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ARCHITECTURE FOR HIGH-ORDER MULTIDIMENSIONAL CONVOLUTION USING polynomial-transforms

ELECTRONICS LETTERS 1990年第12期26卷 801-802页

作者： ARAMBEPOLA, B Department of Electrical Engineering Imperial College of Science Technology and Medicine London UK

An efficient hardware architecture is presented for computing convolutions and correlations with two or more dimensions. This is derived from combining a class of polynomial transforms with currently available VLSI convolution devices. The proposed method is particularly suitable for computing high order convolutions with little or no arithmetic quantisation errors.

关键词： transforms Computer architecture VLSI convolution devices correlations signal processor butterfly processor Integral transforms correlators parallel architectures polynomials hardware architecture polynomial transforms Digital signal processing computerised signal processing 2D data array high-order multidimensional convolution

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Parametric image coding by means of polynomial transforms

Parametric image coding by means of polynomial transforms

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1997 Conference on Visual Communications and Image Processing

作者： ViverosCancino, O EscalanteRamirez, B Univ. Nacional Autonoma de Mexico (Mexico)

ISBN: (纸本)0819424358

In this paper we propose an image coding scheme based on the polynomial transform and multiresolution analysis. The polynomial transform is an image representation model that mimics some properties of the human visual system, and which we use in order to model edges in terms of their characteristic parameters. Based on the polynomial transform, we build a pyramidal hierarchical predictive scheme for image coding. The feature parameters that we encode are: local average, edge orientation, edge position and edge magnitude.

关键词： polynomial transforms image coding parametric representations edge-based modeling multiresolution

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Edge-orientation-based noise-reduction with polynomial transforms 19

Edge-orientation-based noise-reduction with polynomial trans...

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Conference on Applications of Digital Image Processing XIX

作者： EscalanteRamirez, B LopezMiranda, JR National Univ. of Mexico (Mexico)

ISBN: (纸本)0819422355

We present a technique for directional-sensitive image restoration based on the polynomial transform. The polynomial transform is an image description model which incorporates important properties of visual perception, such as the Gaussian-derivative model of early vision. The polynomial transform basically consists of a local description of an image. Localization is achieved by multiplying the image with overlapping window functions. In the case of the discrete polynomial transform, the contents of the image within every position of the analysis window is represented by a finite set of coefficients. These coefficients correspond to the weights in a polynomial expansion that reconstructs the image within the window function. It has been showed how the polynomial transform can be used to design efficient noise-reduction algorithms by adaptively transforming the coefficients of every window according to the image contents(1). Other types of transformations on the polynomial coefficents lead to different image-restoration applications, such as deblurring, coding, and interpolation(2). In all cases, the restored image is obtained by means of an inverse polynomial transform which consists of interpolating the transformed coefficients with pattern functions that are products of a polynomial and a window function. We show in this paper how image restoration, namely noise reduction and deblurring, based on the polynomial transform can be improved by detecting the position and orientation of relevant edges in the image.

关键词： image restoration noise reduction polynomial transforms directional filtering

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Arithmetic Circuits, Structured Matrices and (not so) Deep Learning

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THEORY OF COMPUTING SYSTEMS 2023年第3期67卷 592-626页

作者： Rudra, Atri Univ Buffalo Dept Comp Sci & Engn New York NY 14260 USA

This survey presents a necessarily incomplete (and biased) overview of results at the intersection of arithmetic circuit complexity, structured matrices and deep learning. Recently there has been some research activity in replacing unstructured weight matrices in neural networks by structured ones (with the aim of reducing the size of the corresponding deep learning models). Most of this work has been experimental and in this survey, we formalize the research question and show how a recent work that combines arithmetic circuit complexity, structured matrices and deep learning essentially answers this question. This survey is targeted at complexity theorists who might enjoy reading about how tools developed in arithmetic circuit complexity helped design (to the best of our knowledge) a new family of structured matrices, which in turn seem well-suited for applications in deep learning. However, we hope that folks primarily interested in deep learning would also appreciate the connections to complexity theory.

关键词： Linear arithmetic circuit complexity Neural networks polynomial transforms

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COMPUTING WITH EXPANSIONS IN GEGENBAUER polynomialS

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SIAM JOURNAL ON SCIENTIFIC COMPUTING 2009年第3期31卷 2151-2171页

作者： Keiner, Jens Univ New S Wales Sch Math & Stat Sydney NSW 2052 Australia

We develop fast algorithms for computations involving finite expansions in Gegenbauer polynomials. A method is described to convert any finite expansion between different families of Gegenbauer polynomials. For a degree-n expansion the computational cost is O(n(log(1/epsilon)+vertical bar alpha-beta vertical bar)), where e is the prescribed accuracy, and alpha and beta are the respective Gegenbauer indices. Special cases involving Chebyshev polynomials of first kind are particularly important. In combination with (nonequispaced) discrete cosine transforms, we obtain efficient methods for the evaluation of expansions at prescribed nodes, and for the projection onto a sequence of Gegenbauer polynomials from given function values, respectively.

关键词： Gegenbauer polynomials polynomial transforms fast algorithms

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Minimum multiplicative complexity implementation of the 2-D DCT using Xilinx FPGAs

Minimum multiplicative complexity implementation of the 2-D ...

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Conference on Configurable Computing - Technology and Applications

作者： Dick, C Xilinx Inc San Jose CA 95124 USA

ISBN: (纸本)0819429872

This paper investigates two options for the field programmable gate array (FPGA) implementation of a very high-performance 2-D discrete cosine transform (DCT) processor for real-time applications. The first architecture exploits the transform separability and uses a row-column decomposition. The row and column processors are realized using distributed arithmetic (DA) techniques. The second approach uses a naturally 2-D method based on polynomial transforms. The paper provides an overview of the DCT calculation using DA methods and describes the FPGA implementation. A tutorial overview of a computationally efficient method for computing 2-D DCTs using polynomial transforms is presented. A detailed analysis of the datapath for this approach using an 8 x 8 data-set is given. Comparisons are made that show the polynomial transform approach to require 67% of the logic resources of a DA processor for equal throughputs. The polynomial transform approach is also shown to scale better with increasing block size than the DA approach.

关键词： 2-D DCT FPGA polynomial transforms real-time video

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GEGENBAUER polynomialS AND SEMISEPARABLE MATRICES

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ELECTRONIC TRANSACTIONS ON NUMERICAL ANALYSIS 2008年 30卷 26-53页

作者： Keiner, Jens Med Univ Lubeck Inst Math D-23560 Lubeck Germany

In this paper, we develop a new O(n log n) algorithm for converting coefficients between expansions in different families of Gegenbauer polynomials up to a finite degree n. To this end, we show that the corresponding linear mapping is represented by the eigenvector matrix of an explicitly known diagonal plus upper triangular semiseparable matrix. The method is based on a new efficient algorithm for computing the eigendecomposition of such a matrix. Using fast summation techniques, the eigenvectors of an nxn matrix can be computed explicitly with O(n(2)) arithmetic operations and the eigenvector matrix can be applied to an arbitrary vector at cost O(n logn). All algorithms are accurate up to a prefixed accuracy epsilon. We provide brief numerical results.

关键词： Gegenbauer polynomials polynomial transforms semiseparable matrices eigendecomposition spectral divide-and-conquer methods

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POLES AND ZEROS OF MULTIVARIABLE LINEAR TIME-VARYING SYSTEMS

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IFAC Proceedings Volumes 2002年第1期35卷 261-266页

作者： Kai Zenger Raimo Ylinen Helsinki University of Technology Control Engineering Laboratory P.O.Box 5400 FIN-02015 HUT Finland. Phone: +358 9 4515204 Fax: +358 9 4515208 University of Oulu Systems Engineering Lab oratory Finland

In the paper it is shown that the pole set of a linear system can be calculated by suitable polynomial factorizations. The theoretical issues related to poles and zeros of time-varying systems are discussed. Further, it is shown how the poles and zeros can be defined starting from a state-space realization of the input-output system.

关键词： Time-varying systems Linear systems Poles polynomial transforms

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FORMAL LINEARIZATION OF NONLINEAR TIME-VARYING DYNAMIC SYSTEMS USING CHEBYSHEV AND LAGUERRE polynomialS

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IFAC Proceedings Volumes 2005年第1期38卷 982-987页

作者： Hitoshi Takata Kazuo Komatsu Hideki Sano Kagoshima University Kumamoto National College of Technology

This paper is concerned with a formal linearization problem for a general class of nonlinear time-varying dynamic systems. To a given system, a linearization function is made up of Chebyshev polynomials about its state variables. The nonlinear time-varying system is transformed into a linear time-varying system in terms of the linearization function using Chebyshev interpolation to state variables and Laguerre expansion to time variable. An error bound formula of this linearization which is derived in this paper explains that the accuracy of this algorithm is improved as the order of Chebyshev and Laguerre polynomials increases. As its application, a nonlinear observer is designed to demonstrate the usefulness of this formal linearization approach.

关键词： Nonlinear system Linearization Time-varying system polynomial transforms Observers

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NOTIONS OF EQUIVALENCE FOR DISCRETE TIME AR-REPRESENTATIONS

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IFAC Proceedings Volumes 2002年第1期35卷 29-34页

作者： N. Karampetakis S. Vologiannidis A.I. Vardulakis Department of Mathematics Aristotle University of Thessaloniki Thessaloniki 54006 Greece

A transformation of polynomial matrices which preserves both the finite and infinite elementary divisor structure is presented and related to other known transformations.

关键词： polynomial transforms Autoregressive models discrete time systems

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