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Positivity of refinable functions defined by nonnegative finite masks

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2009年第2期27卷 133-156页

作者： Zhou, Xinlong Univ Duisburg Essen Dept Math D-47048 Duisburg Germany

Let (a(j) vertical bar j = 0, 1, ..., N) with a(0), a(N) not equal 0 be a given nonnegative mask. Assume that the subdivision scheme with this mask is convergent. Let the associated refinable function be phi. So the support of phi is contained in [0, N]. Melkman conjectured in 1997 that unless the scheme is interpolatory and N > 2 the refinable function phi is positive on (0, N). In the present paper we confirm this conjecture. A lower bound of phi on [2(-m), N - 2(-m)] is also given. (C) 2009 Elsevier Inc. All rights reserved.

关键词： cascade algorithm Connection of directed graph Dilation equation Positivity of refinable function Subdivision scheme

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cascade algorithm and multiresolution analysis on the Heisenberg group

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Progress in Natural Science:Materials International 2005年第7期15卷 602-608页

作者： LIU Heping, LIU Yu, PENG Lizhong and CHU Xiaoyong (LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, China) LMAM School of Mathematical Sciences Peking University Beijing 100871 China

In this paper we investigate the relationship between the convergence of cascade algorithm and orthogonal (or biorthogo-nal) multiresolution analysis on the Heisenberg group. It is proved that the (strong) convergence of cascade algorithm together with the perfect reconstruction condition induces an orthogonal multiresolution analysis and vice versa . Similar results are also proved for biorthogo-nal multiresolution analysis.

关键词： Heisenberg group cascade algorithm multiresolution analysis refinable functions.

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Pointwise convergence of bounded cascade sequences

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PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY 2007年第1期135卷 181-189页

作者： Chen, Di-Rong Han, Min Beijing Univ Aeronaut & Astronaut Dept Math Beijing 100083 Peoples R China Beijing Univ Aeronaut & Astronaut LMIB Beijing 100083 Peoples R China

The cascade algorithm plays an important role in computer graphics and wavelet analysis. For an initial function phi(0), a cascade sequence (phi(n))(n=0)(infinity) is constructed by the iteration phi(n) = C-a phi(n-1), n = 1, 2,..., where C-a is defined by C(a)g = Sigma(alpha is an element of Z) a(alpha)g(2 (.) -alpha), g is an element of L-p(R). In this paper, under a condition that the sequence (phi n)(n=0)(infinity) is bounded in L-infinity(R), we prove that the following three statements are equivalent: (i) (phi n)(n=0)(infinity) converges a.e. x is an element of R. (ii) For a.e. x is an element of R, there exist a positive constant c and a constant gamma is an element of (0, 1) such that vertical bar phi(n+1)(x) - phi(n)(x)vertical bar <= c gamma(n) for all n = 1, 2,.... (iii) For some p is an element of [1,8), (phi(n))(n=0)(infinity) converges in L-p(R). An example is presented to illustrate our result.

关键词： refinable function cascade algorithm subdivision scheme pointwise convergence refinable curve joint spectral radius

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Computational methods for integrals involving functions and Daubechies wavelets

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APPLIED MATHEMATICS AND COMPUTATION 2007年第2期189卷 1828-1840页

作者： Maleknejad, K. Yousefi, M. Nouri, K. Iran Univ Sci & Technol Sch Math Tehran 16844 Iran Islamic Azad Univ Tehran Iran Iran Univ Sci & Technol Sch Math Tehran 16844 Iran

When wavelets are used as basis functions in Galerkin approach to solve the integral equations, Integrals of the form integral(supp(theta j,k))f(x)theta(j,k)(x) dx occur. By a change of variable, these integrals can be translated into integrals involving only theta. In this paper, we find quadrature rule on the supp(theta) for the integrals of the form integral(supp(theta))f(x)theta dx, theta is an element of {phi,psi}. Wavelets in this article are those discovered by Daubechies [I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988) 909-996], where phi is the scaling function and psi is the wavelet function. (c) 2007 Elsevier Inc. All rights reserved.

关键词： gauss quadrature integral equation daubechies wavelets bezout technique cascade algorithm

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Multiscale edge detection of noisy images using wavelets

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Journal of Harbin Institute of Technology(New Series) 2007年第5期14卷 737-740页

作者： FAROOQ M College of Automation Engineering Nanjing University of Aeronautics & Astronautics

The classical edge detectors work fine with the high quality pictures, but often are not good enough for noisy images because they cannot distinguish edges of different significance. The paper presented a novel approach to multiscale edge detection for noisy images using wavelet transforms based on Lipschitz regularity coefficients and a cascade algorithm. The relationship between wavelet transform and Lipschitz regularity was established. The proposed wavelet based edge detection algorithm combined the coefficients of wavelet transforms along with a cascade algorithm which significantly improves the result. The comparison between the proposed method and the classical edge detectors was carried out. The algorithm was applied to various images and its performance was discussed. The results of edge detection of contaminated images using the proposed algorithm show that it works better than the classical edge detectors.

关键词： edge detection wavelet lipschitz regularity cascade algorithm

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Convergence of cascade algorithms by frequency approach

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2006年第1期314卷 335-344页

作者： Chen, DR Beijing Univ Aeronaut & Astronaut Dept Appl Math Beijing 100083 Peoples R China

Starting with an initial function phi(0), the cascade algorithm generates a sequence {Q(a)(n)phi(0)}(infinity)(n=1) by a cascade operator Q(a) defined by Qa f = Sigma(alpha is an element of Zd) a(alpha) f (M (.)-alpha). A function phi is refinable if it satisfies Q(a)phi = phi. The refinable functions play an important role in wavelet analysis and computer graphics. The cascade algorithm is the main approach to approximate the refinable functions and to study their properties. This note establishes a sufficient condition, in terms of Fourier transforms of the initial function phi(0) and the refinable function phi, for the convergence of cascade algorithm. Our results apply to the case where neither the initial function is compactly supported nor the refinement mask is finitely supported. As a byproduct, we prove that any compactly supported refinable function has a positive Sobolev regularity exponent provided it is in L-2. (c) 2005 Published by Elsevier Inc.

关键词： refinable function cascade algorithm stability transition operator

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A cascade optimization methodology for automatic parameter identification and shape/process optimization in metal forming simulation

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COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING 2006年第41-43期195卷 5472-5508页

作者： Ponthot, J-P. Kleinermann, J-P. Univ Liege LTAS MC&T B-4000 Liege Belgium

Computer simulations of metal forming processes using the finite element method (FEM) are, today, well established. This form of simulation uses an increasing number of sophisticated geometrical and material models, relying on a certain number of input data, which are not always readily available. The aim of inverse problems, which will be considered here, is to determine one or more of the input data relating to these forming process simulations, thereby leading to a desired result. In this paper, we will focus on two categories of such inverse problems. The first category consists of parameter identification inverse problems. These involve evaluating the material parameters for material constitutive models that would lead to the most accurate results with respect to physical experiments, i.e. minimizing the difference between experimental results and FEM simulations. The second category consists of shape/process optimization inverse problems. These involve determining the initial geometry of the specimen and/or the shape of the forming tools, as well as some parameters of the process itself, in order to provide the desired final geometry after the forming process. These two categories of inverse problems can be formulated as optimization problems in a similar way, i.e. by using identical optimization algorithms. In this paper, we intend firstly to solve these two types of optimization problems by using different non-linear gradient based optimization methods and secondly to compare their efficiency and robustness in a variety of numerical applications. (c) 2005 Elsevier B.V. All rights reserved.

关键词： finite element method (FEM) large strains inverse problems parameter identification forming processes optimization methods cascade algorithm

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Solutions in Sobolev spaces of vector refinement equations with a general dilation matrix

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2006年第1-4期24卷 375-403页

作者： Han, B Univ Alberta Dept Math & Stat Sci Edmonton AB T6G 2G1 Canada

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W-p(k)(R-s) ( 1 0. This paper generalizes the results in R. Q. Jia, K. S. Lau and D. X. Zhou (J. Fourie... 详细信息

In this paper, we present a necessary and sufficient condition for the existence of solutions in a Sobolev space W-p(k)(R-s) ( 1 <= p <= infinity) to a vector refinement equation with a general dilation matrix. The criterion is constructive and can be implemented. Rate of convergence of vector cascade algorithms in a Sobolev space W-p(k)(R-s) will be investigated. When the dilation matrix is isotropic, a characterization will be given for the L-p (1 <= p <= infinity) critical smoothness exponent of a refinable function vector without the assumption of stability on the refinable function vector. As a consequence, we show that if a compactly supported function vector phi is an element of L-p(R-s) (phi is an element of C(R-s) when p = infinity) satisfies a refinement equation with a finitely supported matrix mask, then all the components of phi must belong to a Lipschitz space Lip(nu, L-p(R-s)) for some nu > 0. This paper generalizes the results in R. Q. Jia, K. S. Lau and D. X. Zhou (J. Fourier Anal. Appl. 7 ( 2001) 143 - 167) in the univariate setting to the multivariate setting.

关键词： vector refinement equation refinable function vector Sobolev space rate of convergence smoothness cascade algorithm

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On refinable functions and subdivision with positive masks

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ADVANCES IN COMPUTATIONAL MATHEMATICS 2006年第1-4期24卷 281-295页

作者： De Villiers, J Univ Stellenbosch Dept Math ZA-7602 Matieland South Africa

We consider aspects of the analysis of refinement equations with positive mask coefficients. First we derive, explicitly in terms of the mask, estimates for the geometric convergence rate of both the cascade algorithm and the corresponding subdivision scheme, as well as the Holder continuity exponent of the resulting refinable function. Moreover, we show that the subdivision scheme converges for a class of unbounded initial sequences. Finally, we present a regularity result containing sufficient conditions on the mask for the refinable function to possess continuous derivatives up to a given order.

关键词： refinable functions subdivision wavelets cascade algorithm

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The collocation method for Hammerstein equations by Daubechies wavelets

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APPLIED MATHEMATICS AND COMPUTATION 2006年第2期172卷 846-864页

作者： Maleknejad, K Derili, H Iran Univ Sci & Technol Sch Math Tehran 16844 Iran Islamic Azad Univ Fac Sci Karaj Unit Dept Math Karaj *** Iran

The numerical solutions to the nonlinear integral equations of Hammerstein-type y(t) = f(t)+ integral(1)(0) k(t, s)g(s, y(s))ds, t is an element of [0, 1] with using Daubechies wavelets are investigated. A general kernel scheme basing on Daubechies wavelets combined with a collocation method is presented. The approach of creating Daubechies interval wavelets and their main properties are briefly mentioned. Also we present an algorithm for computing of Daubechies wavelets in collocation points. The rate of approximation solution converging to the exact solution is given. Finally we also give some numerical examples for showing efficiency of the method. (c) 2005 Published by Elsevier Inc.

关键词： cascade algorithm Daubechies wavelets Bezout polynomials collocation method nonlinear integral equations

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