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Factoring wavelet transforms into lifting steps
 J. Fourier Anal. Appl
, 1998
"... ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This dec ..."
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Cited by 434 (7 self)
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ABSTRACT. This paper is essentially tutorial in nature. We show how any discrete wavelet transform or two band subband filtering with finite filters can be decomposed into a finite sequence of simple filtering steps, which we call lifting steps but that are also known as ladder structures. This decomposition corresponds to a factorization of the polyphase matrix of the wavelet or subband filters into elementary matrices. That such a factorization is possible is wellknown to algebraists (and expressed by the formula); it is also used in linear systems theory in the electrical engineering community. We present here a selfcontained derivation, building the decomposition from basic principles such as the Euclidean algorithm, with a focus on applying it to wavelet filtering. This factorization provides an alternative for the lattice factorization, with the advantage that it can also be used in the biorthogonal, i.e, nonunitary case. Like the lattice factorization, the decomposition presented here asymptotically reduces the computational complexity of the transform by a factor two. It has other applications, such as the possibility of defining a waveletlike transform that maps integers to integers. 1.
The Lifting Scheme: A Construction Of Second Generation Wavelets
, 1997
"... . We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to ..."
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Cited by 373 (16 self)
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. We present the lifting scheme, a simple construction of second generation wavelets, wavelets that are not necessarily translates and dilates of one fixed function. Such wavelets can be adapted to intervals, domains, surfaces, weights, and irregular samples. We show how the lifting scheme leads to a faster, inplace calculation of the wavelet transform. Several examples are included. Key words. wavelet, multiresolution, second generation wavelet, lifting scheme AMS subject classifications. 42C15 1. Introduction. Wavelets form a versatile tool for representing general functions or data sets. Essentially we can think of them as data building blocks. Their fundamental property is that they allow for representations which are efficient and which can be computed fast. In other words, wavelets are capable of quickly capturing the essence of a data set with only a small set of coefficients. This is based on the fact that most data sets have correlation both in time (or space) and frequenc...
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 181 (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...
Building Your Own Wavelets at Home
"... Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what mak ..."
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Cited by 127 (13 self)
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Wavelets have been making an appearance in many pure and applied areas of science and engineering. Computer graphics with its many and varied computational problems has been no exception to this rule. In these notes we will attempt to motivate and explain the basic ideas behind wavelets and what makes them so successful in application areas. The main
Wavelet Families Of Increasing Order In Arbitrary Dimensions
, 1997
"... . We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its ..."
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Cited by 46 (0 self)
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. We build compactly supported biorthogonal wavelets and perfect reconstruction filter banks for any lattice in any dimension with any number of primal and dual vanishing moments. The resulting scaling functions are interpolating. Our construction relies on the lifting scheme and inherits all of its advantages: fast transform, inplace calculation, and integerto integer transforms. We show that two lifting steps suffice: predict and update. The predict step can be built using multivariate polynomial interpolation, while update is a multiple of the adjoint of predict. Submitted to IEEE Transactions on Image Processing Over the last decade several constructions of compactly supported wavelets have originated both from signal processing and mathematical analysis. In signal processing, critically sampled wavelet transforms are known as filter banks or subband transforms [32, 43, 54, 56]. In mathematical analysis, wavelets are defined as translates and dilates of one fixed function and ar...
Wavelet shrinkage for correlated data and inverse problems: adaptivity results
 Statist. Sinica
, 1999
"... Abstract: Johnstone and Silverman (1997) described a leveldependent thresholding method for extracting signals from correlated noise. The thresholds were chosen to minimize a data based unbiased risk criterion. Here we show that in certain asymptotic models encompassing short and long range depende ..."
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Cited by 34 (1 self)
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Abstract: Johnstone and Silverman (1997) described a leveldependent thresholding method for extracting signals from correlated noise. The thresholds were chosen to minimize a data based unbiased risk criterion. Here we show that in certain asymptotic models encompassing short and long range dependence, these methods are simultaneously asymptotically minimax up to constants over a broad range of Besov classes. We indicate the extension of the methods and results to a class of linear inverse problems possessing a wavelet vaguelette decomposition. Key words and phrases: Adaptation, correlated data, fractional brownian motion, linear inverse problems, long range dependence, mixing conditions, oracle inequalities, rates of convergence, unbiased risk estimate, wavelet vaguelette decomposition, wavelet shrinkage, wavelet thresholding. 1.
Exact Risk Analysis of Wavelet Regression
, 1995
"... Wavelets have motivated development of a host of new ideas in nonparametric regression smoothing. Here we apply the tool of exact risk analysis, to understand the small sample behavior of wavelet estimators, and thus to check directly the conclusions suggested by asymptotics. Comparisons between som ..."
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Cited by 24 (2 self)
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Wavelets have motivated development of a host of new ideas in nonparametric regression smoothing. Here we apply the tool of exact risk analysis, to understand the small sample behavior of wavelet estimators, and thus to check directly the conclusions suggested by asymptotics. Comparisons between some wavelet bases, and also between hard and soft thresholding are given from several viewpoints. Our results provide insight as to why the viewpoints and conclusions of Donoho and Johnstone differ from those of Hall and Patil. 1 Introduction In a series of papers, Donoho and Johnstone (1992 [9],1994a [10], 1995 [13]) and Donoho, Johnstone, Kerkyacharian and Picard (1995) [14] developed nonlinear wavelet shrinkage technology in nonparametric regression. For other work relating wavelets and nonparametric estimation, see Doukhan (1988) [15], Kerkyacharian and Picard, (1992) [21], Antoniadis (1994) [1] and Antoniadis, Gregoire and McKeague (1994) [2]. These papers have both introduced a new clas...
A New Class Of Unbalanced Haar Wavelets That Form An Unconditional Basis For L p On GENERAL MEASURE SPACES
 J. Fourier Anal. Appl
, 1995
"... . Given a complete separable oefinite measure space (X; \Sigma; ¯) and nested partitions of X, we construct unbalanced Haarlike wavelets on X that form an unconditional basis for Lp(X; \Sigma; ¯) where 1 ! p ! 1. Our construction and proofs build upon ideas of Burkholder and Mitrea. We show that i ..."
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Cited by 23 (5 self)
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. Given a complete separable oefinite measure space (X; \Sigma; ¯) and nested partitions of X, we construct unbalanced Haarlike wavelets on X that form an unconditional basis for Lp(X; \Sigma; ¯) where 1 ! p ! 1. Our construction and proofs build upon ideas of Burkholder and Mitrea. We show that if (X; \Sigma; ¯) is not purely atomic, then the unconditional basis constant of our basis is (max(p; q) \Gamma 1). We derive a fast algorithm to compute the coefficients. 1. Introduction Our goal is, given a measure space (X; \Sigma; ¯) and nested partitions of X, to construct unbalanced Haarlike wavelets on X that form an unconditional basis for L p j L p (X; \Sigma; ¯) where 1 ! p ! 1. Wavelets are traditionally defined on Euclidean spaces. They usually are the translates and dilates of one particular function and are orthogonal or biorthogonal with respect to the Lebesgue measure. However, we work on a general measure space, which need not even have a vector space structure, so transla...
WaveletBased Numerical Homogenization
 SIAM J. NUMER. ANAL
, 1998
"... A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in fine and coarsescale components followed by the elimination of the finescale contributions. If the operator is in divergence form, this is preserve ..."
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Cited by 23 (2 self)
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A numerical homogenization procedure for elliptic differential equations is presented. It is based on wavelet decompositions of discrete operators in fine and coarsescale components followed by the elimination of the finescale contributions. If the operator is in divergence form, this is preserved by this homogenization procedure. For periodic problems, results similar to classical effective coefficient theory are proved. The procedure can be applied to problems that are not cellperiodic and can be viewed as a direct solver using approximate and sparse block Gauss elimination.