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Norms concerning subdivision sequences and their applications in wavelets

APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 2001年第3期11卷 329-346页

作者： Zhou, DX City Univ Hong Kong Dept Math Kowloon Hong Kong Peoples R China

Let a := (a(alpha))(alpha is an element ofZ)s be a finitely supported sequence of r x r matrices and M be a dilation matrix. The subdivision sequence {(a(n)(alpha))(alpha is an element ofZ)s : n is an element of N} is defined by a(1) = a and [GRAPHICS] Let 1 less than or equal to p less than or equal to infinity and f = (f(l),...,f(r))(T) be a vector of compactly supported functions in L-p (RI). The stability is not assumed for f. The purpose of this paper is to give a formula for the asymptotic behavior of the L-p-norms of the combinations of the shifts of f with the subdivision sequence coefficients: [GRAPHICS] Such an asymptotic behavior plays an essential role in the investigation of wavelets and subdivision schemes. In this paper we show some applications in the convergence of cascade algorithms, construction of inhomogeneous multiresolution analyzes, and smoothness analysis of refinable functions. Some examples are provided to illustrate the method. (C) 2001 Academic Press.

关键词： subdivision sequence joint spectral radius cascade algorithm inhomogeneous refinement equation wavelets smoothness

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Spline interpolation and wavelet construction

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APPLIED AND COMPUTATIONAL HARMONIC ANALYSIS 1998年第3期5卷 249-276页

作者： Lee, SL Sharma, A Tan, HH Natl Univ Singapore Dept Math Singapore 119260 Singapore Univ Alberta Dept Math Edmonton AB T6G 2G1 Canada

The method of Dubuc and Deslauriers on symmetric interpolatory subdivision is extended to study the relationship between interpolation processes and wavelet construction. Refinable and interpolatory functions are constructed in stages from B-splines. Their method constructs the filter sequence (its Laurent polynomial) of the interpolatory function as a product of Laurent polynomials. This provides a natural way of splitting the filter for the construction of orthonormal and biorthogonal scaling functions leading to orthonormal and biorthogonal wavelets. Their method also leads to a class of filters which includes the minimal length Daubechies compactly supported orthonormal wavelet coefficients. Examples of ''good'' filters are given together with results of numerical experiments conducted to test the performance of these filters in data compression. (C) 1998 Academic Press.

关键词： refinable function wavelets interpolatory function uniform B-spline Euler-Frobenius polynomial Riesz basis biorthogonal basis orthonormal basis transition operator subdivision algorithm cascade algorithm condition E

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Direct algorithm for computation of derivatives of the Daubechies basis functions

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APPLIED MATHEMATICS AND COMPUTATION 2005年第2期170卷 1006-1013页

作者： Lin, WB Kovvali, N Carin, L Duke Univ Dept Elect & Comp Engn Durham NC 27708 USA

We extend the direct algorithm for computing the derivatives of the compactly supported Daubechies N-vanishing-moment basis functions. The method yields exact values at dyadic rationals for the nth derivative (0 <= n <= N - 1) of the basis functions, when it exists. Example results are shown for the first derivatives of the basis functions from the Daubechies N-vanishing-moment extremal phase orthonormal family (for N = 3,4, and 5), and the CDF(2, N) spline-based biorthogonal family (for N = 6,8 and 10). (c) 2005 Elsevier Inc. All rights reserved.

关键词： wavelets derivatives Daubechies basis functions cascade algorithm direct algorithm

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Asymptotic expansions for linear homogeneous divide-and-conquer recurrences: Algebraic and analytic approaches collated

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THEORETICAL COMPUTER SCIENCE 2014年第C期548卷 25-53页

作者： Dumas, Philippe Inria Saclay SpecFun Team Palaiseau France

Among all sequences that satisfy a divide-and-conquer recurrence, those which are rational with respect to a numeration system are certainly the most basic and the most essential. Nevertheless, until recently this specific class of sequences has not been systematically studied from the asymptotic standpoint. We recall how a mechanical process designed by the author permits to compute their asymptotic expansions. The process is based on linear algebra, and involves computing Jordan normal forms, joint spectral radii, and solving dilation equations. The main contribution of the present article is the comparison between our algebraic method and the classical analytic number theory approach. Moreover, we develop new ways to compute the Fourier series of the periodic functions involved in the expansion. The article comes with an extended bibliography. (C) 2014 Elsevier B.V. All rights reserved.

关键词： Divide-and-conquer recurrence Radix-rational sequence Spectral radius Dilation equation cascade algorithm Dirichlet series Fourier series

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Positivity of refinable functions defined by nonnegative finite masks

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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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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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Refinable function vectors

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 1998年第1期29卷 235-250页

作者： Shen, ZW Natl Univ Singapore Dept Math Singapore 119260 Singapore

Refinable function vectors are usually given in the form of an infinite product of their refinement (matrix) masks in the frequency domain and approximated by a cascade algorithm in both time and frequency domains. We provide necessary and sufficient conditions for the convergence of the cascade algorithm. We also give necessary and sufficient conditions for the stability and orthonormality of refinable function vectors in terms of their refinement matrix masks. Regularity of function vectors gives smoothness orders in the time domain and decay rates at infinity in the frequency domain. Regularity criteria are established in terms of the vanishing moment order of the matrix mask.

关键词： refinable function vectors stable basis cascade algorithm regularity

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Design of Time-Frequency Optimal Three-Band Wavelet Filter Banks with Unit Sobolev Regularity Using Frequency Domain Sampling

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CIRCUITS SYSTEMS AND SIGNAL PROCESSING 2016年第12期35卷 4501-4531页

作者： Bhati, Dinesh Sharma, Manish Pachori, Ram Bilas Nair, Sujath S. Gadre, Vikram M. Indian Inst Technol Dept Elect Engn Bombay Maharashtra India Indian Inst Technol Indore Discipline Elect Engn Indore Madhya Pradesh India

In this paper, we design three-band time-frequency-localized orthogonal wavelet filter banks having single vanishing moment. We propose new expressions to compute mean and variances in time and frequency from the samples of the Fourier transform of the asymmetric band-pass compactly supported wavelet functions. We determine discrete-time filter of length eight that generates the time-frequency optimal time-limited scaling and wavelet functions using cascade algorithm. Time-frequency product (TFP) of a function is defined as the product of its time variance and frequency variance. The TFP of the designed functions is close to 0.25 with unit Sobolev regularity. Three-band filter banks are designed by minimizing a weighted combination of TFPs of wavelets and scaling functions. Interestingly, empirical results show that time-frequency optimal, filter banks of length nine, designed with the proposed methodology, have unit Sobolev regularity, which is maximum achievable with single vanishing moment. Design examples for length six and length nine filter banks are given to demonstrate the effectiveness of the proposed design methodology.

关键词： Dyadic factorization Sobolev regularity Time-frequency localization Vanishing moment cascade algorithm Frequency domain sampling

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Some Sufficient Conditions for Convergent Multivariate Subdivision Schemes with Nonnegative Finite Masks

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Acta Mathematica Sinica,English Series 2012年第7期28卷 1411-1420页

作者： Min WU Jia Li ZHOU Department of Mathematics Zhejiang University of Science and Technology Department of Mathematics Zhejiang University of Technology

It is well known that the convergence of multivariate subdivision schemes with finite masks can be characterized via joint spectral radius. For nonnegative masks, we will present in this paper some computable simply s... 详细信息

关键词： cascade algorithm dilation equation joint spectral radius mask subdivision scheme

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The spline filter: A regularization approach for the Gaussian filter

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PRECISION ENGINEERING-JOURNAL OF THE INTERNATIONAL SOCIETIES FOR PRECISION ENGINEERING AND NANOTECHNOLOGY 2012年第4期36卷 586-592页

作者： Zhang, Hao Yuan, Yibao Piao, Weiying Nanjing Forestry Univ Coll Mech & Elect Engn Nanjing Jiangsu Peoples R China Harbin Inst Technol Sch Elect Engn & Automat Harbin 150006 Peoples R China

The Gaussian filter described in the ISO 11562 standard has become the most widely used filtering technique in surface metrology. However, this filter is always plagued by the large distortions called end effects at the boundaries of the filtered result. In order to alleviate the end effects, the spline filter based on natural cubic splines is incorporated into ISO standard as a substitute. So there exist two kinds of linear profile filters with different transmission characteristics which also lead to different mean lines for the assessment of the same surface. A new spline algorithm for determining the Gaussian filtered mean line is deduced using the central limit theorem. The filter uses the cascade method of the approximating spline filter, and therefore can approximate the transmission characteristic of the Gaussian filter with high accuracy. It is proved that the transmission characteristic relative deviation of the cascade approximating spline filter from the Gaussian filter is only 0.3% when the cascade order approaches infinity. With this theorem, it is easy to achieve the unification of the international standard ISO 16610-21 and ISO 16610-22. (C) 2012 Elsevier Inc. All rights reserved.

关键词： Approximating spline filter cascade algorithm Gaussian filter Surface metrology

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