Fast Algorithms For The Digital Computation Of Linear Canonical Transforms

Author : Aykut Koc
Publisher : Stanford University
Page : 173 pages
File Size : 50,9 Mb
Release : 2011
Category : Electronic
ISBN : STANFORD:fq782pt6225

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Fast Algorithms for the Digital Computation of Linear Canonical Transforms by Aykut Koc Pdf

Although it is straightforward to determine the relationship between the in-focus image and the object of a simple optical system such as a lens, it is far more challenging to compute the input/output relationships of general first-order and astigmatic optical systems. Such optical systems are known as quadratic-phase systems (QPS) and they include the Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by representing the physical system with a general mathematical framework of integrations against kernels and then distilling the entire system into one input-output relationship that can be represented by a linear integral transform. The underlying mathematical integral transforms can be applied to a wider field of signal processing where they are known as the linear canonical transform (LCT) of a signal. Conventional numerical integration methods have a computational complexity of O(N^2) where N is the space-bandwidth product of the sampling scheme, e.g. the number of pixels in the field for an optical system. The algorithms described here yield a complexity of only O(Nlog N). The key is the use of different decompositions (or factorizations) of a given input/output relationship into simpler ones. Instead of following the general physical subparts in cascaded systems and computing input-output relations separately, these algorithms use the simplest possible decompositions to represent the entire system in terms of least possible number of steps. The algorithms are Fast Fourier Transform (FFT) based methods and the only essential deviation from exactness arises from approximating a continuous Fourier transform (FT) with the discrete Fourier transform (DFT). Thus the algorithms work with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Unlike conventional techniques these algorithms also track and control the space-bandwidth products, in order to achieve information that is theoretically sufficient but not wastefully redundant.

Author : Aykut Koc
Publisher : Unknown
Page : 128 pages
File Size : 50,5 Mb
Release : 2011
Category : Electronic
ISBN : OCLC:712637678

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Fast Algorithms for the Digital Computation of Linear Canonical Transforms by Aykut Koc Pdf

Although it is straightforward to determine the relationship between the in-focus image and the object of a simple optical system such as a lens, it is far more challenging to compute the input/output relationships of general first-order and astigmatic optical systems. Such optical systems are known as quadratic-phase systems (QPS) and they include the Fresnel propagation in free space, propagation in graded-index media, passage through thin lenses, and arbitrary concatenations of any number of these, including anamorphic, astigmatic, nonorthogonal elements. Such computation is accomplished by representing the physical system with a general mathematical framework of integrations against kernels and then distilling the entire system into one input-output relationship that can be represented by a linear integral transform. The underlying mathematical integral transforms can be applied to a wider field of signal processing where they are known as the linear canonical transform (LCT) of a signal. Conventional numerical integration methods have a computational complexity of O(N^2) where N is the space-bandwidth product of the sampling scheme, e.g. the number of pixels in the field for an optical system. The algorithms described here yield a complexity of only O(Nlog N). The key is the use of different decompositions (or factorizations) of a given input/output relationship into simpler ones. Instead of following the general physical subparts in cascaded systems and computing input-output relations separately, these algorithms use the simplest possible decompositions to represent the entire system in terms of least possible number of steps. The algorithms are Fast Fourier Transform (FFT) based methods and the only essential deviation from exactness arises from approximating a continuous Fourier transform (FT) with the discrete Fourier transform (DFT). Thus the algorithms work with a performance similar to that of the fast Fourier transform algorithm in computing the Fourier transform, both in terms of speed and accuracy. Unlike conventional techniques these algorithms also track and control the space-bandwidth products, in order to achieve information that is theoretically sufficient but not wastefully redundant.

Author : John J. Healy,M. Alper Kutay,Haldun M. Ozaktas,John T. Sheridan
Publisher : Springer
Page : 463 pages
File Size : 47,8 Mb
Release : 2015-11-26
Category : Science
ISBN : 9781493930289

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Linear Canonical Transforms by John J. Healy,M. Alper Kutay,Haldun M. Ozaktas,John T. Sheridan Pdf

This book provides a clear and accessible introduction to the essential mathematical foundations of linear canonical transforms from a signals and systems perspective. Substantial attention is devoted to how these transforms relate to optical systems and wave propagation. There is extensive coverage of sampling theory and fast algorithms for numerically approximating the family of transforms. Chapters on topics ranging from digital holography to speckle metrology provide a window on the wide range of applications. This volume will serve as a reference for researchers in the fields of image and signal processing, wave propagation, optical information processing and holography, optical system design and modeling, and quantum optics. It will be of use to graduate students in physics and engineering, as well as for scientists in other areas seeking to learn more about this important yet relatively unfamiliar class of integral transformations.

Author : Richard E. Blahut
Publisher : Addison Wesley Publishing Company
Page : 464 pages
File Size : 52,8 Mb
Release : 1985
Category : Mathematics
ISBN : UOM:39015048118247

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Fast Algorithms for Digital Signal Processing by Richard E. Blahut Pdf

Introduction to abstract algebra. Fast algorithms for short convolutions. Fast algorithms for the discrete Fourier transform. Number theory and algebraic field theory. Computation in surrogate fields. Fast algorithms and multidimensional convolutions. Fast algorithms and multidimensional transforms. Architecture of filters and transforms. Fast algorithms based on doubling strategies. Fast algorithms for solving Toeplitz systems. Fast algorithms for Trellis and tree search. A collection of cyclic convolution algorithms. A collection of Winograd small FFT algorithms.

Author : H.J. Nussbaumer
Publisher : Springer Science & Business Media
Page : 260 pages
File Size : 55,6 Mb
Release : 2013-03-08
Category : Mathematics
ISBN : 9783662005514

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Fast Fourier Transform and Convolution Algorithms by H.J. Nussbaumer Pdf

This book presents in a unified way the various fast algorithms that are used for the implementation of digital filters and the evaluation of discrete Fourier transforms. The book consists of eight chapters. The first two chapters are devoted to background information and to introductory material on number theory and polynomial algebra. This section is limited to the basic concepts as they apply to other parts of the book. Thus, we have restricted our discussion of number theory to congruences, primitive roots, quadratic residues, and to the properties of Mersenne and Fermat numbers. The section on polynomial algebra deals primarily with the divisibility and congruence properties of polynomials and with algebraic computational complexity. The rest of the book is focused directly on fast digital filtering and discrete Fourier transform algorithms. We have attempted to present these techniques in a unified way by using polynomial algebra as extensively as possible. This objective has led us to reformulate many of the algorithms which are discussed in the book. It has been our experience that such a presentation serves to clarify the relationship between the algorithms and often provides clues to improved computation techniques. Chapter 3 reviews the fast digital filtering algorithms, with emphasis on algebraic methods and on the evaluation of one-dimensional circular convolutions. Chapters 4 and 5 present the fast Fourier transform and the Winograd Fourier transform algorithm.

Author : Thierry Fournel,Bahram Javidi
Publisher : Springer Science & Business Media
Page : 285 pages
File Size : 42,8 Mb
Release : 2010-11-01
Category : Science
ISBN : 9781441973801

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Information Optics and Photonics by Thierry Fournel,Bahram Javidi Pdf

This book will address the advances, applications, research results, and emerging areas of optics, photonics, computational approaches, nano-photonics, bio-photonics, with applications in information systems. The objectives are to bring together novel approaches, analysis, models, and technologies that enhance sensing, measurement, processing, interpretation, and visualization of information. The book will concentrate on new approaches to information systems, including integration of computational algorithms, bio-inspired models, photonics technologies, information security, bio-photonics, and nano-photonics. Applications include bio-photonics, digitally enhanced sensing and imaging systems, multi-dimensional optical imaging and image processing, bio-inspired imaging, 3D visualization, 3D displays, imaging on nano-scale, quantum optics, super resolution imaging, photonics for biological applications, microscopy, information optics, and holographic information systems.

Author : C. Sidney Burrus
Publisher : Lulu.com
Page : 256 pages
File Size : 53,5 Mb
Release : 2012-11-30
Category : Technology & Engineering
ISBN : 9781300461647

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Fast Fourier Transforms by C. Sidney Burrus Pdf

This book uses an index map, a polynomial decomposition, an operator factorization, and a conversion to a filter to develop a very general and efficient description of fast algorithms to calculate the discrete Fourier transform (DFT). The work of Winograd is outlined, chapters by Selesnick, Pueschel, and Johnson are included, and computer programs are provided.

Author : Charles Van Loan
Publisher : SIAM
Page : 286 pages
File Size : 52,6 Mb
Release : 1992-01-01
Category : Mathematics
ISBN : 1611970997

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Computational Frameworks for the Fast Fourier Transform by Charles Van Loan Pdf

The most comprehensive treatment of FFTs to date. Van Loan captures the interplay between mathematics and the design of effective numerical algorithms--a critical connection as more advanced machines become available. A stylized Matlab notation, which is familiar to those engaged in high-performance computing, is used. The Fast Fourier Transform (FFT) family of algorithms has revolutionized many areas of scientific computation. The FFT is one of the most widely used algorithms in science and engineering, with applications in almost every discipline. This volume is essential for professionals interested in linear algebra as well as those working with numerical methods. The FFT is also a great vehicle for teaching key aspects of scientific computing.

Author : Fionn Murtagh
Publisher : SPIE-International Society for Optical Engineering
Page : 324 pages
File Size : 45,8 Mb
Release : 2005
Category : Computers
ISBN : UOM:39015061014828

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Imaging and Vision by Fionn Murtagh Pdf

Proceedings of SPIE present the original research papers presented at SPIE conferences and other high-quality conferences in the broad-ranging fields of optics and photonics. These books provide prompt access to the latest innovations in research and technology in their respective fields. Proceedings of SPIE are among the most cited references in patent literature.

Author : Okan K. Ersoy
Publisher : Prentice Hall
Page : 550 pages
File Size : 48,5 Mb
Release : 1997
Category : Computers
ISBN : UOM:39015041052948

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Fourier-related Transforms, Fast Algorithms, and Applications by Okan K. Ersoy Pdf

Presenting an introduction to all Fourier-related transforms, this work includes a number of applications in the different markets. The accompanying disk provides C and Fortran routines that can be implemented.

Author : Douglas F. Elliott,K. Ramamohan Rao
Publisher : Elsevier
Page : 448 pages
File Size : 40,7 Mb
Release : 1983-03-09
Category : Mathematics
ISBN : 9780080918068

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Fast Transforms Algorithms, Analyses, Applications by Douglas F. Elliott,K. Ramamohan Rao Pdf

This book has grown from notes used by the authors to instruct fast transform classes. One class was sponsored by the Training Department of Rockwell International, and another was sponsored by the Department of Electrical Engineering of The University of Texas at Arlington. Some of the material was also used in a short course sponsored by the University of Southern California. The authors are indebted to their students for motivating the writing of this book and for suggestions to improve it.

Author : Vladimir Britanak,Patrick C. Yip,K. R Rao
Publisher : Elsevier
Page : 364 pages
File Size : 53,9 Mb
Release : 2010-07-28
Category : Mathematics
ISBN : 9780080464640

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Discrete Cosine and Sine Transforms by Vladimir Britanak,Patrick C. Yip,K. R Rao Pdf

The Discrete Cosine Transform (DCT) is used in many applications by the scientific, engineering and research communities and in data compression in particular. Fast algorithms and applications of the DCT Type II (DCT-II) have become the heart of many established international image/video coding standards. Since then other forms of the DCT and Discrete Sine Transform (DST) have been investigated in detail. This new edition presents the complete set of DCT and DST discrete trigonometric transforms, including their definitions, general mathematical properties, and relations to the optimal Karhunen-Loéve transform (KLT), with the emphasis on fast algorithms (one-dimensional and two-dimensional) and integer approximations of DCTs and DSTs for their efficient implementations in the integer domain. DCTs and DSTs are real-valued transforms that map integer-valued signals to floating-point coefficients. To eliminate the floating-point operations, various methods of integer approximations have been proposed to construct and flexibly generate a family of integer DCT and DST transforms with arbitrary accuracy and performance. The integer DCTs/DSTs with low-cost and low-powered implementation can replace the corresponding real-valued transforms in wireless and satellite communication systems as well as portable computing applications. The book is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations and integer aproximations. It is hoped that the book will serve as a valuable reference for industry, academia and research institutes in developing integer DCTs and DSTs as well as an inspiration source for further advanced research. Presentation of the complete set of DCTs and DSTs in context of entire class of discrete unitary sinusoidal transforms: the origin, definitions, general mathematical properties, mutual relationships and relations to the optimal Karhunen-Loéve transform (KLT) Unified treatment with the fast implementations of DCTs and DSTs: the fast rotation-based algorithms derived in the form of recursive sparse matrix factorizations of a transform matrix including one- and two-dimensional cases Detailed presentation of various methods and design approaches to integer approximation of DCTs and DSTs utilizing the basic concepts of linear algebra, matrix theory and matrix computations leading to their efficient multiplierless real-time implementations, or in general reversible integer-to-integer implementations Comprehensive list of additional references reflecting recent/latest developments in the efficient implementations of DCTs and DSTs mainly one-, two-, three- and multi-dimensional fast DCT/DST algorithms including the recent active research topics for the time period from 1990 up to now

Author : Henri J Nussbaumer
Publisher : Unknown
Page : 292 pages
File Size : 41,8 Mb
Release : 1982-09-01
Category : Electronic
ISBN : 3642818986

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Fast Fourier Transform and Convolution Algorithms by Henri J Nussbaumer Pdf

Author : Lokenath Debnath,Firdous A. Shah
Publisher : Birkhäuser
Page : 220 pages
File Size : 40,6 Mb
Release : 2017-09-05
Category : Mathematics
ISBN : 9783319594330

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Lecture Notes on Wavelet Transforms by Lokenath Debnath,Firdous A. Shah Pdf

This book provides a systematic exposition of the basic ideas and results of wavelet analysis suitable for mathematicians, scientists, and engineers alike. The primary goal of this text is to show how different types of wavelets can be constructed, illustrate why they are such powerful tools in mathematical analysis, and demonstrate their use in applications. It also develops the required analytical knowledge and skills on the part of the reader, rather than focus on the importance of more abstract formulation with full mathematical rigor. These notes differs from many textbooks with similar titles in that a major emphasis is placed on the thorough development of the underlying theory before introducing applications and modern topics such as fractional Fourier transforms, windowed canonical transforms, fractional wavelet transforms, fast wavelet transforms, spline wavelets, Daubechies wavelets, harmonic wavelets and non-uniform wavelets. The selection, arrangement, and presentation of the material in these lecture notes have carefully been made based on the authors’ teaching, research and professional experience. Drafts of these lecture notes have been used successfully by the authors in their own courses on wavelet transforms and their applications at the University of Texas Pan-American and the University of Kashmir in India.

Author : Keith Jones
Publisher : Springer Science & Business Media
Page : 232 pages
File Size : 45,9 Mb
Release : 2010-03-10
Category : Mathematics
ISBN : 9789048139170

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The Regularized Fast Hartley Transform by Keith Jones Pdf

Most real-world spectrum analysis problems involve the computation of the real-data discrete Fourier transform (DFT), a unitary transform that maps elements N of the linear space of real-valued N-tuples, R , to elements of its complex-valued N counterpart, C , and when carried out in hardware it is conventionally achieved via a real-from-complex strategy using a complex-data version of the fast Fourier transform (FFT), the generic name given to the class of fast algorithms used for the ef?cient computation of the DFT. Such algorithms are typically derived by explo- ing the property of symmetry, whether it exists just in the transform kernel or, in certain circumstances, in the input data and/or output data as well. In order to make effective use of a complex-data FFT, however, via the chosen real-from-complex N strategy, the input data to the DFT must ?rst be converted from elements of R to N elements of C . The reason for choosing the computational domain of real-data problems such N N as this to be C , rather than R , is due in part to the fact that computing equ- ment manufacturers have invested so heavily in producing digital signal processing (DSP) devices built around the design of the complex-data fast multiplier and accumulator (MAC), an arithmetic unit ideally suited to the implementation of the complex-data radix-2 butter?y, the computational unit used by the familiar class of recursive radix-2 FFT algorithms.