Results 1  10
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10
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 798 (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.
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 ..."
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Cited by 103 (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 ...
ForWaRD: FourierWavelet Regularized Deconvolution for IllConditioned Systems
 IEEE Trans. on Signal Processing
, 2002
"... We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) al gorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inhere ..."
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Cited by 90 (2 self)
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We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) al gorithm that performs noise regularization via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the Fourier transform's sparse representation of the colored noise inherent in deconvolution, while the wavelet shrinkage exploits the wavelet do main's sparse representation of piecewise smooth signals and images. We derive the optimal balance between the amount of Fourier and wavelet regularization by optimizing an approxi mate meansquarederror (MSE) metric and find that signals with sparser wavelet representa tions require less Fourier shrinkage. ForWaRD is applicable to all illconditioned deconvolution problems, unlike the purely waveletbased Wavelet Vaguelette Deconvolution (WVD), and its es timate features minimal ringing, unlike purely Fourierbased Wiener deconvolution. We analyze ForWaRD's MSE decay rate as the number of samples increases and demonstrate its improved performance compared to the optimal WVD over a wide range of practical samplelengths.
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 56 (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.
Inverse Problems in Image Processing
, 2003
"... Inverse problems involve estimating parameters or data from inadequate observations; the observations are often noisy and contain incomplete information about the target parameter or data due to physical limitations of the measurement devices. Consequently, solutions to inverse problems are nonuni ..."
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Cited by 2 (2 self)
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Inverse problems involve estimating parameters or data from inadequate observations; the observations are often noisy and contain incomplete information about the target parameter or data due to physical limitations of the measurement devices. Consequently, solutions to inverse problems are nonunique. To pin down a solution, we must exploit the underlying structure of the desired solution set. In this thesis, we formulate novel solutions to three image processing inverse problems: deconvolution, inverse halftoning, and JPEG compression history estimation for color images. Deconvolution aims to extract crisp images from blurry observations. We propose an efficient, hybrid FourierWavelet Regularized Deconvolution (ForWaRD) algorithm that comprises blurring operator inversion followed by noise attenuation via scalar shrinkage in both the Fourier and wavelet domains. The Fourier shrinkage exploits the structure of the colored noise inherent in deconvolution, while the wavelet shrinkage exploits the piecewise smooth structure of realworld signals and images. ForWaRD yields stateoftheart meansquarederror (MSE) performance in practice. Further, for certain problems, ForWaRD guarantees an optimal rate of MSE decay with increasing resolution. Halftoning is a
Restoration Of Binary Images For Bernoullian Noise Models
, 1996
"... In this paper the problem of the restoration of binary images from Bernoulli random noise is investigated. Assuming that an image has regular boundary, we show that linear averaging estimators are nearly optimal up to a logarithmic factor of the number of pixels. Moreover, we provide examples showi ..."
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In this paper the problem of the restoration of binary images from Bernoulli random noise is investigated. Assuming that an image has regular boundary, we show that linear averaging estimators are nearly optimal up to a logarithmic factor of the number of pixels. Moreover, we provide examples showing that for any estimator there exist images with regular boundary such that an estimation error is of the order of the size of this boundary. Furthermore we introduce a class of hierarchical threshold estimators which are wavelet based estimators achieving this lower bound. In terms of Besov norms these estimators have optimal convergence rates. Some simulation examples illustrate the performance of these estimators.
Nonparametric Wavelet based Minimax Estimation in non Gaussian Models
"... We investigate nonparametric estimation procedures for images and functions with discontinuities in the presence of additive noise using wavelet decomposition methods. For functions and images belonging to Besov type spaces we derive bounds for the minimax risk in the class of linear and the class o ..."
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We investigate nonparametric estimation procedures for images and functions with discontinuities in the presence of additive noise using wavelet decomposition methods. For functions and images belonging to Besov type spaces we derive bounds for the minimax risk in the class of linear and the class of arbitrary estimators for non Gaussian additive error models. The bounds are sharp for the linear minimax risks under second order moment assumptions on the noise. They are sharp up to a logarithmic factor for the general minimax risks under exponential moment conditions for the noise. Applications are made to quantization and image compression problems.
WInliD: Waveletbased Inverse Halftoning via Deconvolution
, 2002
"... We propose the Waveletbased Inverse Halftoning via Deconvolution (WInliD) algorithm to perform inverse halftoning of errordiffused halftones. WInliD is motivated by our realization that inverse halftoning can be formulated as a deconvolution problem under Kite et al.'s linear approximation mode ..."
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We propose the Waveletbased Inverse Halftoning via Deconvolution (WInliD) algorithm to perform inverse halftoning of errordiffused halftones. WInliD is motivated by our realization that inverse halftoning can be formulated as a deconvolution problem under Kite et al.'s linear approximation model for error diffusion halftoning. Under the linear model, the errordiffused halftone comprises the original grayscale image blurred by a convolution operator and colored noise; the convolution operator and noise coloring are determined by the error diffusion tech nique. WInliD performs inverse halftoning by first inverting the modelspecified convolution operator and then attenuating the residual noise using scalar waveletdomain shrinkage. Since WInliD is modelbased, it is easily adapted to different error diffusion halftoning techniques. Using
ECONOMIC RESEARCH REPORTS Wavelets in Economics and Finance: Past and Future
, 2002
"... In this paper I review what insights we have gained about economic and financial relationships from the use of wavelets and speculate on what further insights we may gain in the future. Wavelets are treated as a “lens” that enables the researcher to explore relationships that previously were unobser ..."
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In this paper I review what insights we have gained about economic and financial relationships from the use of wavelets and speculate on what further insights we may gain in the future. Wavelets are treated as a “lens” that enables the researcher to explore relationships that previously were unobservable.