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47
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...
Stability and Orthonormality of Multivariate Refinable Functions
 SIAM J. Math. Anal
, 1997
"... This paper characterizes the stability and orthonormality of the shifts of a multidimensional (M; c) refinable function OE in terms of the eigenvalues and eigenvectors of the transition operator W cau defined by the autocorrelation c au of its refinement mask c; where M is an arbitrary dilation matr ..."
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Cited by 43 (16 self)
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This paper characterizes the stability and orthonormality of the shifts of a multidimensional (M; c) refinable function OE in terms of the eigenvalues and eigenvectors of the transition operator W cau defined by the autocorrelation c au of its refinement mask c; where M is an arbitrary dilation matrix. Another consequence is that if the shifts of OE form a Riesz basis, then W cau has a unique eigenvector of eigenvalue 1; and all its other eigenvalues lie inside the unit circle. The general theory is applied to twodimensional nonseparable (M, c) refinable functions whose masks are constructed from Daubechies' conjugate quadrature filters.
Multiwavelets: Theory and Applications
, 1996
"... A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gam ..."
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Cited by 42 (4 self)
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A function OE(t) is refinable if it satisfies a dilation equation OE(t) = P k C k OE(2t \Gamma k). A refinable function (scaling function) generates a multiresolution analysis (MRA): Set of nested subspaces : : : V \Gamma1 ae V 0 ae V 1 : : : such that S 1 j=\Gamma1 V j = L 2 (R), T 1 j=\Gamma1 V j = f0g, and translates OE(t \Gamma k) constitute a basis of V 0 . Then a basis fw jk : w jk = w(2 j t \Gamma k) j; k 2 Zg of L 2 (R) is generated by a wavelet w(t), whose translates w(t \Gamma k) form a basis of W 0 , V 1 = V 0 \Phi W 0 . A standard (scalar) MRA assumes that there is only one scaling function. We make a step forward and allow several scaling functions OE 0 (t); : : : ; OE r\Gamma1 (t) to generate a basis of V 0 . The vector OE(t) = [OE 0 (t) : : : OE r\Gamma1 (t)] T satisfies a dilation equation with matrix coefficients C k . Associated with OE(t) is a multiwavelet w(t) = [w 0 (t) : : : w r\Gamma1 (t)] T . Unlike the scalar case, construction of a multiwave...
Frames in Hilbert C*modules and C*algebras
 J. OPERATOR THEORY
, 2000
"... We present a general approach to a module frame theory in C*algebras and Hilbert C*modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*modules over unital C*algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections ..."
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Cited by 36 (12 self)
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We present a general approach to a module frame theory in C*algebras and Hilbert C*modules. The investigations rely on the idea of geometric dilation to standard Hilbert C*modules over unital C*algebras that possess orthonormal Hilbert bases, and of reconstruction of the frames by projections and other bounded module operators with suitable ranges. We obtain frame representation and decomposition theorems, as well as similarity and equivalence results. Hilbert space frames and quasibases for conditional expectations of finite index on C*algebras appear as special cases. Using a canonical categorical equivalence of Hilbert C*modules over commutative C*algebras and (F)Hilbert bundles the results find a reinterpretation for frames in vector and (F)Hilbert bundles.
On Existence and Weak Stability of Matrix Refinable Functions
, 1996
"... : We consider the existence of distributional (or L 2 ) solutions of the matrix refinement equation b \Phi = P(\Delta=2) b \Phi(\Delta=2); where P is an r \Theta r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a ..."
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Cited by 35 (5 self)
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: We consider the existence of distributional (or L 2 ) solutions of the matrix refinement equation b \Phi = P(\Delta=2) b \Phi(\Delta=2); where P is an r \Theta r matrix with trigonometric polynomial entries. One of the main results of this paper is that the above matrix refinement equation has a compactly supported distributional solution if and only if the matrix P(0) has an eigenvalue of the form 2 n , n 2 ZZ + . A characterization of the existence of L 2 solutions of the above matrix refinement equation in terms of the mask is also given. A concept of L 2 weak stability of a (finite) sequence of function vectors is introduced. In the case when the function vectors are solutions of a matrix refinement equation, we characterize this weak stability in terms of the mask. AMS Subject Classification: Primary 42C15, 42B05, 41A30 Secondary 39B62, 42B10 Keywords: Refinable function vectors, stable basis y Permanent address: Department of Mathematics, Peking University. 1. Introducti...
Symmetric wavelet tight frames with two generators
 Applied and Computational Harmonic Analysis
, 2004
"... This paper uses the UEP approach for the construction of wavelet tight frames with two (anti) symmetric wavelets, and provides some results and examples that complement recent results by Q. Jiang. A description of a family of solutions when the lowpass scaling filter is of evenlength is provided. ..."
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Cited by 20 (1 self)
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This paper uses the UEP approach for the construction of wavelet tight frames with two (anti) symmetric wavelets, and provides some results and examples that complement recent results by Q. Jiang. A description of a family of solutions when the lowpass scaling filter is of evenlength is provided. When one wavelet is symmetric and the other is antisymmetric, the wavelet filters can be obtained by a simple procedure based on matching the roots of associated polynomials. The design examples in this paper begin with the construction of a lowpass filter h0(n) that is designed so as to ensure that both wavelets have at least a specified number of vanishing moments.
Characterizations of Scaling Functions: Continuous Solutions
 SIAM J. Matrix Anal. Appl
, 1994
"... A dilation equation is a functional equation of the form f(t) = � N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper ..."
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Cited by 20 (4 self)
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A dilation equation is a functional equation of the form f(t) = � N k=0 ck f(2t − k), and any nonzero solution of such an equation is called a scaling function. Dilation equations play an important role in several fields, including interpolating subdivision schemes and wavelet theory. This paper obtains sharp bounds for the Hölder exponent of continuity of any continuous, compactly supported scaling function in terms of the joint spectral radius of two matrices determined by the coefficients {c0,..., cN}. The arguments lead directly to a characterization of all dilation equations that have continuous, compactly supported solutions.
Regularity of Multivariate Refinable Functions
 CONSTR. APPROX
, 1998
"... The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A, although factorization techniques, which are typically use ..."
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Cited by 18 (1 self)
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The regularity of a univariate compactly supported refinable function is known to be related to the spectral properties of an associated transfer operator. In the case of multivariate refinable functions with a general dilation matrix A, although factorization techniques, which are typically used in the univariate setting, are no longer applicable, we derive similar results that also depend on the spectral properties of A.
The Spectral Theory of Multiresolution Operators and Applications
, 1994
"... this article we explore the notion of the multiresolution operator, its spectral theory, and applications. This operator (also called the transition operator) has appeared as a fundamental tool in several aspects of wavelet theory, such as the LawtonCohen theorem on wavelet orthonormal bases [2, 18 ..."
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Cited by 17 (2 self)
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this article we explore the notion of the multiresolution operator, its spectral theory, and applications. This operator (also called the transition operator) has appeared as a fundamental tool in several aspects of wavelet theory, such as the LawtonCohen theorem on wavelet orthonormal bases [2, 18, 19], and the work of Eirola [8] and others [1, 3, 4, 5, 23] on the Sobolev smoothness of wavelet scaling functions. After introducing the multiresolution operator and observing its connection with the convolution and downsampling operations of multirate signal processing, we present a review of the work of Lawton and that of Eirola. Throughout the paper we work in the setting of rank m wavelet systems (m is the integer dilation factor, not necessarily 2). This involves some generalization of previous work, and yields initial results on the differentiability of wavelet scaling functions for rank m ? 2. In particular, we find that the minimal support rank 3 scaling functions with