a wide range of problems. Over the last decade, two relaxations of optimal transport have been studied in depth: unbalanced transport, which is robust to the presence of outliers and can be used when distributions don't have the same total mass; entropy-regularized transport, which is robust to sampling noise and lends itself to fast computations using the Sinkhorn algorithm. This paper combines both lines of work to put robust optimal transport on solid ground. Our main contribution is a generalization of the Sinkhorn algorithm to unbalanced transport: our method alternates between the standard Sinkhorn updates and the pointwise application of a contractive function. This implies that entropic transport solvers on grid images, point clouds and sampled distributions can all be modified easily to support unbalanced transport, with a proof of linear convergence that holds in all settings. We then show how to use this method to define pseudo-distances on the full space of positive measures that satisfy key geometric axioms: (unbalanced) Sinkhorn divergences are differentiable, positive, definite, convex, statistically robust and avoid any "entropic bias" towards a shrinkage of the measures' supports

Sparse reconstruction from a limited projection number of the coronary artery tree in X-ray rotational imaging

International audienceThis paper deals with the 3D reconstruction of sparse data in X-ray rotational imaging. Due to the cardiac motion, the number of available projections for this reconstruction is equal to four, which leads to a strongly undersampled reconstruction problem. We address thus this illness problem through a regularized iterative method. The whole algorithm is divided into two steps. Firstly, a minimal path segmentation step extracts artery tree boundaries. Secondly, a MAP reconstruction comparing L0-norm and L1-norm priors is applied on this extracted coronary tree. The reconstruction optimization process relies on a separable paraboloidal (SPS) algorithm. Some preliminary results are provided on simulated rotational angiograms

Interpolating between Optimal Transport and MMD using Sinkhorn Divergences

Comparing probability distributions is a fundamental problem in data sciences. Simple norms and divergences such as the total variation and the relative entropy only compare densities in a point-wise manner and fail to capture the geometric nature of the problem. In sharp contrast, Maximum Mean Discrepancies (MMD) and Optimal Transport distances (OT) are two classes of distances between measures that take into account the geometry of the underlying space and metrize the convergence in law. This paper studies the Sinkhorn divergences, a family of geometric divergences that interpolates between MMD and OT. Relying on a new notion of geometric entropy, we provide theoretical guarantees for these divergences: positivity, convexity and metrization of the convergence in law. On the practical side, we detail a numerical scheme that enables the large scale application of these divergences for machine learning: on the GPU, gradients of the Sinkhorn loss can be computed for batches of a million samples

Signification géodynamique des calcaires de plate-forme en cours de subduction sous l'arc des Nouvelles-Hébrides (Sud-Ouest de l'océan Pacifique)

Note présentée par Jean DercourtInternational audienceThe analysis of carbonates from New Hébrides Trench shows that three main épisodes of shallow water carbonate déposition occurred during Late Eocene,Late Oligocene-Early Miocène,Mio-Pliocene-Quaternary, controlled by eustatism and tectonic.L'analyse de carbonates issus de la fosse des Nouvelles-Hébrides a permis de reconnaître trois périodes favorables au développement de plates-formes(Éocène supérieur,Oligocène supérieur-Miocène inférieur,Mio-Pliocène-Quaternaire)contrôlé par l'eustatisme et la tectonique