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29
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.
The application of multiwavelet filter banks to image processing
 IEEE Trans. Image Process
, 1999
"... Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrixvalued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2channel wavelet systems. After reviewing this recently d ..."
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Cited by 59 (5 self)
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Multiwavelets are a new addition to the body of wavelet theory. Realizable as matrixvalued filter banks leading to wavelet bases, multiwavelets offer simultaneous orthogonality, symmetry, and short support, which is not possible with scalar 2channel wavelet systems. After reviewing this recently developed theory, we examine the use of multiwavelets in a filter bank setting for discretetime signal and image processing. Multiwavelets differ from scalar wavelet systems in requiring two or more input streams to the multiwavelet filter bank. We describe two methods (repeated row and approximation/deapproximation) for obtaining such a vector input stream from a onedimensional signal. Algorithms for symmetric extension of signals at boundaries are then developed, and naturally integrated with approximationbased preprocessing. We describe an additional algorithm for multiwavelet processing of twodimensional signals, two rows at a time, and develop a new family of multiwavelets (the constrained pairs) that is wellsuited to this approach. This suite of novel techniques is then applied to two basic signal processing problems, denoising via waveletshrinkage, and data compression. After developing the approach via model problems in one dimension, we applied multiwavelet processing to images, frequently obtaining performance superior to the comparable scalar wavelet transform.
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 35 (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...
Balanced Multiwavelets Theory and Design
 IEEE TRANS. SIGNAL PROCESSING
, 1998
"... This correspondence deals with multiwavelets, which are a recent generalization of wavelets in the context of timevarying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. ..."
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Cited by 34 (0 self)
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This correspondence deals with multiwavelets, which are a recent generalization of wavelets in the context of timevarying filter banks and with their applications to signal processing and especially compression. By their inherent structure, multiwavelets are fit for processing multichannel signals. This is the main issue in which we will be interested here. The outline of the correspondence is as follows. First, we will review material on multiwavelets and their links with multifilter banks and, especially, timevarying filter banks. Then, we will have a close look at the problems encountered when using multiwavelets in applications, and we will propose new solutions for the design of multiwavelets filter banks by introducing the socalled balanced multiwavelets.
Regularity of Multiwavelets
 Adv. Comput. Math
, 1997
"... this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for Bsplines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vecto ..."
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Cited by 30 (2 self)
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this paper is an interesting observation made by Plonka concerning the factorization of the matrix symbol associated with the refinement equation for Bsplines with equally spaced multiple knots at integers and subsequent developments which relate this factorization to regularity of refinable vector fields over the real line. Our intention is to contribute to this train of ideas which is partially driven by the importance of refinable vector fields in the construction of multiwavelets. The use of subdivision methods will allow us to consider the problem almost entirely in the spatial domain and leads to exact characterizations of differentiability and Holder regularity in arbitrary Lp spaces. We first study the close relationship between vector subdivision schemes and a generalized notion of scalar subdivision schemes based on biinfinite matrices with certain periodicity properties. For the latter type of subdivision scheme we will derive criteria for convergence and Holder regularity of the limit function, which mainly depend on the spectral of a biinfinite matrix induced by the subdivision operator, and we will show that differentiability of the limit functions can be characterized by factorization properties of the subdivision operator. By switching back to vector subdivision we will transfer these results to refinable vectors fields and obtain characterizations of regularity by factorization and spectral radius properties of the symbol associated to the refinable vector field. Finally, we point out how multiwavelets can be generated from orthonormal refinable biinfinite vector fields.
Approximation Properties and Construction of Hermite Interpolants and Biorthogonal Multiwavelets
, 2001
"... ..."
Biorthogonal Multiwavelets on the Interval: Cubic Hermite Splines
 CONSTR. APPROX
, 1998
"... Starting with Hermite cubic splines as primal multigenerator, first a dual multigenerator on R is constructed which consists of continuous functions, has small support and is exact of order two. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the ..."
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Cited by 25 (8 self)
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Starting with Hermite cubic splines as primal multigenerator, first a dual multigenerator on R is constructed which consists of continuous functions, has small support and is exact of order two. We then derive multiresolution sequences on the interval while retaining the polynomial exactness on the primal and dual side. This guarantees moment conditions of the corresponding wavelets. The concept of stable completions [CDP] is then used to construct corresponding primal and dual multiwavelets on the interval as follows. An appropriate variation of what is known as hierarchical basis in finite element methods is shown to be an initial completion. This is then in a second step projected into the desired complements spanned by compactly supported biorthogonal multiwavelets. The masks of all multigenerators and multiwavelets are finite so that decomposition and reconstruction algorithms are simple and efficient. Furthermore, in addition to Jackson estimates which follow from the exactness,...
Multiwavelet Prefilters: Orthogonal Prefilters Preserving Approximation Order p<=3
, 1997
"... CONTENTS Page DEDICATION PAGE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES : : : : : : : : : : : : : : : ..."
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Cited by 24 (2 self)
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CONTENTS Page DEDICATION PAGE : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : ii ACKNOWLEDGMENTS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : iii LIST OF FIGURES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : v Chapter I. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Preliminaries : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1 Synopsis of Main Theorems : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6 II. MULTIWAVELETS AND SCALING VECTORS : : : : : : : : : : : : : : : : : : : : 11 Orthonormal Scaling Vector of Legendre Polynomials : : : : : : : :
Signal and Image Denoising via Wavelet Thresholding: Orthogonal and Biorthogonal, Scalar and Multiple Wavelet Transforms
 In Nonlinear and Nonstationary Signal Processing
, 1998
"... The method of signal denoising... In this paper we study wavelet thresholding in the context of scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. Two types of multiwavelet thresholding are considered: scalar and vector. Both of them take into a ..."
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Cited by 20 (2 self)
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The method of signal denoising... In this paper we study wavelet thresholding in the context of scalar orthogonal, scalar biorthogonal, multiple orthogonal and multiple biorthogonal wavelet transforms. Two types of multiwavelet thresholding are considered: scalar and vector. Both of them take into account the covariance structure of the transform. The form of the universal threshold is carefully formulated and is the key to the excellent results obtained in the extensive numerical simulations of signal and image denoising reported here. Sections 2 to 5 are concerned with the deterministic formulation of relevant components of multiwavelet analysis. In Section 2 we give a summary of multiresolution analysis and semiorthogonal, orthogonal and biorthogonal multiwavelet functions. The rest of the paper concentrates on the mostused practical cases of multiplicity 1 (scalar) and 2 (two scaling functions). Section 3 introduces two classes of orthogonal multiwavelets which may be used in multifilter banks  the GeronimoHardinMassopust (GHM) and ChuiLian (CL) classes, while Section 4 discusses biorthogonal multifilter banks, denoted BiGHM and BiHermite. For multifilter banks the given scalar input signal must be associated with a sequence of length2 vectors: oversampling and critical sampling preprocessing is carefully studied in Section 5. Sections 6 and 7 look at two important aspects of the processing of 1D stochastic signals, the covariance structure of the output, and denoising via thresholding. If stochastic noise is the input to preprocessing followed by discrete multiwavelet transform, very variable covariance structures of the output can result; in particular the average input variance of white noise can be inflated or deflated on output as shown in Section 6 for the GHM and C...