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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 541 (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...
Symmetric orthonormal scaling functions and wavelets with dilation factor 4
 Adv. Comput. Math
, 1998
"... Abstract. It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling f ..."
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Cited by 26 (10 self)
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Abstract. It is well known that in the univariate case, up to an integer shift and possible sign change, there is no dyadic compactly supported symmetric orthonormal scaling function except for the Haar function. In this paper we are concerned with the construction of symmetric orthonormal scaling functions with dilation factor d = 4. Several examples of such scaling functions are provided in this paper. In particular, two examples of C1 orthonormal scaling functions, which are symmetric about 0 and 1, respectively, are presented. We will then discuss how to construct symmetric 6 wavelets from these scaling functions. We explicitly construct the corresponding orthonormal symmetric wavelets for all the examples given in this paper.
Construction of Multivariate Biorthogonal Wavelets by CBC Algorithm
 Adv. Comput. Math
, 1998
"... In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavele ..."
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Cited by 23 (11 self)
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In applications, it is well known that short support, high vanishing moments and reasonable smoothness are the three most important properties of a biorthogonal wavelet. Based on our previous work on analysis and construction of optimal fundamental refinable functions and optimal biorthogonal wavelets, in this paper, we shall discuss the mutual relations among these three properties. For example, we shall see that any orthogonal scaling function, which is supported on [0; 2r \Gamma 1] s for some positive integer r and has accuracy order r, has Lp (1 p 1) smoothness not exceeding that of the univariate Daubechies orthogonal scaling function which is supported on [0; 2r \Gamma 1]. Similar results hold true for fundamental refinable functions and biorthogonal wavelets. Then, we shall discuss the relation between symmetry and the smoothness of a refinable function. Next, we discuss the coset by coset (CBC) algorithm reported in Han [29] to construct biorthogonal wavelets with arbitrar...
Construction of Compactly Supported MBand Wavelets
 Appl. Comput. Harmonic Anal
, 1995
"... In this paper, we consider the asymptotic regularity of Daubechies scaling functions and construct examples of Mband scaling functions which are both orthonormal and cardinal for M 3. Keywords Multiresolution, Mband scaling function, Mband wavelets, cardinal function, orthonormality. 1 The aut ..."
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Cited by 16 (10 self)
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In this paper, we consider the asymptotic regularity of Daubechies scaling functions and construct examples of Mband scaling functions which are both orthonormal and cardinal for M 3. Keywords Multiresolution, Mband scaling function, Mband wavelets, cardinal function, orthonormality. 1 The author is partially supported by the National Science Foundation of China and the Zhejiang Provincial Science Foundation of China. 1. Introduction Let M 2 be a fixed positive integer. A family of closed subspaces V j ; j 2 Z; of L 2 , the space of all square integrable functions on the real line, is said to be a multiresolution of L 2 if the following conditions hold: (i) V j ae V j+1 , and f 2 V j if and only if f(M \Delta) 2 V j+1 for all j 2 Z; (ii) [ j2Z V j is dense in L 2 and " j2Z V j = ;; (iii) There exists a function OE in V 0 such that fOE(\Delta \Gamma k); k 2 Zg is a Riesz basis of V 0 . Here we say that fOE(\Delta \Gamma k); k 2 Zg is a Riesz basis of V 0 if there exis...
MBand Scaling Function With Filter Having Vanishing Moments Two And Minimal Length
 J. Math. Anal. Appl
, 1998
"... . In this paper, we consider the Holder continuity, local linearity, linear independence and interpolation problem of the Mband scaling function with its filter having vanishing moments two and minimal length, and explicit construction of wavelets. Especially we find some new properties which is no ..."
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Cited by 9 (7 self)
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. In this paper, we consider the Holder continuity, local linearity, linear independence and interpolation problem of the Mband scaling function with its filter having vanishing moments two and minimal length, and explicit construction of wavelets. Especially we find some new properties which is not true when M = 2, such as local linearity, local linear dependence, differentiability at adjoint Madic points and interpolation problem at integer knots. 1. Introduction Let M 2 be a fixed positive integer. A compactly supported and square integrable function OE is called a scaling function if it satisfies R R OE(x)dx = 1, R R OE(x)OE(x \Gamma k) = ffi k and a refinement equation OE(x) = X k2Z c k OE(Mx \Gamma k); (1) where fc k g is a sequence with finite length and satisfying P k2Z c k = M , and ffi k is the Kronecker symbol defined by ffi k = 1 when k = 0 and ffi k = 0 when k 6= 0. For a sequence fc k g with finite length, define H(z) = 1 M X k2Z c k z k : Then H...
A simple construction for the mband dualtree complex wavelet transform
 Digital Signal Processing Workshop, 12th  Signal Processing Education Workshop, 4th
, 2006
"... The dualtree complex wavelet transform (DTCWT) which utilizes two 2band discrete wavelet transforms (DWT) was recently extended to Mband by Chaux et al. In this paper, we provide a simple construction method for an Mband DTCWT, with M = rd where r, d ∈ Z. In particular, we show how to extend a ..."
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Cited by 3 (1 self)
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The dualtree complex wavelet transform (DTCWT) which utilizes two 2band discrete wavelet transforms (DWT) was recently extended to Mband by Chaux et al. In this paper, we provide a simple construction method for an Mband DTCWT, with M = rd where r, d ∈ Z. In particular, we show how to extend a given rband DTCWT to an rdband one. For convenience, the case where r = 2, d = 2 is considered. However, the scheme can be extended to general {r, d} pairs straightforwardly. The extension to 2D which achieves a directional analysis is also provided. Index Terms — Mband dualtree, 2band dualtree, Hilbert transform pairs, directional wavelets. 1.
An Algorithm for the Construction of Symmetric and AntiSymmetric MBand Wavelets
 in Wavelet Applications in Signal and Image Processing VIII, Proceedings of SPIE 4119
, 2000
"... In this paper, we give an algorithm to construct semiorthogonal symmetric and antisymmetric Mband wavelets. As an application, some semiorthogonal symmetric and antisymmetric Mband spline wavelets are constructed explicitly. Also we show that if we want to construct symmetric or antisymmetric ..."
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Cited by 2 (1 self)
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In this paper, we give an algorithm to construct semiorthogonal symmetric and antisymmetric Mband wavelets. As an application, some semiorthogonal symmetric and antisymmetric Mband spline wavelets are constructed explicitly. Also we show that if we want to construct symmetric or antisymmetric Mband wavelets from a multiresolution, then that multiresolution has a symmetric scaling function. Keywords: Symmetric and Antisymmetric Wavelets, M Band Wavelets, Spline Wavelets 1. INTRODUCTION Fix an integer M larger than 2. A multiresolution 4 is a family of nested subspaces fV j g j2Z of L 2 such that ffl " j2Z V j = f0g and [ j2Z V j is dense in L 2 . ffl f 2 V j () f(M \Delta) 2 V j+1 for all j 2 Z. ffl V j ae V j+1 for all j 2 Z. ffl There exists a compactly supported function OE 2 V 0 such that fOE(\Delta \Gamma k) : k 2 Zg is a Riesz basis of V 0 . We remark that the function OE in the multiresolution of this paper is assumed to be compactly supported instead of OE 2 ...
Sobolev Regularity For Rank M Wavelets
 CML Rep., Rice Univ
, 1997
"... . This paper explores the Sobolev regularity of rank M wavelets and refinement schemes. We find that the regularity of orthogonal wavelets with maximal vanishing moments grows at most logarithmically with filter length when M is odd, but linearly for even M . When M = 3 and M = 4, we show that the r ..."
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Cited by 2 (0 self)
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. This paper explores the Sobolev regularity of rank M wavelets and refinement schemes. We find that the regularity of orthogonal wavelets with maximal vanishing moments grows at most logarithmically with filter length when M is odd, but linearly for even M . When M = 3 and M = 4, we show that the regularity does achieve these upper bounds for asymptotic growth, complementing earlier results for M = 2. A new class of wavelet filters is introduced, by asserting zeros of the wavelet symbol at preperiodic points of the mapping ø : ! !M! mod 2ß. While this class includes the generalized Daubechies wavelets, numerical experiments demonstrate that the class also includes wavelet families with greater smoothness for a given filter length. Finally, members of the class of wavelets that have maximal Sobolev regularity for a given filter length are found as the solution to an optimization problem. Key words. wavelets, refinement equations, Sobolev regularity, smoothness, filter design AMS subj...
Orthogonal pwavelets on R+
"... In this paper we give a general construction of compactly supported orthogonal pwavelets in L2(R+) arising from scaling ¯lters with p n many terms. For all integer p ¸ 2 these wavelets are identi¯ed with certain lacunary Walsh series on R+. The case where p = 2 was studied by W.C. Lang mainly from ..."
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Cited by 1 (0 self)
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In this paper we give a general construction of compactly supported orthogonal pwavelets in L2(R+) arising from scaling ¯lters with p n many terms. For all integer p ¸ 2 these wavelets are identi¯ed with certain lacunary Walsh series on R+. The case where p = 2 was studied by W.C. Lang mainly from the point of view of the wavelet analysis on the Cantor dyadic group (the dyadic or 2series local ¯eld). Our approach is connected with the Walsh { Fourier transform and the elements of Mband wavelet theory.