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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...
Wavelet transforms versus Fourier transforms
 Department of Mathematics, MIT, Cambridge MA
, 213
"... Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and t ..."
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Cited by 82 (2 self)
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Abstract. This note is a very basic introduction to wavelets. It starts with an orthogonal basis of piecewise constant functions, constructed by dilation and translation. The "wavelet transform " maps each f(x) to its coefficients with respect to this basis. The mathematics is simple and the transform is fast (faster than the Fast Fourier Transform, which we briefly explain), but approximation by piecewise constants is poor. To improve this first wavelet, we are led to dilation equations and their unusual solutions. Higherorder wavelets are constructed, and it is surprisingly quick to compute with them — always indirectly and recursively. We comment informally on the contest between these transforms in signal processing, especially for video and image compression (including highdefinition television). So far the Fourier Transform — or its 8 by 8 windowed version, the Discrete Cosine Transform — is often chosen. But wavelets are already competitive, and they are ahead for fingerprints. We present a sample of this developing theory. 1. The Haar wavelet To explain wavelets we start with an example. It has every property we hope for, except one. If that one defect is accepted, the construction is simple and the computations are fast. By trying to remove the defect, we are led to dilation equations and recursively defined functions and a small world of fascinating new problems — many still unsolved. A sensible person would stop after the first wavelet, but fortunately mathematics goes on. The basic example is easier to draw than to describe: W(x)
Subdivision schemes in Lp spaces
 Adv. Comput. Math
, 1995
"... Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes in Lp spaces (1 ≤ p ≤ ∞). We characterize the Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of two ..."
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Cited by 62 (22 self)
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Subdivision schemes play an important role in computer graphics and wavelet analysis. In this paper we are mainly concerned with convergence of subdivision schemes in Lp spaces (1 ≤ p ≤ ∞). We characterize the Lpconvergence of a subdivision scheme in terms of the pnorm joint spectral radius of two matrices associated with the corresponding mask. We also discuss various properties of the limit function of a subdivision scheme, such as stability, linear independence, and smoothness.
Wavelet theory demystified
 IEEE Trans. Signal Process
, 2003
"... Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to red ..."
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Cited by 57 (26 self)
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Abstract—In this paper, we revisit wavelet theory starting from the representation of a scaling function as the convolution of a Bspline (the regular part of it) and a distribution (the irregular or residual part). This formulation leads to some new insights on wavelets and makes it possible to rederive the main results of the classical theory—including some new extensions for fractional orders—in a selfcontained, accessible fashion. In particular, we prove that the Bspline component is entirely responsible for five key wavelet properties: order of approximation, reproduction of polynomials, vanishing moments, multiscale differentiation property, and smoothness (regularity) of the basis functions. We also investigate the interaction of wavelets with differential operators giving explicit time domain formulas for the fractional derivatives of the basis functions. This allows us to specify a corresponding dual wavelet basis and helps us understand why the wavelet transform provides a stable characterization of the derivatives of a signal. Additional results include a new peeling theory of smoothness, leading to the extended notion of wavelet differentiability in thesense and a sharper theorem stating that smoothness implies order. Index Terms—Approximation order, Besov spaces, Hölder smoothness, multiscale differentiation, splines, vanishing moments, wavelets. I.
Wavelets on Irregular Point Sets
 Phil. Trans. R. Soc. Lond. A
, 1999
"... this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included. ..."
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Cited by 48 (0 self)
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this article we review techniques for building and analyzing wavelets on irregular point sets in one and two dimensions. We discuss current results both on the practical and theoretical side. In particular we focus on subdivision schemes and commutation rules. Several examples are included.
Characterization of smoothness of multivariate refinable functions in Sobolev spaces
 Trans. Amer. Math. Soc
, 1999
"... Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spect ..."
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Cited by 45 (4 self)
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Abstract. Wavelets are generated from refinable functions by using multiresolution analysis. In this paper we investigate the smoothness properties of multivariate refinable functions in Sobolev spaces. We characterize the optimal smoothness of a multivariate refinable function in terms of the spectral radius of the corresponding transition operator restricted to a suitable finite dimensional invariant subspace. Several examples are provided to illustrate the general theory. 1.
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.
Spectral analysis of the transition operator and its applications to smoothness analysis of wavelets
 SIAM J. Matrix. Anal. Appl
, 2001
"... The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compac ..."
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Cited by 43 (17 self)
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The purpose of this paper is to investigate spectral properties of the transition operator associated to a multivariate vector refinement equation and their applications to the study of smoothness of the corresponding refinable vector of functions. Let Φ = (φ1,..., φr) T be an r × 1 vector of compactly supported functions in L2(IR s) satisfying the refinement equation Φ = � α ∈ Zs a(α)Φ(M · − α), where M is an expansive integer matrix. We assume that M is isotropic, i.e., M is similar to a diagonal matrix diag(σ1,..., σs) with σ1  = · · · = σs. For µ = (µ1,..., µs) ∈ IN s 0, define. The smoothness of Φ is measured by the critical exponent σ −µ: = σ −µ1