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45
Multivariate Compactly Supported Fundamental Refinable Functions, Duals and Biorthogonal Wavelets
, 1997
"... In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function OE with sufficient regularity which is fundamental for interpolation (that means, OE(0) = 1 and OE(ff) = 0 for all ff 2 Z s nf0g). Low regularity examples of such functions have been ..."
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Cited by 34 (13 self)
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In areas of geometric modeling and wavelets, one often needs to construct a compactly supported refinable function OE with sufficient regularity which is fundamental for interpolation (that means, OE(0) = 1 and OE(ff) = 0 for all ff 2 Z s nf0g). Low regularity examples of such functions have been obtained numerically by several authors and a more general numerical scheme was given in [RiS1]. This paper presents several schemes to construct compactly supported fundamental refinable functions with higher regularity directly from a given continuous compactly supported refinable fundamental function OE. Asymptotic regularity analyses of the functions generated by the constructions are given. The constructions provide the basis for multivariate interpolatory subdivision algorithms that generate highly smooth surfaces. A very important consequence of the constructions is a natural formation of pairs of dual refinable functions, a necessary element in the construction of biorthogonal wav...
Mathematical properties of the JPEG2000 wavelet filters
 In
, 2003
"... to special prominence because they were selected for inclusion in the JPEG2000 standard. Here, we determine their key mathematical features: Riesz bounds, order of approximation, and regularity (Hölder and Sobolev). We give approximation theoretic quantities such as the asymptotic constant for the L ..."
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Cited by 32 (2 self)
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to special prominence because they were selected for inclusion in the JPEG2000 standard. Here, we determine their key mathematical features: Riesz bounds, order of approximation, and regularity (Hölder and Sobolev). We give approximation theoretic quantities such as the asymptotic constant for the L 2 error and the angle between the analysis and synthesis spaces which characterizes the loss of performance with respect to an orthogonal projection. We also derive new asymptotic error formulæ that exhibit bound constants that are proportional to the magnitude of the first nonvanishing moment of the wavelet. The Daubechies 9/7 stands out because it is very close to orthonormal, but this turns out to be slightly detrimental to its asymptotic performance when compared to other wavelets with four vanishing moments. I.
The Sobolev regularity of refinable functions
, 1997
"... Refinable functions underlie the theory and constructions of wavelet systems on the one hand, and the theory and convergence analysis of uniform subdivision algorithms. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system, and, in the su ..."
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Cited by 28 (6 self)
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Refinable functions underlie the theory and constructions of wavelet systems on the one hand, and the theory and convergence analysis of uniform subdivision algorithms. The regularity of such functions dictates, in the context of wavelets, the smoothness of the derived wavelet system, and, in the subdivision context, the smoothness of the limiting surface of the iterative process. Since the refinable function is, in many circumstances, not known analytically, the analysis of its regularity must be based on the explicitly known mask. We establish in this paper a formula that computes, for isotropic dilation and in any number of variables, the sharp L2regularity of the refinable function φ in terms of the spectral radius of the restriction of the associated transfer operator to a specific invariant subspace. For a compactly supported refinable function φ, the relevant invariant space is proved to be finite dimensional, and is completely characterized in terms of the dependence relations among the shifts of φ together with the polynomials that these shifts reproduce. The previously known formula for this compact support case requires the further assumptions that the mask is finitely supported, and that the shifts of φ are stable. Adopting a stability assumption (but without assuming the finiteness of the mask), we derive that known formula from our general one. Moreover, we show that in the absence of stability, the lower bound provided by that previously known formula may
Triangular √ 3subdivision schemes: the regular case
 J. Comput. Appl. Math
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Compactly supported wavelet bases for Sobolev spaces
, 2003
"... In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and ˜φ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk: = 2j/2ψ(2j ·− ..."
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Cited by 22 (8 self)
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In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and ˜φ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk: = 2j/2ψ(2j ·−k) (j,k ∈ Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space H m (R), then the derivatives 2j/2ψ (m) (2j ·−k) (j,k ∈ Z) also form a Riesz basis for L2(R). Consequently, {ψjk: j,k ∈ Z} is a stable wavelet basis for the Sobolev space H m (R). The pair of φ and ˜φ are not required to be biorthogonal or semiorthogonal. In particular, φ and ˜φ can be a pair of Bsplines. The added flexibility on φ and ˜φ allows us to construct wavelets with relatively small supports.
Smoothness of nonlinear medianinterpolation subdivision
 Adv. in Comput. Math
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Fully Discrete Wavelet Galerkin Schemes
, 2002
"... The present paper is intended to give a survey of the developments of the wavelet Galerkin boundary element method. Using appropriate wavelet bases for the discretization of boundary integral operators yields numerically sparse system matrices. These system matrices can be compressed to... ..."
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Cited by 14 (3 self)
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The present paper is intended to give a survey of the developments of the wavelet Galerkin boundary element method. Using appropriate wavelet bases for the discretization of boundary integral operators yields numerically sparse system matrices. These system matrices can be compressed to...
Convergence of the cascade algorithm at irregular scaling functions, in preparation
"... Abstract. The spectral properties of the Ruelle transfer operator which arises from a given polynomial wavelet filter are related to the convergence question for the cascade algorithm for approximation of the corresponding wavelet scaling function. ..."
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Cited by 14 (9 self)
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Abstract. The spectral properties of the Ruelle transfer operator which arises from a given polynomial wavelet filter are related to the convergence question for the cascade algorithm for approximation of the corresponding wavelet scaling function.
Computing the Sobolev Regularity of Refinable Functions by the Arnoldi Method
 SIAM J. Matrix Anal. Appl
, 2001
"... The recent paper [J. Approx. Theory, 106 (2000), pp. 185225] provides a complete characterization of the L 2 smoothness of a refinable function in terms of the spectrum of an associated operator. Based on this theory, we devise in this paper a numerically stable algorithm for calculating that smo ..."
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Cited by 9 (1 self)
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The recent paper [J. Approx. Theory, 106 (2000), pp. 185225] provides a complete characterization of the L 2 smoothness of a refinable function in terms of the spectrum of an associated operator. Based on this theory, we devise in this paper a numerically stable algorithm for calculating that smoothness parameter, employing the deflated Arnoldi method to this end. The algorithm is coded in Matlab, and details of the numerical implementation are discussed, together with some of the numerical experiments. The algorithm is designed to handle large masks, as well as masks of refinable functions with unstable shifts. This latter case is particularly important, in view of the recent developments in the area of wavelet frames. Key words. refinable functions, wavelets, smoothness, regularity, transition operators, transfer operators, Arnoldi's method AMS subject classifications. Primary, 42C15; Secondary, 39B99, 46E35 PII. S0895479899363010 1.