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In this paper, we investigate various connections between wavelet shrinkage methods in image processing and Bayesian estimation using Generalized Gaussian priors. We present fundamental properties of the shrinkage rules implied by Generalized Gaussian and other heavy-tailed priors. This allows us to show a simple relationship between differentiability of the log-prior at zero and the sparsity of the estimates, as well as an equivalence between universal thresholding schemes and Bayesian estimation using a certain Generalized Gaussian prior.

this article a wavelet shrinkage by coherent

this paper, weintroduce nonlinear regularized wavelet estimators for estimating nonparametric regression functions when sampling points are not uniformly spaced. The approach can apply readily to many other statistical contexts. Various new penalty functions are proposed. The hard-thresholding and soft-thresholding estimators of Donoho and Johnstone (1994) are specic members of nonlinear regularized wavelet estimators. They correspond to the lower and upper bound of a class of the penalized least-squares estimators. Necessary conditions for penalty functions are given for regularized estimators to possess thresholding properties. Oracle inequalities and universal thresholding parameters are obtained for a large class of penalty functions. The sampling properties of nonlinear regularized wavelet estimators are established, and are shown to be adaptively minimax. To eciently solve penalized least-squares problems, Nonlinear Regularized Sobolev Interpolators (NRSI) are proposed as initial estimators, which are shown to have good sampling properties. The NRSI is further ameliorated by Regularized One-Step Estimators (ROSE), which are the one-step estimators of the penalized least-squares problems using the NRSI as initial estimators. Two other approaches, the graduated nonconvexity algorithm and wavelet networks, are also introduced to handle penalized least-squares problems. The newly introduced approaches are also illustrated by a few numerical examples. ####### ########## ## ########## ########### ## ############# ## ####### ######################### ##### ######## ##### ## ####### ######## ### ## ########## ########## ## ########### ########## ## ########### ### ######## ## ########## ### ### ####### ########## ## #### ##### ##### ########### ######### ######### ## ###...

The sparseness and decorrelation properties of the discrete wavelet transform have been exploited to develop powerful denoising methods. However, most of these methods have free parameters which have to be adjusted or estimated. In this paper, we propose a wavelet-based denoising technique without any free parameters; it is, in this sense, a "universal" method. Our approach uses empirical Bayes estimation based on a Jeffreys' noninformative prior; it is a step toward objective Bayesian wavelet-based denoising. The result is a remarkably simple fixed nonlinear shrinkage/thresholding rule which performs better than other more computationally demanding methods.

Abstract—Image deconvolution is formulated in the wavelet domain under the Bayesian framework. The well-known sparsity of the wavelet coefficients of real-world images is modeled by heavy-tailed priors belonging to the Gaussian scale mixture (GSM) class; i.e., priors given by a linear (finite of infinite) combination of Gaussian densities. This class includes, among others, the generalized Gaussian, the Jeffreys, and the Gaussian mixture priors. Necessary and sufficient conditions are stated under which the prior induced by a thresholding/shrinking denoising rule is a GSM. This result is then used to show that the prior induced by the “nonnegative garrote ” thresholding/shrinking rule, herein termed the garrote prior, is a GSM. To compute the maximum a posteriori estimate, we propose a new generalized expectation maximization (GEM) algorithm, where the missing variables are the scale factors of the GSM densities. The maximization step of the underlying expectation maximization algorithm is replaced with a linear stationary second-order iterative method. The result is a GEM algorithm of ( log) computational complexity. In a series of benchmark tests, the proposed approach outperforms or performs similarly to state-of-the art methods, demanding comparable (in some cases, much less) computational complexity. Index Terms—Bayesian, deconvolution, expectation maximization (EM), generalized expectation maximization (GEM), Gaussian scale mixtures (GSM), heavy-tailed priors, wavelet. I.

In recent years there has been a considerable development in the use of wavelet methods in statistics. As a result, we are now at the stage where it is reasonable to consider such methods to be another standard tool of the applied statistician rather than a research novelty. With that in mind, this article is intended to give a relatively accessible introduction to standard wavelet analysis and to provide an up to date review of some common uses of wavelet methods in statistical applications. It is primarily orientated towards the general statistical audience who may be involved in analysing data where the use of wavelets might be e ective, rather than to researchers already familiar with the eld. Given that objective, we do not emphasise mathematical generality or rigour in our exposition of wavelets and we restrict our discussion to the more frequently employed wavelet methods in statistics. We provide extensive references where the ideas and concepts discussed can be followed up in...

In this paper, we combine Donoho and Johnstone's Wavelet Shrinkage denoising technique (known as WaveShrink) with Breiman's non-negative garrote. We show that the non-negative garrote shrinkage estimate enjoys the same asymptotic convergence rate as the hard and the soft shrinkage estimates. Simulations are used to demonstrate that garrote shrinkage offers advantages over both hard shrinkage (generally smaller meansquare -error and less sensitivity to small perturbations in the data) and soft shrinkage (generally smaller bias and overall mean-square-error). The minimax thresholds for the non-negative garrote are derived and the threshold selection procedure based on Stein's Unbiased Risk Estimate (SURE) is studied. We also propose a threshold selection procedure based on combining Coifman and Donoho's cycle-spinning and SURE. The procedure is called SPINSURE. We use examples to show that SPINSURE is more stable than SURE: smaller standard deviation and smaller range. Key Words and Phra...

Recently there has been significant development in the use of wavelet methods in various data mining processes. However, there has been written no comprehensive survey available on the topic. The goal of this is paper to fill the void. First, the paper presents a high-level data-mining framework that reduces the overall process into smaller components. Then applications of wavelets for each component are reviewd. The paper concludes by discussing the impact of wavelets on data mining research and outlining potential future research directions and applications.

In this chapter, we will provide an overview of the current status of research involving Bayesian inference in wavelet nonparametric problems. In many statistical applications, there is a need for procedures to (i) adapt to data and (ii) use prior information. The interface of wavelets and the Bayesian paradigm provide a natural terrain for both of these goals.