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Neural Network Approximation of Refinable Functions

IEEE TRANSACTIONS ON INFORMATION THEORY 2023年第1期69卷 482-495页

作者： Daubechies, Ingrid De Vore, Ronald Dym, Nadav Faigenbaum-Golovin, Shira Kovalsky, Shahar Z. Lin, Kung-Chin Park, Josiah Petrova, Guergana Sober, Barak Duke Univ Dept Math Durham NC 27708 USA Texas A&M Univ Dept Math College Stn TX 77843 USA Technion Israel Inst Technol Fac Math IL-3200003 Haifa Israel Duke Univ Rhodes Informat Initiat Durham NC 27708 USA Univ North Carolina Chapel Hill Dept Math Chapel Hill NC 27599 USA Hebrew Univ Jerusalem Dept Stat & Data Sci & Digital Humanities IL-9190501 Jerusalem Israel

In the desire to quantify the success of neural networks in deep learning and other applications, there is a great interest in understanding which functions are efficiently approximated by the outputs of neural networks. By now, there exists a variety of results which show that a wide range of functions can be approximated with sometimes surprising accuracy by these outputs. For example, it is known that the set of functions that can be approximated with exponential accuracy (in terms of the number of parameters used) includes, on one hand, very smooth functions such as polynomials and analytic functions and, on the other hand, very rough functions such as the Weierstrass function, which is nowhere differentiable. In this paper, we add to the latter class of rough functions by showing that it also includes refinable functions. Namely, we show that refinable functions are approximated by the outputs of deep ReLU neural networks with a fixed width and increasing depth with accuracy exponential in terms of their number of parameters. Our results apply to functions used in the standard construction of wavelets as well as to functions constructed via subdivision algorithms in Computer Aided Geometric Design.

关键词： Artificial neural networks Standards Deep learning Approximation algorithms Upper bound Stress Terminology Neural networks neural network approximation refinable functions exponential accuracy cascade algorithm

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Human colour skin detection in CMYK colour space

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IET IMAGE PROCESSING 2015年第9期9卷 751-757页

作者： Sawicki, Dariusz J. Miziolek, Weronika Warsaw Univ Technol Inst Theory Elect Engn Measurement & Informat Sys Warsaw Poland

Fast detection of skin colour in images is an important task for computer vision and image processing. In this study, a new thresholding method for detecting skin colour has been introduced. The proposed method uses CMYK colour space, which, although not popular in image processing, especially in skin detecting algorithm, turned out to be a good choice. Its properties seem the most appropriate for the task of skin colour detection. Some experiments have been worked out using Compaq Database - a large database of skin and non-skin photos. A comparison of the new method against well-known thresholding methods in RGB, YCbCr and HSV spaces were prepared. In this comparison, the proposed method in CMYK colour space wins. This good result was further improved using a simple cascade algorithm, where two spaces RGB and CMYK were used.

关键词： image colour analysis object detection skin computer vision image segmentation visual databases human colour skin detection CMYK colour space computer vision image processing thresholding method Compaq Database RGB space YCbCr space HSV space cascade algorithm

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M-channel lifting factorization of perfect reconstruction filter banks and reversible M-band wavelet transforms

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IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II-EXPRESS BRIEFS 2003年第12期50卷 963-976页

作者： Chen, YJ Amaratunga, KS MIT Dept Civil & Environm Engn Intelligent Engn Syst Lab Wavelet Grp Cambridge MA 02139 USA

An intrinsic M-channel lifting factorization of perfect reconstruction filter banks (PRFBs) is presented as an extension of Sweldens' conventional two-channel lifting scheme. Given a polyphase matrix E(z) of a finite-impulse response (FIR) M-channel PRFB with det(E(z))=z(-K), K is an element of Z, a systematic M-channel lifting factorization is derived based on the Monic Euclidean algorithm. The M-channel lifting structure provides an efficient factorization and implementation;examples include optimizing the factorization for the number of lifting steps, delay elements, and dyadic coefficients. Specialization to paraunitary building blocks enables the design of paraunitary filter banks based on lifting. We show how to achieve reversible, possibly multiplierless, implementations under finite precision, through the unit diagonal scaling property of the Monic Euclidean algorithm. Furthermore, filter-bank regularity of a desired order can be imposed on the lifting structure, and PRFBs with a prescribed admissible scaling filter are conveniently parameterized.

关键词： cascade algorithm discrete cosine transform (DCT) greatest common divisor laurent polynomials lifting factorization M-band wavelets Monic Euclidean algorithm multiplierless approximations perfect reconstruction filter banks (PRFBs) regularity reversible transforms

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Inhomogeneous refinement equations

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 1998年第6期4卷 733-747页

作者： Strang, G Zhou, DX MIT Dept Math Cambridge MA 02139 USA City Univ Hong Kong Dept Math Kowloon Hong Kong

Equations with two time scales (refinement equations or dilation equations) are central to wavelet theory. Several applications also include an inhomogeneous forcing term F(t). We develop here a parr of the existence theory for the inhomogeneous refinement equation [GRAPHICS] where a(k) is a finite sequence and F is a compactly supported distribution on R. The existence of compactly supported distributional solutions to an inhomogeneous refinement equation is characterized in terms of conditions on the pair (a, F). To have L-p solutions from F is an element of L-p(R), we construct by the cascade algorithm a sequence of functions {phi(n)} from a compactly supported initial function phi(o) is an element of L-p (R) as [GRAPHICS] A necessary and sufficient condition for the sequence {phi(n)} to converge in L-p(R)(1 less than or equal to p less than or equal to infinity) is given by the p-norm joint spectral radius of two matrices derived from the mask a. A convexity property of the p-norm joint spectral radius (1 less than or equal to p less than or equal to infinity) is presented. Finally, the general theory is applied to some examples and multiple refinable functions.

关键词： inhomogeneous refinement equation wavelets joint spectral radius cascade algorithm multiple refinable function

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Some new discrete biorthogonal wavelets constructed with Laguerre polynomials

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SN APPLIED SCIENCES 2020年第9期2卷 1470页

作者： Peachap, Atemangoh Bruno Tchiotsop, Daniel Louis-Dorr, Valerie Wolf, Didier Univ Dschang Unite Rech Mat Condensee Elect & Traitement Signa Fac Sci POB 69 Dschang Cameroon Univ Dschang Unite Rech Automat & Informat Appl LAIA IUT FV Bandjoun POB 134 Bandjoun Cameroon Univ Lorraine Ctr Rech Automat Nancy CRAN UMR CNRS 7039 ENSEM Nancy France

In this study, we present a new family of discrete wavelets which are constructed with the help of Laguerre polynomials and the Daubechies biorthogonal wavelets construction method. Our aim is to propose the discrete version of some previously constructed continuous Laguerre wavelets and also to present a method of discrete wavelets construction by several iterations. With this scheme, we use two different sets of finite impulse response filters for the signal decomposition and their duals for reconstruction. The quadruplet finite impulse response filters respect the a nti-aliasing and the perfect reconstruction conditions, and at the same time, they resemble as much as possible the continuous Laguerre wavelets when using the cascade algorithm. We use the mean squared error, the maximum deviation, and the standard deviation to quantify the similarity between the continuous Laguerre wavelets and the constructed discrete Laguerre *** results show that, they are both the same wavelets due to the small nature of these parameters. Our method is important because, it can permit the determination of the finite impulse response filter coefficients corresponding to many other continuous wavelets.

关键词： Laguerre wavelets Biorthogonal wavelets cascade algorithm FIR filters

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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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Subdivision schemes with nonnegative masks

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MATHEMATICS OF COMPUTATION 2005年第250期74卷 819-839页

作者： Zhou, XL China Jiliang Univ Dept Math Hangzhou Peoples R China Univ Duisburg Essen Inst Math D-47057 Duisburg Germany

The conjecture concerning the characterization of a convergent univariate subdivision algorithm with nonnegative finite mask is confirmed.

关键词： cascade algorithm joint spectral radius nonnegative mask subdivision scheme

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Generalized Pseudo-Butterworth Refinable Functions

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JOURNAL OF FOURIER ANALYSIS AND APPLICATIONS 2013年第2期19卷 296-311页

作者： Luo, Zhongxuan Qi, Wanfeng Zhou, Mengmeng Zhang, Jielin Dalian Univ Technol Sch Math Sci Dalian Liaoning Peoples R China

In this paper we propose the generalized pseudo-Butterworth refinable functions which involve pseudo-splines of type I and II, Butterworth refinable functions, pseudo-Butterworth refinable functions, and almost all symmetric and causal fractional B-splines. Furthermore, the convergence of cascade algorithms associated with the new masks is proved, and Riesz wavelet bases in L (2)(a"e) corresponding to the parameters are constructed. The regularity of the generalized pseudo-Butterworth refinable functions is also analyzed by Fourier analysis.

关键词： Pseudo-Butterworth masks cascade algorithm Riesz wavelet basis Sobolev exponent

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A complete solution characterizing smooth refinable functions

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SIAM JOURNAL ON MATHEMATICAL ANALYSIS 2000年第6期31卷 1332-1350页

作者： Protasov, V Erasmus Univ Dept Econ NL-3062 PA Rotterdam Netherlands Inst Adv Study Princeton NJ 08540 USA

In this paper we study univariate two-scale refinement equations phi(x) = Sigma(k is an element of Z)c(k)phi (2x - k), where the coefficients c(k) is an element of C satisfy an exponential decay assumption. We show that any refinement equation that has a smooth solution can be reduced to the well-studied case of complete sum rules: Sigma(k)(-1)(k)k(n)c(k) = 0, n = 0,..., L, where L depends on regularity of the solution. This result makes it possible to extend previously known results on refinable functions and subdivision schemes from the case of complete sum rules to the general case. As a corollary we obtain sharp necessary conditions for the existence of smooth refinable functions and the convergence of corresponding cascade algorithms. Other applications concern polynomial spaces spanned by integer translates of a refinable function and one special property of linear operators associated to refinement equations.

关键词： refinement equations regularity reducibility tree cascade algorithm

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