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96
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 543 (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...
Sampling—50 years after Shannon
 Proceedings of the IEEE
, 2000
"... This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the math ..."
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Cited by 341 (27 self)
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This paper presents an account of the current state of sampling, 50 years after Shannon’s formulation of the sampling theorem. The emphasis is on regular sampling, where the grid is uniform. This topic has benefited from a strong research revival during the past few years, thanks in part to the mathematical connections that were made with wavelet theory. To introduce the reader to the modern, Hilbertspace formulation, we reinterpret Shannon’s sampling procedure as an orthogonal projection onto the subspace of bandlimited functions. We then extend the standard sampling paradigm for a representation of functions in the more general class of “shiftinvariant” functions spaces, including splines and wavelets. Practically, this allows for simpler—and possibly more realistic—interpolation models, which can be used in conjunction with a much wider class of (antialiasing) prefilters that are not necessarily ideal lowpass. We summarize and discuss the results available for the determination of the approximation error and of the sampling rate when the input of the system is essentially arbitrary; e.g., nonbandlimited. We also review variations of sampling that can be understood from the same unifying perspective. These include wavelets, multiwavelets, Papoulis generalized sampling, finite elements, and frames. Irregular sampling and radial basis functions are briefly mentioned. Keywords—Bandlimited functions, Hilbert spaces, interpolation, least squares approximation, projection operators, sampling,
Short Wavelets and Matrix Dilation Equations
, 1995
"... Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a twoband orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling funct ..."
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Cited by 84 (10 self)
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Scaling functions and orthogonal wavelets are created from the coefficients of a lowpass and highpass filter (in a twoband orthogonal filter bank). For "multifilters" those coefficients are matrices. This gives a new block structure for the filter bank, and leads to multiple scaling functions and wavelets. Geronimo, Hardin, and Massopust constructed two scaling functions that have extra properties not previously achieved. The functions \Phi 1 and \Phi 2 are symmetric (linear phase) and they have short support (two intervals or less), while their translates form an orthogonal family. For any single function \Phi, apart from Haar's piecewise constants, those extra properties are known to be impossible. The novelty is to introduce 2 by 2 matrix coefficients while retaining orthogonality. This note derives the properties of \Phi 1 and \Phi 2 from the matrix dilation equation that they satisfy. Then our main step is to construct associated wavelets: two wavelets for two scaling functions....
Refinable Function Vectors
 SIAM J. Math. Anal
"... Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade alg ..."
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Cited by 77 (9 self)
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Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade algorithm. We also give necessary and sufficient conditions for the stability and orthonormality of refinable function vectors in terms of their refinement matrix masks. Regularity of function vectors gives smoothness orders in the time domain, and decay rates at infinity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.
Approximation By Translates Of Refinable Functions
, 1996
"... . The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth ..."
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Cited by 70 (14 self)
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. The functions f 1 (x); : : : ; fr (x) are refinable if they are combinations of the rescaled and translated functions f i (2x \Gamma k). This is very common in scientific computing on a regular mesh. The space V 0 of approximating functions with meshwidth h = 1 is a subspace of V 1 with meshwidth h = 1=2. These refinable spaces have refinable basis functions. The accuracy of the computations depends on p, the order of approximation, which is determined by the degree of polynomials 1; x; : : : ; x p\Gamma1 that lie in V 0 . Most refinable functions (such as scaling functions in the theory of wavelets) have no simple formulas. The functions f i (x) are known only through the coefficients c k in the refinement equationscalars in the traditional case, r \Theta r matrices for multiwavelets. The scalar "sum rules" that determine p are well known. We find the conditions on the matrices c k that yield approximation of order p from V 0 . These are equivalent to the StrangFix condition...
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 70 (17 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.
Design of prefilters for discrete multiwavelet transforms
 IEEE Trans. Signal Processing
, 1996
"... AbstractThe pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using treestructured multirate filter banks. In this paper, we propose a general algorithm to compute multiwavelet transform coeficients by adding proper pre ..."
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Cited by 66 (4 self)
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AbstractThe pyramid algorithm for computing single wavelet transform coefficients is well known. The pyramid algorithm can be implemented by using treestructured multirate filter banks. In this paper, we propose a general algorithm to compute multiwavelet transform coeficients by adding proper premultirate filter banks before the vector filter banks that generate multiwavelets. The proposed algorithm can be thought of as a discrete vectorvalued wavelet transform for certain discretetime vectorvalued signals. The proposed algorithm can be also thought of as a discrete multiwavelet transform for discretetime signals. We then present some numerical experiments to illustrate the performance of the algorithm, which indicates that the energy compaction for discrete multiwavelet transforms may be better than the one for conventional discrete wavelet transforms. I.
Approximation Order Provided by Refinable Function Vectors
 CONSTR. APPROX.
, 1995
"... In this paper, we consider Lp{approximation byinteger translates of a finite set of functions ( =0�:::�r; 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector = ( ) r;1 =0 is refinable, necessary and sufficient conditions for the refinem ..."
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Cited by 60 (6 self)
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In this paper, we consider Lp{approximation byinteger translates of a finite set of functions ( =0�:::�r; 1) which are not necessarily compactly supported, but have a suitable decay rate. Assuming that the function vector = ( ) r;1 =0 is refinable, necessary and sufficient conditions for the refinement mask are derived. In particular, if algebraic polynomials can be exactly reproduced by integer translates of, then a factorization of the refinement mask of can be given. This result is a natural generalization of the result for a single function, where the refinement mask
Matrix Refinement Equations: Existence and Uniqueness
 J. Fourier Anal. Appl
, 1996
"... . Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vectorvalued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of sca ..."
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Cited by 58 (3 self)
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. Matrix refinement equations are functional equations of the form f(x) = P N k=0 c k f(2x \Gamma k), where the coefficients c k are matrices and f is a vectorvalued function. Refinement equations play key roles in wavelet theory and approximation theory. Existence and uniqueness properties of scalar refinement equations (where the coefficients c k are scalars) are known. This paper considers analogous questions for matrix refinement equations. Conditions for existence and uniqueness of compactly supported distributional solutions are given in terms of the convergence properties of an infinite product of the matrix \Delta = 1 2 P c k with itself. Fundamental differences between solutions of matrix equations and scalar refinement equations are examined. In particular, it is shown that "constrained" solutions of the matrix refinement equation can exist even when the infinite product diverges. The existence of constrained solutions is related to the eigenvalue structure of \Delta; so...
An algorithm for matrix extension and wavelet construction
 Math. Comp
, 1996
"... Abstract. This paper gives a practical method of extending an n × r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, toa square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a para ..."
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Cited by 56 (8 self)
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Abstract. This paper gives a practical method of extending an n × r matrix P(z), r ≤ n, with Laurent polynomial entries in one complex variable z, toa square matrix also with Laurent polynomial entries. If P(z) has orthonormal columns when z is restricted to the torus T, it can be extended to a paraunitary matrix. If P(z)hasrankrfor each z ∈ T, it can be extended to a matrix with nonvanishing determinant on T. The method is easily implemented in the computer. It is applied to the construction of compactly supported wavelets and prewavelets from multiresolutions generated by several univariate scaling functions with an arbitrary dilation parameter. 1.