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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 233 (14 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...
Time Invariant Orthonormal Wavelet Representations
"... A simple construction of an orthonormal basis starting with a so called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable ..."
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Cited by 70 (9 self)
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A simple construction of an orthonormal basis starting with a so called mother wavelet, together with an efficient implementation gained the wavelet decomposition easy acceptance and generated a great research interest in its applications. An orthonormal basis may not, however, always be a suitable representation of a signal, particularly when time (or space) invariance is a required property. The conventional way around this problem is to use a redundant decomposition. In this paper, we address the time invariance problem for orthonormal wavelet transforms and propose an extension to wavelet packet decompositions. We show that it is possible to achieve time invariance and preserve the orthonormality. We subsequently propose an efficient approach to obtain such a decomposition. We demonstrate the importance of our method by considering some application examples in signal reconstruction and time delay estimation.
Wavelet Analysis and Its Statistical Applications
, 1999
"... 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 ..."
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Cited by 61 (13 self)
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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...
Choice of Thresholds for Wavelet Shrinkage Estimate of the Spectrum
, 1990
"... We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t j;n depending on sample size n, wavelet basis and resolution level j. At fine resolution levels (j = 1; 2;:::), we propose ..."
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Cited by 39 (1 self)
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We study the problem of estimating the log spectrum of a stationary Gaussian time series by thresholding the empirical wavelet coefficients. We propose the use of thresholds t j;n depending on sample size n, wavelet basis and resolution level j. At fine resolution levels (j = 1; 2;:::), we propose t j;n = ff j log n; where fff j g are leveldependent constants and at coarse levels (j AE 1) t
Wavelet Smoothing of Evolutionary Spectra By NonLinear Thresholding
 Appl. Comput. Harm. Anal
, 1994
"... We consider wavelet estimation of the timedependent (evolutionary) power spectrum of a locally stationary time series. Hereby, wavelets are used to provide an adaptive local smoothing of a shorttime periodogram in the timefrequency plane. For this, in contrast to classical nonparametric (linea ..."
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Cited by 33 (12 self)
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We consider wavelet estimation of the timedependent (evolutionary) power spectrum of a locally stationary time series. Hereby, wavelets are used to provide an adaptive local smoothing of a shorttime periodogram in the timefrequency plane. For this, in contrast to classical nonparametric (linear) approaches we use nonlinear thresholding of the empirical wavelet coe#cients. We show how these techniques allow for both adaptively reconstructing the local structure in the timefrequency plane and for denoising the resulting estimates. To this end a threshold choice is derived which results into a nearoptimal L 2 minimax rate for the resulting spectral estimator. Our approach is based on a 2d orthogonal wavelet transform modified by using a cardinal Lagrange interpolation function on the finest scale. As an example, we apply our procedure to a timevarying spectrum motivated from mobile radio propagation. 1 Introduction Estimating power spectra which (slowly) change over ...
Speech enhancement based on wavelet thresholding the multitaper spectrum
 IEEE Trans. Speech Audio Process
, 2004
"... ..."
Wavelet thresholding via mdl for natural images
 IEEE Transactions on Information Theory
, 2000
"... We study the application of Rissanen's Principle of Minimum Description Length (MDL) to the problem of wavelet denoising and compression for natural images. After making a connection between thresholding and model selection, we derive an MDL criterion based on a Laplacian model for noiseless w ..."
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Cited by 28 (1 self)
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We study the application of Rissanen's Principle of Minimum Description Length (MDL) to the problem of wavelet denoising and compression for natural images. After making a connection between thresholding and model selection, we derive an MDL criterion based on a Laplacian model for noiseless wavelet coe cients. We nd that this approach leads to an adaptive thresholding rule. While achieving mean squared error performance comparable with other popular thresholding schemes, the MDL procedure tends to keep far fewer coe cients. From this property, we demonstrate that our method is an excellent tool for simultaneous denoising and compression. We make this claim precise by analyzing MDL thresholding in two optimality frameworks; one in which we measure rate and distortion based on quantized coe cients and one in which we do not quantize, but instead record rate simply as the number of nonzero coe cients.
The What, How, and Why of Wavelet Shrinkage Denoising
 IEEE Computing in Science & Engineering
, 2000
"... Principles of wavelet shrinkage denoising are reviewed. Both 1D and 2D examples are demonstrated. The performance of various ideal and practical Fourier and wavelet based denoising procedures are evaluated and compared in a new Monte Carlo simulation experiment. Finally, recommendations for the ..."
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Cited by 26 (1 self)
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Principles of wavelet shrinkage denoising are reviewed. Both 1D and 2D examples are demonstrated. The performance of various ideal and practical Fourier and wavelet based denoising procedures are evaluated and compared in a new Monte Carlo simulation experiment. Finally, recommendations for the practitioner are discussed. 1 Some Opposing Viewpoints Applied scientists and engineers who work with data obtained from the real world know that signals do not exist without noise. Under ideal conditions, this noise may decrease to such negligible levels, while the signal increases to such significant levels, that for all practical purposes denoising is not necessary. Unfortunately, the noise corrupting the signal, more often than not, must be removed in order to recover the signal and proceed with further data analysis. Should this noise removal take place in the original signal (timespace) domain or in a transform domain? If the latter, should it be the timefrequency domain via the F...
Spectrum Estimation by Wavelet Thresholding of Multitaper Estimators
, 1995
"... Current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor estimate when the true spectrum has a high dynamic range and/or is rapidly varying. Also, because the distribut ..."
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Cited by 24 (2 self)
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Current methods for power spectrum estimation by wavelet thresholding use the empirical wavelet coefficients derived from the log periodogram. Unfortunately, the periodogram is a very poor estimate when the true spectrum has a high dynamic range and/or is rapidly varying. Also, because the distribution of the log periodogram is markedly nonGaussian, complicated waveletdependent thresholding schemes are needed. These difficulties can be bypassed by starting with a multitaper spectrum estimator. The logarithm of this estimator is close to Gaussian distributed provided a moderate number (>= 5) of tapers are used. In contrast to the log periodogram, log multitaper estimates are not approximately pairwise uncorrelated at the Fourier frequencies, but the form of the correlation can be accurately and simply approximated. For scaleindependent thresholding the correlation acts in accordance with the wavelet shrinkage paradigm to strongly suppress `noise spikes' while leaving informative coarse sca...
A Wavelet Analysis for Time Series
 J. NONPARAMETR. STATIST
, 1997
"... In this paper we develop a wavelet spectral analysis for a stationary discrete process. Some basic ideas on wavelets are given and the concept of wavelet spectrum is introduced. Asymptotic properties of the discrete wavelet transform of a sample of observed values from the process are derived and th ..."
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Cited by 20 (5 self)
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In this paper we develop a wavelet spectral analysis for a stationary discrete process. Some basic ideas on wavelets are given and the concept of wavelet spectrum is introduced. Asymptotic properties of the discrete wavelet transform of a sample of observed values from the process are derived and the wavelet periodogram is considered as an estimator of the wavelet spectrum. Applications to real and simulated series are given.