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38
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 377 (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...
On the Construction of Multivariate (pre)wavelets
, 1992
"... : A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L 2 (IR d ) onto these spaces, and requires neither decay nor stability of the scaling function. F ..."
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Cited by 78 (11 self)
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: A new approach for the construction of wavelets and prewavelets on IR d from multiresolution is presented. The method uses only properties of shiftinvariant spaces and orthogonal projectors from L 2 (IR d ) onto these spaces, and requires neither decay nor stability of the scaling function. Furthermore, this approach allows a simple derivation of previous, as well as new, constructions of wavelets, and leads to a complete resolution of questions concerning the nature of the intersection and the union of a scale of spaces to be used in a multiresolution. AMS (MOS) Subject Classifications: primary: 41A63, 46C99; secondary: 41A30, 41A15, 42B99, 46E20. Key Words and phrases: wavelets, multiresolution, shiftinvariant spaces, box splines. Authors' affiliation and address: 1 Center for Mathematical Sciences University of WisconsinMadison 610 Walnut St. Madison WI 53705 and 2 Department of Mathematics University of South Carolina Columbia SC 29208 This work was carried out while t...
Stability and linear independence associated with wavelet decompositions
 Proc. Amer. Math. Soc
, 1993
"... Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask ..."
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Cited by 60 (14 self)
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Wavelet decompositions are based on basis functions satisfying refinement equations. The stability, linear independence and orthogonality of the integer translates of basis functions play an essential role in the study of wavelets. In this paper we characterize these properties in terms of the mask sequence in the refinement equation satisfied by the basis function.
Multiresolution and wavelets
 Proc. Edinburgh Math. Soc
, 1994
"... Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general ..."
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Cited by 48 (24 self)
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Multiresolution is investigated on the basis of shiftinvariant spaces. Given a finitely generated shiftinvariant subspace S of L2(IR d), let Sk be the 2 kdilate of S (k ∈ Z). A necessary and sufficient condition is given for the sequence {Sk}k ∈ Z to form a multiresolution of L2(IR d). A general construction of orthogonal wavelets is given, but such wavelets might not have certain desirable properties. With the aid of the general theory of vector fields on spheres, it is demonstrated that the intrinsic properties of the scaling function must be used in constructing orthogonal wavelets with a certain decay rate. When the scaling function is skewsymmetric about some point, orthogonal wavelets and prewavelets are constructed in such a way that they possess certain attractive properties. Several examples are provided to illustrate the general theory.
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
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Cited by 24 (12 self)
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The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
Characterizations of Linear Independence and Stability of the Shifts of a Univariate Refinable Function in Terms of Its Refinement Mask
, 1992
"... : Characterizations of the linear independence and stability properties of the integer translates of a compactly supported univariate refinable function in terms of its mask are established. The results extend analogous ones of Jia and Wang which were derived for dyadic refinements and finite masks. ..."
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Cited by 19 (6 self)
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: Characterizations of the linear independence and stability properties of the integer translates of a compactly supported univariate refinable function in terms of its mask are established. The results extend analogous ones of Jia and Wang which were derived for dyadic refinements and finite masks. AMS (MOS) Subject Classifications: primary: 39B32, 41A15, 46C99; secondary: 42A99, 46E20. Key Words and phrases: wavelets, multiresolution, shiftinvariant spaces, refinement equation, stability, linear independence. Author's affiliation and address: Computer Science Department University of WisconsinMadison 1210 W. Dayton St. Madison WI 53706 email: amos@cs.wisc.edu Supported in part by the United States Army (Contract DAAL03G900090) and by the National Science Foundation (grants DMS9000053 and DMS9102857). Characterizations of linear independence and stability of the shifts of a univariate refinable function in terms of its refinement mask Amos Ron 1. The problem Let ...
On the Support Properties of Scaling Vectors
 Appl. Comput. Harmonic Anal
, 1996
"... In Chui and Wang [3], support properties are derived for a scaling function generating a function space V 0 ` L 2 (IR). Motivated by this work, we consider support properties for scaling vectors. In [9], Goodman and Lee derive necessary and sufficient conditions for the scaling vector fOE 1 ; : : ..."
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Cited by 18 (1 self)
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In Chui and Wang [3], support properties are derived for a scaling function generating a function space V 0 ` L 2 (IR). Motivated by this work, we consider support properties for scaling vectors. In [9], Goodman and Lee derive necessary and sufficient conditions for the scaling vector fOE 1 ; : : : ; OE r g, r 1, to form a Riesz basis for V 0 and develop a general theory for spline wavelets of multiplicity r ? 1. We consider conditions under which linear combinations of scaling functions generate V 0 . These conditions also characterize all other scaling vectors that generate the same V 0 . In addition we describe the scaling vectors of minimal support for V 0 . Next, we give sufficient conditions on the twoscale symbol for scaling vectors under which a given matrix refinement equation can be solved. A splinewavelet example illustrates these results. For the single scaling function OE, the support of OE is characterized by the degree of the twoscale symbol. The situation is more ...