We show that various inverse problems in signal recovery can be formulated as the generic problem of minimizing the sum of two convex functions with certain regularity properties. This formulation makes it possible to derive existence, uniqueness, characterization, and stability results in a unified and standardized fashion for a large class of apparently disparate problems. Recent results on monotone operator splitting methods are applied to establish the convergence of a forward-backward algorithm to solve the generic problem. In turn, we recover, extend, and provide a simplified analysis for a variety of existing iterative methods. Applications to geometry/texture image decomposition schemes are also discussed. A novelty of our framework is to use extensively the notion of a proximity operator, which was introduced by Moreau in the 1960s.

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Signal recovery by proximal forward-backward splitting ∗

This paper describes the undecimated wavelet transform and its reconstruction. In the first part, we show the relation between two well known undecimated wavelet transforms, the standard undecimated wavelet transform and the isotropic undecimated wavelet transform. Then we present new filter banks specially designed for undecimated wavelet decompositions which have some useful properties such as being robust to ringing artifacts which appear generally in wavelet-based denoising methods. A range of examples illustrates the results

/pdf/the-undecimated-wavelet-decomposition-and-its-reconstruction-80p5bbb7vh.pdf

The Undecimated Wavelet Decomposition and its Reconstruction

Publisher Summary This chapter presents an alternative deterministic methodology, based on sparsity, toward the problem of morphological component analysis (MCA) and anchors this method with some conclusive theoretical results, essentially guaranteeing successful separation under some conditions. The chapter also demonstrates the use of MCA in several applications for images. A major role in the application of the MCA method is played by the dictionaries chosen for decomposition. A wide survey of possible fast-implementation dictionaries taken from the wavelet theory is presented in the chapter, along with ways to use these dictionaries in linear and nonlinear settings. The combination of multi-scale transforms leads to a powerful method in the MCA framework. For some applications such de-noising or de-convolution, MCA is, however, not the best way to combine the different transforms and to benefit from the advantages of each of them.

/pdf/redundant-multiscale-transforms-and-their-application-for-3ry2h9p1yy.pdf

Redundant Multiscale Transforms and Their Application for Morphological Component Separation

These notes give a mathematical introduction to compressive sensing focusing on recovery using `1-minimization and structured random matrices. An emphasis is put on techniques for proving probabilistic estimates for condition numbers of structured random matrices. Estimates of this type are key to providing conditions that ensure exact or approximate recovery of sparse vectors using `1-minimization.

http://www.mathc.rwth-aachen.de/~rauhut/files/LinzRauhut.pdf

Compressive Sensing with structured random matrices

In Monte Carlo methods quadrupling the sample size halves the error. In simulations of stochastic partial differential equations (SPDEs), the total work is the sample size times the solution cost of an instance of the partial differential equation. A Multi-level Monte Carlo method is introduced which allows, in certain cases, to reduce the overall work to that of the discretization of one instance of the deterministic PDE. The model problem is an elliptic equation with stochastic coefficients. Multi-level Monte Carlo errors and work estimates are given both for the mean of the solutions and for higher moments. The overall complexity of computing mean fields as well as k-point correlations of the random solution is proved to be of log-linear complexity in the number of unknowns of a single Multi-level solve of the deterministic elliptic problem. Numerical examples complete the theoretical analysis.

/pdf/multi-level-monte-carlo-finite-element-method-for-elliptic-5cjy4s9lmx.pdf

Multi-level Monte Carlo Finite Element method for elliptic PDEs with stochastic coefficients

Introduction. Notations. 1. Basic examples. 1.1 Introduction. 1.2 The Haar system. 1.3 The Schauder hierarchical basis. 1.4 Multivariate constructions. 1.5 Adaptive approximation. 1.6 Multilevel preconditioning. 1.7 Conclusions. 1.8 Historical notes. 2. Multiresolution approximation. 2.1 Introduction. 2.2 Multiresolution analysis. 2.3 Refinable functions. 2.4 Subdivision schemes. 2.5 Computing with refinable functions. 2.6 Wavelets and multiscale algorithms. 2.7 Smoothness analysis. 2.8 Polynomial exactness. 2.9 Duality, orthonormality and interpolation. 2.10 Interpolatory and orthonormal wavelets. 2.11 Wavelets and splines. 2.12 Bounded domains and boundary conditions. 2.13 Point values, cell averages, finite elements. 2.14 Conclusions. 2.15 Historical notes. 3. Approximation and smoothness. 3.1 Introduction. 3.2 Function spaces. 3.3 Direct estimates. 3.4 Inverse estimates. 3.5 Interpolation and approximation spaces. 3.6 Characterization of smoothness classes. 3.7 Lp-unstable approximation and 0 1. 3.8 Negative smoothness and Lp-spaces. 3.9 Bounded domains. 3.10 Boundary conditions. 3.11 Multilevel preconditioning. 3.12 Conclusions. 3.13 Historical notes. 4. Adaptivity. 4.1 Introduction. 4.2 Nonlinear approximation in Besov spaces. 4.3 Nonlinear wavelet approximation in Lp. 4.4 Adaptive finite element approximation. 4.5 Other types of nonlinear approximations. 4.6 Adaptive approximation of operators. 4.7 Nonlinear approximation and PDE's. 4.8 Adaptive multiscale processing. 4.9 Adaptive space refinement. 4.10 Conclusions. 4.11 Historical notes. References. Index.

Numerical Analysis of Wavelet Methods

Objective. In early infancy the infant9s thumb is not infrequently enclosed within the palm, ie, thumb-in-fist (TIF). This posture has received scant attention in the neurodevelopmental literature. Its prevalence, resolution, and clinical associations were investigated in this study. Methodology. Two hundred sequentially born, apparently healthy full-term newborn infants comprised the cohort. The whole study group was followed up until the disappearance of the TIF occurred. In the first 150 of the cohort, additional data on development and the neurological status were obtained at 12 months of life. Results. In 125 infants (62.5%) of the total cohort, a TIF was noted. The mean age of disappearance was 1.5 months, and no TIF persisted after 7 months old. No relationship was noted between the TIF resolution and abnormal neurological signs or gross or fine motor development. The only association noted between age of resolution of the TIF and the neurodevelopmental status was a delay in language attainment at the 12-month screening. Conclusions. The TIF posture in infancy was noted in 65% of our cohort, and it had resolved in all infants by 7 months old. Therefore, a TIF posture after this age should alert the clinician to the possibility of possible neurological dysfunction. An unanticipated association of a delay in the 12-month language milestone attainment was noted in those infants with later resolution of the TIF posture. No data on language development in this group were obtained after 12 months old; therefore, the clinical significance of this finding is not yet elucidated.

Ayala Cohen

Papers

Numerical Analysis of Wavelet Methods

Infants With a Thumb-in-Fist Posture

Acid versus non-acid reflux