Results 1  10
of
12
DeNoising By SoftThresholding
, 1992
"... 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 a ..."
Abstract

Cited by 796 (13 self)
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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.
Wavelet shrinkage: asymptopia
 Journal of the Royal Statistical Society, Ser. B
, 1995
"... Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators bein ..."
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Cited by 238 (35 self)
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Considerable e ort has been directed recently to develop asymptotically minimax methods in problems of recovering in nitedimensional objects (curves, densities, spectral densities, images) from noisy data. A rich and complex body of work has evolved, with nearly or exactly minimax estimators being obtained for a variety of interesting problems. Unfortunately, the results have often not been translated into practice, for a variety of reasons { sometimes, similarity to known methods, sometimes, computational intractability, and sometimes, lack of spatial adaptivity. We discuss a method for curve estimation based on n noisy data; one translates the empirical wavelet coe cients towards the origin by an amount p p 2 log(n) = n. The method is di erent from methods in common use today, is computationally practical, and is spatially adaptive; thus it avoids a number of previous objections to minimax estimators. At the same time, the method is nearly minimax for a wide variety of loss functions { e.g. pointwise error, global error measured in L p norms, pointwise and global error in estimation of derivatives { and for a wide range of smoothness classes, including standard Holder classes, Sobolev classes, and Bounded Variation. This is amuch broader nearoptimality than anything previously proposed in the minimax literature. Finally, the theory underlying the method is interesting, as it exploits a correspondence between statistical questions and questions of optimal recovery and informationbased complexity.
Interpolating Wavelet Transform
, 1992
"... 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 reinterpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function. ..."
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Cited by 122 (13 self)
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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 reinterpret the empirical wavelet transform, i.e. the common practice of applying pyramid filters to samples of a function.
Nonlinear Wavelet Shrinkage With Bayes Rules and Bayes Factors
 Journal of the American Statistical Association
, 1998
"... this article a wavelet shrinkage by coherent ..."
Nonlinear Wavelet Methods for Recovery of Signals, Densities, and Spectra from Indirect and Noisy Data
 In Proceedings of Symposia in Applied Mathematics
, 1993
"... . 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 i ..."
Abstract

Cited by 102 (5 self)
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. 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 DeNoising. (2) Wavelet Approaches to Linear Inverse Problems. (4) Wavelet Packet DeNoising. (5) Segmented MultiResolutions. (6) Nonlinear Multiresolutions. 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 ...
Smooth Wavelet Decompositions with Blocky Coefficient Kernels
, 1993
"... We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approac ..."
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Cited by 54 (12 self)
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We describe bases of smooth wavelets where the coefficients are obtained by integration against (finite combinations of) boxcar kernels rather than against traditional smooth wavelets. Bases of this type were first developed in work of Tchamitchian and of Cohen, Daubechies, and Feauveau. Our approach emphasizes the idea of averageinterpolation  synthesizing a smooth function on the line having prescribed boxcar averages  and the link between averageinterpolation and DubucDeslauriers interpolation. We also emphasize characterizations of smooth functions via their coefficients. We describe boundarycorrected expansions for the interval, which have a simple and revealing form. We use these results to reinterpret the empirical wavelet transform  i.e. finite, discrete wavelet transforms of data arising from boxcar integrators (e.g. CCD devices).
On minimum entropy segmentation
 In
, 1994
"... We describe segmented multiresolution analyses of [0; 1]. Such multiresolution analyses lead to segmented wavelet bases which are adapted to discontinuities, cusps, etc., at a given location 2 [0; 1]. Our approach emphasizes the idea of averageinterpolation { synthesizing a smooth function on the l ..."
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Cited by 17 (4 self)
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We describe segmented multiresolution analyses of [0; 1]. Such multiresolution analyses lead to segmented wavelet bases which are adapted to discontinuities, cusps, etc., at a given location 2 [0; 1]. Our approach emphasizes the idea of averageinterpolation { synthesizing a smooth function on the line having prescribed boxcar averages. This particular approach leads to methods with subpixel resolution and to wavelet transforms with the advantage that, for a signal of length n, all n pixellevel segmented wavelet transforms can be computed simultaneously in a total time and space which are both O(n log(n)). We consider the search for a segmented wavelet basis which, among all such segmented bases, minimizes the \entropy " of the resulting coe cients. Fast access to all segmentations enables fast search for a best segmentation. When the \entropy " is Stein's Unbiased Risk Estimate, one obtains a new method of edgepreserving denoising. When the \entropy " is the ` 2energy, one obtains a new multiresolution edge detector, which works not only for step discontinuities but also for cusp and higherorder discontinuities, and in a nearoptimal fashion in the presence of noise. We describe an iterative approach, Segmentation Pursuit, for identifying edges by the fast segmentation algorithm and removing them from the data. 1 Key Words and Phrases. Segmented MultiResolution analysis. EdgePreserving Image processing methods. Edge detection. Subpixel resolution.
Wavelet Shrinkage and W.V.D.: A 10minute tour
 Progress in Wavelet Analysis and Applications
, 1993
"... 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 D ..."
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Cited by 8 (0 self)
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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 9209130. 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. DeNoising by SoftThresholding
Wavelet Thresholding for Unequally Spaced Data
, 1998
"... We consider the problem of nonparametric regression. Given are noisy observations y1; : : : ; yn at time points t1; : : : ; tn which are modelled as yi = f (ti) + "i, where f is the function to be estimated and "1; : : : ; "n i.i.d. Gaussian white Noise with mean 0 and variance oe2. One approach t ..."
Abstract

Cited by 5 (1 self)
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We consider the problem of nonparametric regression. Given are noisy observations y1; : : : ; yn at time points t1; : : : ; tn which are modelled as yi = f (ti) + "i, where f is the function to be estimated and "1; : : : ; "n i.i.d. Gaussian white Noise with mean 0 and variance oe2. One approach to this problem is wavelet shrinkage, but it has the disadvantage that it requires the number of data points to be a power of two and the time points to be regularly spaced, i.e. ti = i=n: In the first chapter we give a short introduction to the thesis and discuss the problem of nonparametric regression as well as the problem with unequally spaced data within the approach of wavelet shrinkage. We review some material on wavelets, the discrete wavelet transform and nonparametric regression in the second chapter. The third chapter deals with a fast O(n) algorithm for calculating the variances of wavelet coefficients that are obtained by applying the discrete wavelet transform to correlated normal noise with mean 0 and covariance matrix \Sigma, where \Sigma is a band matrix. We use this algorithm to introduce a new method for thresholding unequally spaced data in the fourth chapter. We also review contributions of other authors to this topic, present an implementation of the new procedures in the software package WaveThresh which runs under SPlus and apply the new method to two real data sets. In the fifth chapter we analyse different threshold selections by calculating the exact mean square error for four test signals and different choices of parameters like different noise levels, hard and soft thresholding rules etc. In particular we analyse versions of Donoho and Johnstone's VisuShrink and SureShrink. The sixth chapter deals with robust wavelet techniques which perform well even if the the noise is not normally distributed or if the data set is disturbed by several consecutive outliers. Two new algorithms are developed which make use of the new methods for thresholding unequally spaced data and calculating the variances of wavelet coefficients. The last chapter summarises the results of this thesis and gives the prospects for possible future research.