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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 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...
Framelets: MRABased Constructions of Wavelet Frames
, 2001
"... We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spl ..."
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Cited by 131 (54 self)
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We discuss wavelet frames constructed via multiresolution analysis (MRA), with emphasis on tight wavelet frames. In particular, we establish general principles and specific algorithms for constructing framelets and tight framelets, and we show how they can be used for systematic constructions of spline, pseudospline tight frames and symmetric biframes with short supports and high approximation orders. Several explicit examples are discussed. The connection of these frames with multiresolution analysis guarantees the existence of fast implementation algorithms, which we discuss briefly as well.
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 61 (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.
Biorthogonal Wavelet Expansions
 Constr. Approx
"... This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connectio ..."
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Cited by 50 (7 self)
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This paper is concerned with developing conditions on a given finite collection of compactly supported algebraically linearly independent refinable functions that insure the existence of biorthogonal systems of refinable functions with similar properties. In particular we address the close connection of this issue with stationary subdivision schemes. Key Words: Finiteley generated shiftinvariant spaces, stationary subdivision schemes, matrix refinement relations, biorthogonal wavelets. AMS Subject Classification: 39B62, 41A63 1 Introduction During the past few years the construction of multivariate wavelets has received considerable attention. It is quite apparent that multivariate wavelets with good localazition properties in frequency and spatial domains which constitute an orthonormal basis of L 2 (IR s ) are hard to realize. On the other hand, it turns out that in many applications orthogonality is not really important whereas locality, in particular, compact support is very...
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 25 (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 : : : : : : : :
Orthogonal polynomials and the construction of piecewise polynomial smooth wavelets
 SIAM J. Math. Anal
, 1999
"... Abstract. Orthogonal polynomials are used to construct families of C 0 and C 1 orthogonal, compactly supported spline multiwavelets. These families are indexed by an integer which represents the order of approximation. We indicate how to obtain from these multiwavelet bases for L 2 [0,1] and present ..."
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Cited by 14 (3 self)
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Abstract. Orthogonal polynomials are used to construct families of C 0 and C 1 orthogonal, compactly supported spline multiwavelets. These families are indexed by an integer which represents the order of approximation. We indicate how to obtain from these multiwavelet bases for L 2 [0,1] and present a C 2 example.
Multiwavelet Frames from Refinable Function Vectors
, 2001
"... Starting from any two compactly supported drefinable function vectors in L 2 (R) multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual dwavelet frames in L 2 (R) and they achieve the b ..."
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Cited by 14 (8 self)
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Starting from any two compactly supported drefinable function vectors in L 2 (R) multiplicity r and dilation factor d, we show that it is always possible to construct 2rd wavelet functions with compact support such that they generate a pair of dual dwavelet frames in L 2 (R) and they achieve the best possible orders of vanishing moments. When all the components of the two realvalued drefinable function vectors are either symmetric or antisymmetric with their symmetry centers differing by half integers, such 2rd wavelet functions, which generate a pair of dual dwavelet frames, can be realvalued and be either symmetric or antisymmetric with the same symmetry center. Wavelet frames from any drefinable function vector are also considered. This paper generalizes the work in [5, 12, 13] on constructing dual wavelet frames from scalar refinable functions to the multiwavelet case. Examples are provided to illustrate the construction in this paper.
NonUniform Sampling In ShiftInvariant Spaces
, 2000
"... . This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. The reconstruction of a function or signal or image f from its nonuniform samples f(x j ) is a common task in many applications in data or signal or image processing. The n ..."
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Cited by 8 (4 self)
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. This article discusses modern techniques for nonuniform sampling and reconstruction of functions in shiftinvariant spaces. The reconstruction of a function or signal or image f from its nonuniform samples f(x j ) is a common task in many applications in data or signal or image processing. The nonuniformity of the sampling set is often a fact of life and prevents the use of the standard methods from Fourier analysis. The nonuniform sampling problem is usually treated in the context of bandlimited functions and with tools from complex analysis. However, many applied problems impose di#erent a priori constraints on the type of function. These constraints are taken into consideration by investigating the problem in general shiftinvariant spaces rather than for bandlimited functions only. This generalization requires a new set of techniques and ideas. While some of the tools are implicitly used in certain questions in approximation theory, wavelet theory and frame theory, they hav...
A class of orthogonal multiresolution analyses in 2D
"... . A two parameter family of multiresolution analyses of L 2 (R 2 ) each generated by three orthogonal, continuous, compactly supported scaling functions is constructed using fractal interpolation surfaces. The scaling functions remain orthogonal when restricted to certain triangles making th ..."
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Cited by 5 (1 self)
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. A two parameter family of multiresolution analyses of L 2 (R 2 ) each generated by three orthogonal, continuous, compactly supported scaling functions is constructed using fractal interpolation surfaces. The scaling functions remain orthogonal when restricted to certain triangles making them useful for problems with bounded domains. x1. Introduction In the usual construction of a wavelet using a multiresolution analysis (MRA) (V p ) there is a single function OE called the scaling function whose set of integer translates forms a Riesz basis for V 0 . In contrast, multiwavelets are constructed from an MRA that is generated by a finite set of scaling functions OE 1 ; : : : ; OE r . Definition 1. If e 1 ; : : : e n are independent vectors in R n and N is an integer greater than 1, then a nested sequence of closed linear subspaces (V p ) in L 2 (R n ) is called an MRA of multiplicity r with dilation N and lattice e 1 Z \Theta \Delta \Delta \Delta \Theta e nZ if a) ...
A Class of Orthogonal Refinable Functions and Wavelets
"... We give a construction, for any n 2, of a space S of spline functions of degree n 1 with simple knots in 4 ZZ which is generated by a triple of re nable, orthogonal functions with compact support. Indeed the result holds more generally by replacing the Bspline of degree n 1 with simple knots at t ..."
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Cited by 4 (1 self)
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We give a construction, for any n 2, of a space S of spline functions of degree n 1 with simple knots in 4 ZZ which is generated by a triple of re nable, orthogonal functions with compact support. Indeed the result holds more generally by replacing the Bspline of degree n 1 with simple knots at the integers by any continuous re nable function whose mask is a Hurwitz polynomial of degree n. A simple construction is also given for the corresponding wavelets. x1.