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37
The application of multiwavelet filter banks to image processing
 IEEE Trans. Image Process
, 1999
"... Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrixvalued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2channel wavelet systems. After reviewing this recently d ..."
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Cited by 83 (5 self)
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Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrixvalued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discretetime signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a onedimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximationbased preprocessing. We describe an additional algorithm for multiwavelet processing of twodimensional signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is wellsuited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via waveletshrinkage, and data compression. After developing the approach via model problems in one dimension, we applied multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.
Incorporating Information on Neighboring Coefficients into Wavelet Estimation
, 1999
"... In standard wavelet methods, the empirical wavelet coefficients are thresholded term by term, on the basis of their individual magnitudes. Information on other coefficients has no influence on the treatment of particular coefficients. We propose a wavelet shrinkage method that incorporates informati ..."
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Cited by 75 (11 self)
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In standard wavelet methods, the empirical wavelet coefficients are thresholded term by term, on the basis of their individual magnitudes. Information on other coefficients has no influence on the treatment of particular coefficients. We propose a wavelet shrinkage method that incorporates information on neighboring coefficients into the decision making. The coefficients are considered in overlapping blocks; the treatment of coefficients in the middle of each block depends on the data in the whole block. The asymptotic and numerical performances of two particular versions of the estimator are investigated. We show that, asymptotically, one version of the estimator achieves the exact optimal rates of convergence over a range of Besov classes for global estimation, and attains adaptive minimax rate for estimating functions at a point. In numerical comparisons with various methods, both versions of the estimator perform excellently.
A HaarFisz algorithm for Poisson intensity estimation
 J. Comput. Graph. Stat
, 2004
"... This article introduces a new method for the estimation of the intensity of an inhomogeneous onedimensional Poisson process. The HaarFisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkag ..."
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Cited by 74 (20 self)
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This article introduces a new method for the estimation of the intensity of an inhomogeneous onedimensional Poisson process. The HaarFisz transformation transforms a vector of binned Poisson counts to approximate normality with variance one. Hence we can use any suitable Gaussian wavelet shrinkage method to estimate the Poisson intensity. Since the HaarFisz operator does not commute with the shift operator we can dramatically improve accuracy by always cycle spinning before the HaarFisz transform as well as optionally after. Extensive simulations show that our approach usually significantly outperformed stateoftheart competitors but was occasionally comparable. Our method is fast, simple, automatic, and easy to code. Our technique is applied to the estimation of the intensity of earthquakes in northern California. We show that our technique gives visually similar results to the current stateoftheart.
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 64 (14 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...
Signal and Image Denoising via Wavelet Thresholding: Orthogonal and Biorthogonal, Scalar and Multiple Wavelet Transforms
 In Nonlinear and Nonstationary Signal Processing
, 1998
"... The method of signal denoising... In this paper we study wavelet thresholding in the context of scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. Two types of multiwavelet thresholding are considered: scalar and vector. Both of them take into a ..."
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Cited by 24 (2 self)
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The method of signal denoising... In this paper we study wavelet thresholding in the context of scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. Two types of multiwavelet thresholding are considered: scalar and vector. Both of them take into account the covariance structure of the transform. The form of the universal threshold is carefully formulated and is the key to the excellent results obtained in the extensive numerical simulations of signal and image denoising reported here. Sections 2 to 5 are concerned with the deterministic formulation of relevant components of multiwavelet analysis. In Section 2 we give a summary of multiresolution analysis and semiorthogonal, orthogonal and biorthogonal multiwavelet functions. The rest of the paper concentrates on the mostused practical cases of multiplicity 1 (scalar) and 2 (two scaling functions). Section 3 introduces two classes of orthogonal multiwavelets which may be used in multifilter banks  the GeronimoHardinMassopust (GHM) and ChuiLian (CL) classes, while Section 4 discusses biorthogonal multifilter banks, denoted BiGHM and BiHermite. For multifilter banks the given scalar input signal must be associated with a sequence of length2 vectors: oversampling and critical sampling preprocessing is carefully studied in Section 5. Sections 6 and 7 look at two important aspects of the processing of 1D stochastic signals, the covariance structure of the output, and denoising via thresholding. If stochastic noise is the input to preprocessing followed by discrete multiwavelet transform, very variable covariance structures of the output can result; in particular the average input variance of white noise can be inflated or deflated on output as shown in Section 6 for the GHM and C...
Multiwavelet Bases with Extra Approximation Properties
, 1998
"... This paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order. Because multiwavelet bases normally lack important properties that traditional wavelet bases (of equal approximation order) possess, the associated discrete multiwavelet transfor ..."
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Cited by 23 (3 self)
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This paper highlights the differences between traditional wavelet and multiwavelet bases with equal approximation order. Because multiwavelet bases normally lack important properties that traditional wavelet bases (of equal approximation order) possess, the associated discrete multiwavelet transform is less useful for signal processing unless it is preceded by a preprocessing step (prefiltering). This paper examines the properties and design of orthogonal multiwavelet bases, with approximation order b1 that possess those properties that are normally absent. For these “balanced” bases (so named by Lebrun and Vetterli), prefiltering can be avoided. By reorganizing the multiwavelet filter bank, the development in this paper draws from results regarding the approximation order of wband wavelet bases. The main result thereby obtained is a characterization of balanced multiwavelet bases in terms of the divisibility of certain transfer functions by powers of (�0P � 0
TranslationInvariant Denoising Using Multiwavelets
 IEEE Transactions on Signal Processing
, 1998
"... Translation invariant (TI) single wavelet denoising was developed by Coifman and Donoho and they show that TI is better than nonTI single wavelet denoising. On the other hand, Strela et al. have found that nonTI multiwavelet denoising gives better results than nonTI single wavelets. In this pa ..."
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Cited by 20 (0 self)
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Translation invariant (TI) single wavelet denoising was developed by Coifman and Donoho and they show that TI is better than nonTI single wavelet denoising. On the other hand, Strela et al. have found that nonTI multiwavelet denoising gives better results than nonTI single wavelets. In this paper we extend Coifman and Donoho's TI single wavelet denoising scheme to multiwavelets. Experimental results show that TI multiwavelet denoising is better than the single case for soft thresholding. 1 Introduction Over the past decade wavelet transforms have received a lot of attention from researchers in many different areas. Both discrete and continuous wavelet transforms have shown great promises in such diverse fields as image compression, image denoising, signal processing, computer graphics, and pattern recognition to name only a few. In denoising, single orthogonal wavelets with singlemother wavelet function have played an important role (see [2]  [4]). The pioneering work of...
Wavelets in Statistics: Beyond the Standard Assumptions
 Phil. Trans. Roy. Soc. Lond. A
, 1999
"... 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 ..."
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Cited by 9 (2 self)
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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.
Bivariate hard thresholding in wavelet function estimation. Statistica Sinica 16
, 2006
"... Abstract: We propose a generic bivariate hard thresholding estimator of the discrete wavelet coefficients of a function contaminated with i.i.d. Gaussian noise. We demonstrate its good risk properties in a motivating example, and derive upper bounds for its meansquare error. Motivated by the clust ..."
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Cited by 7 (3 self)
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Abstract: We propose a generic bivariate hard thresholding estimator of the discrete wavelet coefficients of a function contaminated with i.i.d. Gaussian noise. We demonstrate its good risk properties in a motivating example, and derive upper bounds for its meansquare error. Motivated by the clustering of large wavelet coefficients in reallife signals, we propose two wavelet denoising algorithms, both of which use specific instances of our bivariate estimator. The BABTE algorithm uses basis averaging, and the BITUP algorithm uses the coupling of "parents" and "children" in the wavelet coefficient tree. We prove the L2 nearoptimality of both algorithms over the usual range of Besov spaces, and demonstrate their excellent finitesample performance. Finally, we propose a robust and effective technique for choosing the parameters of BITUP in a datadriven way.