Donoho and Johnstone (1992a) proposed a method for reconstructing an unknown function f on [0; 1] from noisy data di = f(ti)+ zi, iid i =0;:::;n 1, ti = i=n, zi N(0; 1). The reconstruction fn ^ is de ned in the wavelet domain by translating all the empirical wavelet coe cients of d towards 0 by an amount p 2 log(n) = p n. We prove two results about that estimator. [Smooth]: With high probability ^ fn is at least as smooth as f, in any of a wide variety of smoothness measures. [Adapt]: The estimator comes nearly as close in mean square to f as any measurable estimator can come, uniformly over balls in each of two broad scales of smoothness classes. These two properties are unprecedented in several ways. Our proof of these results develops new facts about abstract statistical inference and its connection with an optimal recovery model.

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We describe several "wavelet transforms" which characterize smoothness spaces and for which the coefficients are obtained by sampling rather than integration. We use them to re-interpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function.

. We describe wavelet methods for recovery of objects from noisy and incomplete data. The common themes: (a) the new methods utilize nonlinear operations in the wavelet domain; (b) they accomplish tasks which are not possible by traditional linear/Fourier approaches to such problems. We attempt to indicate the heuristic principles, theoretical foundations, and possible application areas for these methods. Areas covered: (1) Wavelet De-Noising. (2) Wavelet Approaches to Linear Inverse Problems. (4) Wavelet Packet De-Noising. (5) Segmented MultiResolutions. (6) Nonlinear Multi-resolutions. 1. Introduction. With the rapid development of computerized scientific instruments comes a wide variety of interesting problems for data analysis and signal processing. In fields ranging from Extragalactic Astronomy to Molecular Spectroscopy to Medical Imaging to Computer Vision, one must recover a signal, curve, image, spectrum, or density from incomplete, indirect, and noisy data. What can wavelets ...

is a library of routines for wavelet analysis, wavelet-packet analysis, cosine-packet analysis and matching pursuit. The library is available free of charge over the Internet. Versions are provided for Macintosh, UNIX and Windows machines.

Summary. Increasingly, scientific studies yield functional data, in which the ideal units of observation are curves and the observed data consist of sets of curves that are sampled on a fine grid. We present new methodology that generalizes the linear mixed model to the functional mixed model framework, with model fitting done by using a Bayesian wavelet-based approach. This method is flexible, allowing functions of arbitrary form and the full range of fixed effects structures and between-curve covariance structures that are available in the mixed model framework. It yields nonparametric estimates of the fixed and random-effects functions as well as the various between-curve and within-curve covariance matrices.The functional fixed effects are adaptively regularized as a result of the non-linear shrinkage prior that is imposed on the fixed effects’ wavelet coefficients, and the random-effect functions experience a form of adaptive regularization because of the separately estimated variance components for each wavelet coefficient. Because we have posterior samples for all model quantities, we can perform pointwise or joint Bayesian inference or prediction on the quantities of the model.The adaptiveness of the method makes it especially appropriate for modelling irregular functional data that are characterized by numerous local features like peaks.

Abstract — We present a general wavelet-based denoising scheme for functional magnetic resonance imaging (fMRI) data and compare it to Gaussian smoothing, the traditional denoising method used in fMRI analysis. One-dimensional WaveLab thresholding routines were adapted to two-dimensional images, and applied to 2D wavelet coefficients. To test the effect of these methods on the signal-to-noise ratio (SNR), we compared the SNR of 2D fMRI images before and after denoising, using both Gaussian smoothing and wavelet-based methods. We simulated a fMRI series with a time signal in an active spot, and tested the methods on noisy copies of it. The denoising methods were evaluated in two ways: by the average temporal SNR inside the original activated spot, and by the shape of the spot detected by thresholding the temporal SNR maps. Denoising methods that introduce much smoothness are better suited for low SNRs, but for images of reasonable quality they are not preferable, because they introduce heavy deformations. Wavelet-based denoising methods that introduce less smoothing preserve the sharpness of the images and retain the original shapes of active regions. We also performed statistical parametric mapping (SPM) on the denoised simulated time series, as well as on a real fMRI data set. False discovery rate control was used to correct for multiple comparisons. The results show that the methods that produce smooth images introduce more false positives. The less smoothing wavelet-based methods, although generating more false negatives, produce a smaller total number of errors than Gaussian smoothing or wavelet-based methods with a large smoothing effect. Index Terms — Functional neuroimaging, wavelet-based denoising, Gaussian smoothing, statistical parametric mapping, false discovery rate control. I.

this paper; contact the author at donoho@playfair.stanford.edu. In the discussion I mention work which proves the various theoretical advantages of the new techniques. Based on presentation at the International Conference on Wavelets and Applications, Toulouse, France, June, 1992. Supported by NSF DMS 92-09130. With appreciation to S. Roques for patience and Y. Meyer for encouragement. It is a pleasure to thank Iain Johnstone with whom many of these theoretical results have been derived, and Carl Taswell with whom Johnstone and I have developed the software used here. 2. De-Noising by Soft-Thresholding

Abstract. We introduce a nonlinear refinement subdivision scheme based on median-interpolation. The scheme constructs a polynomial interpolating adjacent block medians of an underlying object. The interpolating polynomial is then used to impute block medians at the next finer triadic scale. Perhaps surprisingly, expressions for the refinement operator can be obtained in closed-form for the scheme interpolating by polynomials of degree D = 2. Despite the nonlinearity of this scheme, convergence and regularity can be established using techniques reminiscent of those developed in analysis of linear refinement schemes. The refinement scheme can be deployed in multiresolution fashion to construct a nonlinear pyramid and an associated forward and inverse transform. In this paper we discuss the basic properties of these transforms and their possible use in removing badly non-Gaussian noise. Analytic and computational results are presented to show that in the presence of highly non-Gaussian noise, the coefficients of the nonlinear transform have much better properties than traditional wavelet coefficients. 1. Introduction. Recent theoretical studies [14, 13] have found that the orthogonal wavelet transform offers a promising approach to noise removal. They assume that one has noisy samples of an underlying function f (1.1)

this paper, attention has been focused on methods that treat coe#cients at least as if they were independent. However, it is intuitively clear that if one coe#cient in the wavelet array is nonzero, then it is more likely #in some appropriate sense# that neighbouring coe#cients will be also. One way of incorporating this notion is by some form of block thresholding, where coe#cients are considered in neighbouring blocks; see for example Hall et al. #1998# and Cai & Silverman #1998#. An obvious question for future consideration is integrate the ideas of block thresholding and related methods within the range of models and methods considered in this paper.