The first part of this paper proposes an adaptive, data-driven threshold for image denoising via wavelet soft-thresholding. The threshold is derived in a Bayesian framework, and the prior used on the wavelet coefficients is the generalized Gaussian distribution (GGD) widely used in image processing applications. The proposed threshold is simple and closed-form, and it is adaptive to each subband because it depends on data-driven estimates of the parameters. Experimental results show that the proposed method, called BayesShrink, is typically within 5% of the MSE of the best soft-thresholding benchmark with the image assumed known. It also outperforms Donoho and Johnstone's SureShrink most of the time. The second part

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...

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.

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We consider model selection in a hierarchical Bayes formulation of the sparse normal linear model in which individual variables have, independently, an unknown prior probability of being included in the model. The focus is on orthogonal designs, which are of particular importance in nonparametric regression via wavelet shrinkage. Empirical Bayes estimates of hyperparameters are easily obtained via the EM algorithm, and this approach is contrasted with a recent conditional likelihood proposal. Our model selection approach yields a straightforward method for data dependent threshold selection in wavelet regression. Performance on standard test sets and data examples is encouraging, especially if a translation invariant form of the estimator is used. Since the method produces separate threshold estimates on each wavelet resolution level, it also comfortably handles stationary correlated error structures.

We consider Bayesian approach to wavelet decomposition. We show how prior knowledge about a function's regularity can be incorporated into a prior model for its wavelet coefficients by establishing a relationship between the hyperparameters of the proposed model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation may be seen as giving insight into the meaning of the Besov space parameters themselves. Furthermore, we consider Bayesian wavelet-based function estimation that gives rise to different types of wavelet shrinkage in non-parametric regression. Finally, we discuss an extension of the proposed Bayesian model by considering random functions generated by an overcomplete wavelet dictionary. 1 Introduction Consider the standard non-parametric regression problem: y i = g(t i ) + ffl i ; i = 1; : : : ; n; (1.1) and suppose we wish to recover the unknown function g from additive noise ffl i given noisy data y i at discrete point...

We present a new kind of second generation wavelets on a rectangular grid. These wavelets are constructed using a 2D lifting scheme which is based on a red-black blocking scheme. Compared to classical tensor product wavelets on the same grid, these new wavelets show less anisotropy. The performance of the new wavelets is compared to tensor product wavelets in an image denoising application.

De-noising algorithms based on wavelet thresholding replace small wavelet coefficients by zero and keep or shrink the coefficients with absolute value above the threshold. The optimal threshold minimizes the error of the result as compared to the unknown, exact data. To estimate this optimal threshold, we use Generalized Cross Validation. This procedure does not require an estimation for the noise energy. Originally, this method assumes uncorrelated noise. In this paper we describe how we can extend it to images with correlated noise.

This article introduces a fast cross-validation algorithm that performs wavelet shrinkage on data sets of arbitrary size and irregular design and also simultaneously selects good values of the primary resolution and number of vanishing moments. We demonstrate the utility of our method by suggesting alternative estimates of the conditional mean of the well-known Ethanol data set. Our alternative estimates outperform the Kovac-Silverman method with a global variance estimate by 25% because of the careful selection of number of vanishing moments and primary resolution. Our alternative estimates are simpler than, and competitive with, results based on the Kovac-Silverman algorithm equipped with a local variance estimate. We include a detailed simulation study that illustrates how our cross-validation method successfully picks good values of the primary resolution and number of vanishing moments for unknown functions based on Walsh functions (to test the response to changing pri...