Results 1  10
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255
Adapting to unknown smoothness via wavelet shrinkage
 JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
, 1995
"... We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: a threshold level is assigned to each dyadic resolution level by the princip ..."
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Cited by 675 (19 self)
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We attempt to recover a function of unknown smoothness from noisy, sampled data. We introduce a procedure, SureShrink, which suppresses noise by thresholding the empirical wavelet coefficients. The thresholding is adaptive: a threshold level is assigned to each dyadic resolution level by the principle of minimizing the Stein Unbiased Estimate of Risk (Sure) for threshold estimates. The computational effort of the overall procedure is order N log(N) as a function of the sample size N. SureShrink is smoothnessadaptive: if the unknown function contains jumps, the reconstruction (essentially) does also; if the unknown function has a smooth piece, the reconstruction is (essentially) as smooth as the mother wavelet will allow. The procedure is in a sense optimally smoothnessadaptive: it is nearminimax simultaneously over a whole interval of the Besov scale; the size of this interval depends on the choice of mother wavelet. We know from a previous paper by the authors that traditional smoothing methods  kernels, splines, and orthogonal series estimates  even with optimal choices of the smoothing parameter, would be unable to perform
Adaptive Wavelet Thresholding for Image Denoising and Compression
 IEEE TRANSACTIONS ON IMAGE PROCESSING
, 2000
"... The first part of this paper proposes an adaptive, datadriven threshold for image denoising via wavelet softthresholding. 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 ..."
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Cited by 230 (4 self)
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The first part of this paper proposes an adaptive, datadriven threshold for image denoising via wavelet softthresholding. 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 closedform, and it is adaptive to each subband because it depends on datadriven estimates of the parameters. Experimental results show that the proposed method, called BayesShrink, is typically within 5% of the MSE of the best softthresholding benchmark with the image assumed known. It also outperforms Donoho and Johnstone's SureShrink most of the time. The second part
Wavelet Thresholding via a Bayesian Approach
 J. R. STATIST. SOC. B
, 1996
"... We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. ..."
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Cited by 204 (27 self)
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We discuss a Bayesian formalism which gives rise to a type of wavelet threshold estimation in nonparametric regression. A prior distribution is imposed on the wavelet coefficients of the unknown response function, designed to capture the sparseness of wavelet expansion common to most applications. For the prior specified, the posterior median yields a thresholding procedure. Our prior model for the underlying function can be adjusted to give functions falling in any specific Besov space. We establish a relation between the hyperparameters of the prior model and the parameters of those Besov spaces within which realizations from the prior will fall. Such a relation gives insight into the meaning of the Besov space parameters. Moreover, the established relation makes it possible in principle to incorporate prior knowledge about the function's regularity properties into the prior model for its wavelet coefficients. However, prior knowledge about a function's regularity properties might b...
Wavelet Threshold Estimators for Data With Correlated Noise
, 1994
"... Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a leveldependent soft threshold to the individual coefficients, allowing the thresholds to depend on the l ..."
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Cited by 182 (13 self)
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Wavelet threshold estimators for data with stationary correlated noise are constructed by the following prescription. First, form the discrete wavelet transform of the data points. Next, apply a leveldependent soft threshold to the individual coefficients, allowing the thresholds to depend on the level in the wavelet transform. Finally, transform back to obtain the estimate in the original domain. The threshold used at level j is s j p 2 log n, where s j is the standard deviation of the coefficients at that level, and n is the overall sample size. The minimax properties of the estimators are investigated by considering a general problem in multivariate normal decision theory, concerned with the estimation of the mean vector of a general multivariate normal distribution subject to squared error loss. An ideal risk is obtained by the use of an `oracle' that provides the optimum diagonal projection estimate. This `benchmark' risk can be considered in its own right as a measure of the s...
Sparsity and smoothness via the fused lasso
 Journal of the Royal Statistical Society Series B
, 2005
"... The lasso (Tibshirani 1996) penalizes a least squares regression by the sum of the absolute values (L1 norm) of the coefficients. The form of this penalty encourages sparse solutions, that is, having many coefficients equal to zero. Here we propose the “fused lasso”, a generalization of the lasso de ..."
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Cited by 132 (11 self)
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The lasso (Tibshirani 1996) penalizes a least squares regression by the sum of the absolute values (L1 norm) of the coefficients. The form of this penalty encourages sparse solutions, that is, having many coefficients equal to zero. Here we propose the “fused lasso”, a generalization of the lasso designed for problems with features that can be ordered in some meaningful way. The fused lasso penalizes both the L1 norm of the coefficients and their successive differences. Thus it encourages both sparsity
The Stationary Wavelet Transform and some Statistical Applications
, 1995
"... Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two ..."
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Cited by 128 (19 self)
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Wavelets are of wide potential use in statistical contexts. The basics of the discrete wavelet transform are reviewed using a filter notation that is useful subsequently in the paper. A `stationary wavelet transform', where the coefficient sequences are not decimated at each stage, is described. Two different approaches to the construction of an inverse of the stationary wavelet transform are set out. The application of the stationary wavelet transform as an exploratory statistical method is discussed, together with its potential use in nonparametric regression. A method of local spectral density estimation is developed. This involves extensions to the wavelet context of standard time series ideas such as the periodogram and spectrum. The technique is illustrated by its application to data sets from astronomy and veterinary anatomy. 1 Introduction In this paper we discuss some aspects of wavelets with a particular view to their statistical application. In particular we shall be conce...
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 ...
Adaptive wavelet estimation: A block thresholding and oracle inequality approach
 Ann. Statist
, 1999
"... We study wavelet function estimation via the approach of block thresholding and ideal adaptation with oracle. Oracle inequalities are derived and serve as guides for the selection of smoothing parameters. Based on an oracle inequality and motivated by the data compression and localization properties ..."
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Cited by 97 (13 self)
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We study wavelet function estimation via the approach of block thresholding and ideal adaptation with oracle. Oracle inequalities are derived and serve as guides for the selection of smoothing parameters. Based on an oracle inequality and motivated by the data compression and localization properties of wavelets, an adaptive wavelet estimator for nonparametric regression is proposed and the optimality of the procedure is investigated. We show that the estimator achieves simultaneously three objectives: adaptivity, spatial adaptivity and computational efficiency. Specifically, it is proved that the estimator attains the exact optimal rates of convergence over a range of Besov classes and the estimator achieves adaptive local minimax rate for estimating functions at a point. The estimator is easy to implement, at the computational cost of O�n�. Simulation shows that the estimator has excellent numerical performance relative to more traditional wavelet estimators. 1. Introduction. Wavelet
Wavelet shrinkage using crossvalidation
, 1996
"... Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold crossvalidation is introduced to ..."
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Cited by 77 (13 self)
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Wavelets are orthonormal basis functions with special properties that show potential in many areas of mathematics and statistics. This article concentrates on the estimation of functions and images from noisy data using wavelet shrinkage. A modified form of twofold crossvalidation is introduced to choose a threshold for wavelet shrinkage estimators operating on data sets of length a power of two. The crossvalidation algorithm is then extended to data sets of any length and to multidimensional data sets. The algorithms are compared to established threshold choosers using simulation. An application to a real data set arising from anaesthesia is presented.
Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences
 Ann. Statist
, 2002
"... An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavytailed density, with the mixing weight chosen by marginal maximum likelihood, in ..."
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Cited by 56 (5 self)
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An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavytailed density, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules using the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall bounds on the risk of the method over the class of signal sequences of length n with normalized ` p norm bounded by , for > 0 and 0 < p 2: Estimation error is measured by mean q loss, for 0 < q 2: For all p and q in (0; 2], the method achieves the optimal estimation rate as n ! 1 and ! 0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q > 1: Simulations show excellent performance. Computationally, the method is tractable and essentially of O(n) complexity, and software is available. The extension to a modi ed thresholding method relevant to the wavelet estimation of derivatives of functions is also considered.