CLEAR: Covariant LEAst-square Re-fitting with applications to image restoration

In this paper, we propose a new framework to remove parts of the systematic errors affecting popular restoration algorithms, with a special focus for image processing tasks. Generalizing ideas that emerged for $\ell_1$ regularization, we develop an approach re-fitting the results of standard methods towards the input data. Total variation regularizations and non-local means are special cases of interest. We identify important covariant information that should be preserved by the re-fitting method, and emphasize the importance of preserving the Jacobian (w.r.t. the observed signal) of the original estimator. Then, we provide an approach that has a "twicing" flavor and allows re-fitting the restored signal by adding back a local affine transformation of the residual term. We illustrate the benefits of our method on numerical simulations for image restoration tasks

Entropic Regularization Approach for Mathematical Programs with Equilibrium Constraints

A new smoothing approach based on entropic perturbationis proposed for solving mathematical programs withequilibrium constraints. Some of the desirableproperties of the smoothing function are shown. Theviability of the proposed approach is supported by acomputationalstudy on a set of well-known test problems.mathematical programs with equilibrium constraints;entropic regularization;smoothing approach

Robust Cardiac Motion Estimation using Ultrafast Ultrasound Data: A Low-Rank-Topology-Preserving Approach

Cardiac motion estimation is an important diagnostic tool to detect heart diseases and it has been explored with modalities such as MRI and conventional ultrasound (US) sequences. US cardiac motion estimation still presents challenges because of the complex motion patterns and the presence of noise. In this work, we propose a novel approach to estimate the cardiac motion using ultrafast ultrasound data. -- Our solution is based on a variational formulation characterized by the L2-regularized class. The displacement is represented by a lattice of b-splines and we ensure robustness by applying a maximum likelihood type estimator. While this is an important part of our solution, the main highlight of this paper is to combine a low-rank data representation with topology preservation. Low-rank data representation (achieved by finding the k-dominant singular values of a Casorati Matrix arranged from the data sequence) speeds up the global solution and achieves noise reduction. On the other hand, topology preservation (achieved by monitoring the Jacobian determinant) allows to radically rule out distortions while carefully controlling the size of allowed expansions and contractions. Our variational approach is carried out on a realistic dataset as well as on a simulated one. We demonstrate how our proposed variational solution deals with complex deformations through careful numerical experiments. While maintaining the accuracy of the solution, the low-rank preprocessing is shown to speed up the convergence of the variational problem. Beyond cardiac motion estimation, our approach is promising for the analysis of other organs that experience motion.Comment: 15 pages, 10 figures, Physics in Medicine and Biology, 201

The convergence of a one-step smoothing Newton method for P0-NCP based on a new smoothing NCP-function

AbstractThe nonlinear complementarity problem (denoted by NCP(F)) can be reformulated as the solution of a nonsmooth system of equations. By introducing a new smoothing NCP-function, the problem is approximated by a family of parameterized smooth equations. A one-step smoothing Newton method is proposed for solving the nonlinear complementarity problem with P0-function (P0-NCP) based on the new smoothing NCP-function. The proposed algorithm solves only one linear system of equations and performs only one line search per iteration. Without requiring strict complementarity assumption at the P0-NCP solution, the proposed algorithm is proved to be convergent globally and superlinearly under suitable assumptions. Furthermore, the algorithm has local quadratic convergence under mild conditions

A squared smoothing Newton method for nonsmooth matrix equations and its applications in semidefinite optimization problems

10.1137/S1052623400379620SIAM Journal on Optimization143783-80

On the finite termination of an entropy function based smoothing Newton method for vertical linear complementarity problems

By using a smooth entropy function to approximate the non-smooth max-type function, a vertical linear complementarity problem (VLCP) can be treated as a family of parameterized smooth equations. A Newton-type method with a testing procedure is proposed to solve such a system. We show that the proposed algorithm finds an exact solution of VLCP in a finite number of iterations, under some conditions milder than those assumed in literature. Some computational results are included to illustrate the potential of this approach.Newton method;Finite termination;Entropy function;Smoothing approximation;Vertical linear complementarity problems

A Quasi-Newton Subspace Trust Region Algorithm for Least-square Problems in Min-max Optimization

The first-order optimality conditions of convexly constrained nonconvex-nonconcave min-max optimization problems formulate variational inequality problems, which are equivalent to a system of nonsmooth equations. In this paper, we propose a quasi-Newton subspace trust region (QNSTR) algorithm for the least-square problem defined by the smoothing approximation of the nonsmooth equation. Based on the structure of the least-square problem, we use an adaptive quasi-Newton formula to approximate the Hessian matrix and solve a low-dimensional strongly convex quadratic program with ellipse constraints in a subspace at each step of QNSTR algorithm. According to the structure of the adaptive quasi-Newton formula and the subspace technique, the strongly convex quadratic program at each step can be solved efficiently. We prove the global convergence of QNSTR algorithm to an $\epsilon$-first-order stationary point of the min-max optimization problem. Moreover, we present numerical results of QNSTR algorithm with different subspaces for the mixed generative adversarial networks in eye image segmentation using real data to show the efficiency and effectiveness of QNSTR algorithm for solving large scale min-max optimization problems