This paper reviews a significant component of the rich field of statistical multiresolution (MR) modeling and processing. These MR methods have found application and permeated the literature of a widely scattered set of disciplines, and one of our principal objectives is to present a single, coherent picture of this framework. A second goal is to describe how this topic fits into the even larger field of MR methods and concepts–in particular making ties to topics such as wavelets and multigrid methods. A third is to provide several alternate viewpoints for this body of work, as the methods and concepts we describe intersect with a number of other fields. The principle focus of our presentation is the class of MR Markov processes defined on pyramidally organized trees. The attractiveness of these models stems from both the very efficient algorithms they admit and their expressive power and broad applicability. We show how a variety of methods and models relate to this framework including models for self-similar and 1/f processes. We also illustrate how these methods have been used in practice. We discuss the construction of MR models on trees and show how questions that arise in this context make contact with wavelets, state space modeling of time series, system and parameter identification, and hidden

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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...

This paper describes a statistical modeling and analysis method for linear inverse problems involving Poisson data based on a novel multiscale framework. The framework itself is founded upon a multiscale analysis associated with recursive partitioning of the underlying intensity, a corresponding multiscale factorization of the likelihood (induced by this analysis), and a choice of prior probability distribution made to match this factorization by modeling the \splits" in the underlying partition. The class of priors used here has the interesting feature that the \non-informative" member yields the traditional maximum likelihood solution; other choices are made to reect prior belief as to the smoothness of the unknown intensity. Adopting the expectation-maximization (EM) algorithm for use in computing the MAP estimate corresponding to our model, we nd that our model permits remarkably simple, closed-form expressions for the EM update equations. The behavior of our EM algorit...

This paper studies the issue of optimal deconvolution density estimation using wavelets. The approach taken here can be considered as orthogonal series estimation in the more general context of the density estimation. We explore the asymptotic properties of estimators based on thresholding of estimated wavelet coefficients. Minimax rates of convergence under the integrated square loss are studied over Besov classes Bσpq of functions for both ordinary smooth and supersmooth convolution kernels. The minimax rates of convergence depend on the smoothness of functions to be deconvolved and the decay rate of the characteristic function of convolution kernels. It is shown that no linear deconvolution estimators can achieve the optimal rates of convergence in the Besov spaces with p < 2 when the convolution kernel is ordinary smooth and super smooth. If the convolution kernel is ordinary smooth, then linear estimators can be improved by using thresholding wavelet deconvolution estimators which are asymptotically minimax within logarithmic terms. Adaptive minimax properties of thresholding wavelet deconvolution estimators are also discussed. Keywords. Adaptive estimation, Besov spaces, Kullback-Leibler information, linear estimators, minimax estimation, thresholding, wavelet bases.

Thresholding algorithms in an orthonormal basis are studied to estimate noisy discrete signals degraded by a linear operator whose inverse is not bounded. For signals in a set Theta, sufficient conditions are established on the basis to obtain a maximum risk with minimax rates of convergence. Deconvolutions with kernels having a Fourier transform which vanishes at high frequencies are examples of unstable inverse problems, where a thresholding in a wavelet basis is a suboptimal estimator. A new "mirror wavelet" basis is constructed to obtain a deconvolution risk which is proved to be asymptotically equivalent to the minimax risk over bounded variation signals. This thresholding estimator is used to restore blurred satellite images.

Because the Radon transform is a smoothing transform, any noise in the Radon data becomes magnified when the inverse Radon transform is applied. Among the methods used to deal with this problem is the Wavelet-Vaguelette Decomposition (WVD) coupled with Wavelet Shrinkage, as introduced by David L. Donoho. We extend several results of Donoho and others here. First, we introduce a new sufficient condition on wavelets to generate a WVD. For a general homogeneous operator, which class includes the Radon transform, we show that a variant of Donoho's method for solving inverse problems can be derived as the exact minimizer of a variational problem that uses a Besov norm as the smoothing functional. We give a new proof of the rate of convergence of wavelet shrinkage that allows us to estimate rather sharply the best shrinkage parameter needed to recover an image from noise-corrupted data. We conduct tomographic reconstruction computations that support the hypothesis that near-optimal shrinkage pa...

What mathematicians, scientists, engineers, and statisticians mean by "inverse problem" differs. For a statistician, an inverse problem is an inference or estimation problem...

.................................................................viii CHAPTER 1 INTRODUCTION ..............................................1 CHAPTER 2 PRELIMINARIES ..............................................5 2.1 Definitions and Notations ..............................................5 2.2 Theorems and Inequalities..............................................6 CHAPTER 3 WAVELETS AND BESOV SPACES ............................9 3.1 Biorthogonal Wavelets .................................................9 3.2 Besov Spaces .........................................................15 CHAPTER 4 EMBEDDING, INTERPOLATION, AND DUALITY BETWEEN BESOV SPACES .............................................................18 4.1 Embedding ...........................................................18 4.2 Interpolation..........................................................21 4.3 Duality ...............................................................23 CHAPTER 5 LINEAR HOMOGENEOUS EQUATION...

Abstract—Tomographic reconstruction from positron emission tomography (PET) data is an ill-posed problem that requires regularization. An attractive approach is to impose an 1-regularization constraint, which favors sparse solutions in the wavelet domain. This can be achieved quite efficiently thanks to the iterative algorithm developed by Daubechies et al., 2004. In this paper, we apply this technique and extend it for the reconstruction of dynamic (spatio-temporal) PET data. Moreover, instead of using classical wavelets in the temporal dimension, we introduce exponential-spline wavelets (E-spline wavelets) that are specially tailored to model time activity curves (TACs) in PET. We show that the exponential-spline wavelets naturally arise from the compartmental description of the dynamics of the tracer distribution. We address the issue of the selection of the “optimal” E-spline parameters (poles and zeros) and we investigate their