Abstract. We propose simple and extremely efficient methods for solving the basis pursuit problem min{ u 1 : Au = f, u ∈ R n }, which is used in compressed sensing. Our methods are based on Bregman iterative regularization, and they give a very accurate solution after solving only a very small number of instances of the unconstrained problem min u∈R n μ u 1 + 1 2 Au − f k 2 2 for given matrix A and vector f k . We show analytically that this iterative approach yields exact solutions in a finite number of steps and present numerical results that demonstrate that as few as two to six iterations are sufficient in most cases. Our approach is especially useful for many compressed sensing applications where matrix-vector operations involving A and A can be computed by fast transforms. Utilizing a fast fixed-point continuation solver that is based solely on such operations for solving the above unconstrained subproblem, we were able to quickly solve huge instances of compressed sensing problems on a standard PC.

Abstract. This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under L 1 data fidelity term yields a self-dual contrast invariant filter. Finally we present some results.

In a recent paper Boykov et al. (LNCS, Vol. 3953, pp. 409-422, 2006) propose an approach for computing curve and surface evolution using a variational approach and the geo-cuts method of Boykov and Kolmogorov (International conference on computer vision, pp. [26][27][28][29][30][31][32][33] 2003). We recall in this paper how this is related to well-known approaches for mean curvature motion, introduced by Alm-

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It is well known that time dependent Hamilton-Jacobi-Isaacs partial differential equations (HJ PDE), play an important role in analyzing continuous dynamic games and control theory problems. An important tool for such problems when they involve geometric motion is the level set method, [41]. This was first used for reachability problems in [36], [37]. The cost of these algorithms, and, in fact, all PDE numerical approximations is exponential in the space dimension and time.In [13], some connections between HJ-PDE and convex optimization in many dimensions are presented. In this work we propose and test methods for solving a large class of the HJ PDE relevant to optimal control problems without the use of grids or numerical approximations. Rather we use the classical Hopf formulas for solving initial value problems for HJ PDE [30]. We have noticed that if the Hamiltonian is convex and positively homogeneous of degree one (which the latter is for all geometrically based level set motion and control and differential game problems) that very fast methods exist to solve the resulting optimization problem. This is very much related to fast methods for solving problems in compressive sensing, based on ℓ 1 optimization [24], [51]. We seem to obtain methods which are polynomial in the dimension. Our algorithm is very fast, requires very low memory and is totally parallelizable. We can evaluate the solution and its gradient in very high dimensions at 10 −4 to 10 −8 seconds per evaluation on a laptop.We carefully explain how to compute numerically the optimal control from the numerical solution of the associated initial valued HJ-PDE for a class of optimal control problems. We show that our algorithms compute all the quantities we need to obtain easily the controller.In addition, as a step often needed in this procedure, we have developed a new and equally fast way to find, in very high dimensions, the closest point y lying in the union of a finite number of compact convex sets Ω to any point x exterior to the Ω. We can also compute the distance to these sets much faster than Dijkstra type "fast methods ", e.g. [15].The term "curse of dimensionality", was coined by Richard Bellman in 1957, [3][4], when considering problems in dynamic optimization.

Abstract-The problem of person recognition and verification based on their hand images has been addressed. The system is based on the images of the right hands of the subjects, captured by a flatbed scanner in an unconstrained pose at 45 dpi. In a preprocessing stage of the algorithm, the silhouettes of hand images are registered to a fixed pose, which involves both rotation and translation of the hand and, separately, of the individual fingers. Two feature sets have been comparatively assessed, Hausdorff distance of the hand contours and independent component features of the hand silhouette images. Both the classification and the verification performances are found to be very satisfactory as it was shown that, at least for groups of about five hundred subjects, hand-based recognition is a viable secure access control scheme.

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