40 IS10 Abstracts

40 IS10 Abstracts IP1 age and video restoration. I will present the framework Approximate Inference in Graphical Models: The and provide numerous examples showing state-of-the-art Fenchel Duality Perspective results. I will then briefly show how to extend this to image classification, deriving energies and optimization pro- Quite a number of problems involving inference from data, cedures that lead to learning non-parametric dictionaries whether visual data or otherwise, can be set as a prob- for sparse representations optimized for classification. I lem of finding the most likely setting of variables of a will conclude by showing results on the extension of this to joint distribution over locally interacting variables. These sensing and the learning of incoherent dictionaries. Models include stereopsis, figure-ground segmentation, model-to- derived from universal coding are presented as well. The image matching, super-resolution, and so forth. Inference work I present in this talk is the result of great collabo- over locally interacting variables can be naturally repre- rations with J. Mairal, F. Rodriguez, J. Martin-Duarte, I. sented by a graphical model, however exact inference over Ramirez, F. Lecumberry, F. Bach, M. Elad, J. Ponce, and general graphs is computationally unwieldy. In the context A. Zisserman. of approximate inference, I will describe a general scheme for message passing update rules based on the framework Guillermo Sapiro of Fenchel duality. Using the framework we derive all University of Minnesota past inference algorithms like the Belief Propagation sum- Dept Electrical & Computer Engineering product and max-product, the Tree-Re-Weighted (TRW) [email protected] model as well as new convergent algorithms for maximum- a-posteriori (MAP) and marginal estimation using ”convex free energies”. Joint work with Tamir Hazan. IP4 From Shape Matching to Shape Mechanics: Geom- Amnon Shashua etry and Dynamics The Hebrew University of Jerusalem, Israel [email protected] During the last decades, the study of shape spaces has been driven by the accelerated development of imaging techniques in biomedical engineering and the emergence of IP2 computational anatomy. Starting from the basic and hard Operator Splitting Techniques in Image Processing problems of shape comparison, shape matching and modeling shape populations, a fascinating landscape is appear- Recently, operator splitting methods have been success- ing involving infinite dimensional manifolds, Lie groups, fully applied to various image processing tasks like the de- Riemannian geometry, Hamiltonian systems and Statistics. noising and deblurring of images also in the presence of Its connection with many existing mathematical theories, non-additive noise, inpainting, sparse recovery problems sometimes outside the usual scope of the imaging commu- and multi-task learning. Splitting methods allow us to de- nity, and the growing need for effective tools in shape anal- compose the original problem into subproblems which are ysis, provide an exciting playground for inter-disciplinary easier to solve.The talk reviews and relates various of these research and collaborations. This talks will visit some optimization methods from the point of view of averaged hot spots in this landscape and some challenges driven by operators, Bregman proximal point methods and primal- anatomical growth modeling. dual Lagrangian approaches. Attention is also paid to mul- tistep methods. Then, various examples are presented how Alain Trouvé splitting algorithms can be successfully applied to image Ecole Normale Supérieure, France recovery problems. [email protected] Gabriele Steidl Institute of Mathematics und Computer Science IP5 Universitaet Mannheim, Germany The Inverse Problem of Seismic Velocities [email protected] To reasonable approximation, the construction of struc- tural images of the earth’s interior, using seismic reflec- IP3 tion data as input, boils down to an inverse problem for Data Don’t Lie: Image Processing Via Learned Ef- the wave equation. This inverse problem requires that the ficient Representations spatially varying wave velocity (a coefficient in the wave equation) be determined from samples of solutions near the In a large number of disciplines, image acquisition has ad- boundary of the space-time domain of propagation. The vanced much faster than image analysis. This permits to data-fitting techniques that have proven effective in other replace a number of pre-defined concepts by learned ones. science and engineering inverse problems encounter funda- In particular, we can learn efficient image representations. mental mathematical obstacles in application to this one. Among these, sparse representations have recently drawn In response, the seismic prospecting industry has devised a much attention from the signal processing and learning collection of methods to extract earth structure from data, communities. The basic underlying model consist of con- that appear at first glance to have little to do with data- sidering that natural images, or signals in general, admit a fitting. I will describe the seismic inverse problem and the sparse decomposition in some redundant dictionary. This qualities that make it resistant to data-fitting, as well as an means that we can find a linear combination of a few atoms underlying mathematical structure that encompasses both from the dictionary that lead to an efficient representation the data-fitting and industrial approaches. This structure of the original signal. Recent results have shown that learn- supports variational principles, more general than the data- ing (overcomplete) non-parametric dictionaries for image fitting or least squares principle, which in some cases have representations, instead of using off-the-shelf ones, sig- proven effective tools for velocity estimation. nificantly improves numerous image and video processing tasks. In this talk, I will first briefly present results on William Symes learning multiscale overcomplete dictionaries for color im- Rice University IS10 Abstracts 41 [email protected] [email protected] Carlos Brito-Loeza IP6 Centre for Mathematical Imaging Techniques and Compressed Sensing in Astronomy Department of Mathematical Sciences, University of Liverpool The spatial infrared astronomical satellite Herschel, [email protected] launched on May 14 2009, is the first satellite which has a compressed sensing (CS) coder included in its on board software. We will present our motivations for using Com- CP1 pressed Sensing in this project. Then we will describe the Method of Micromirrors For Catadioptric Sensor practical implementation of our CS coder/decoder. We Design will show from simulations that CS enables to recover data with a spatial resolution enhanced up to 30% with similar We present a novel method for catadioptric sensor design sensitivity compared to the averaging technique proposed which consists of a micromirror array, asymmetric mirror by ESA, and we will present preliminary results relative and an orthographic camera. The micromirror method al- to the CS performances on real Herschel data. Finally lows construction of any desired projection which was not we will show how other problems in astronomy such in- possible with single-mirror systems. We use Frobenius In- terferometric image deconvolution or gammay ray image tegrability Theorem to write a system of quasilinear PDEs, reconstruction can be also handled differently using CS. whose numerical solution is the camera projection and nu- merically integrate the normal vector field to compute the Jean-Luc Starck mirror surface. Laboratoire AIM, CEA/DSM-CNRS-Universite Paris Diderot Emek Kose Can CEA Saclay, DAPNIA/SEDI-SAP, Service Loyola Marymount University d’Astrophysique [email protected] [email protected] CP1 CP1 Topological Gradient for Image Restoration by Optimal Geometric Transformations of Compact Anisotropic Diffusion Manifolds Topological asymptotic analysis provides tools to detect A fundamental mathematical problem underlying the sub- edges and their orientation. The purpose of this work ject of matching and minimal morphing of images is the is to show the possibilities of anisotropic topological gra- problem of minimal distortion bending and morphing of dient in image restoration. Previous methods based on compact manifolds. We consider cost functionals that mea- the topological gradient used isotropic diffusion to restore sure distortion of geometry produced by a diffeomorphism the image. These methods are improved here by using between two compact n-manifolds. Our cost functionals anisotropic diffusion, a texture detector and differentiat- measure the distortion in volume or the first (metric) and ing between principal and secondary edges. Numerical re- the second (curvature) fundamental forms. We prove the sults are presented, including a comparison with Non-Local existence of the least distortion energy maps. Means method. Oksana Bihun Stanislas Larnier,Jérôme Fehrenbach Concordia College, Moorhead, MN Université Paul Sabatier [email protected] Institut de Mathématiques de Toulouse [email protected], [email protected] CP1 A Fourth Order Nonlinear Pde-Based Image Reg- istration Model and Its Fast Algorithm CP1 Robust Optical Flow for Modeling Dynamical Sys- We first propose a new variational registration model using tems full curvature regularization, leading to superior results, for both smooth and nonsmooth deformations, to competitors Classical optical flow techniques

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