Discrete Cosine and Sine Transforms

Discrete Cosine and Sine Transforms

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 ...

Author: Vladimir Britanak

Publisher: Elsevier

ISBN: 0080464645

Category: Mathematics

Page: 368

View: 430

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
Categories: Mathematics

Discrete Cosine and Sine Transforms

Discrete Cosine and Sine Transforms

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-Loeve transform (KLT), with the emphasis on ...

Author: Vladimir Britanak

Publisher: Academic Press

ISBN: 1493301039

Category:

Page: 368

View: 935

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-Loeve 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. Key Features - 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-Loeve 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."
Categories:

Hardware Implementation of Windowed Discrete Cosine and Discrete Sine Transforms in VHDL

Hardware Implementation of Windowed Discrete Cosine and Discrete Sine Transforms in VHDL

This study is to investigate the hardware implementation of the digital signal processing algorithms namely Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) for image compression and image updating.

Author: Smitha Murthy

Publisher:

ISBN: OCLC:841572931

Category:

Page: 126

View: 898

This study is to investigate the hardware implementation of the digital signal processing algorithms namely Discrete Cosine Transform (DCT) and Discrete Sine Transform (DST) for image compression and image updating. A typical approach when processing a signal or an image using Discrete Cosine Transform (DCT) or Discrete sine Transform (DST) is to extract a portion of the signal by windowing it and then applying DCT or DST of the window contents. The entire signal is processed by shifting the window point by point over the signal, hence limiting the updating algorithms (of DCT & DST) to a single point step between successive windows. In this thesis an algorithm developed to simultaneously update DCT and DST to reflect inclusion of r where 1[less than or equal to] r [less than or equal to] N-1. Additional points and removal of r old points from the signal is implemented in Very High-Speed Integrated Hardware Description Language (VHDL). The algorithm is derived for use with split triangular (trapezoidal) window; in this way the results are extended to handle larger step sizes. This thesis involves the hardware implementation using distributed arithmetic method of implementing DCT and DST in VHDL.
Categories:

Discrete Cosine Transform Second Edition

Discrete Cosine Transform  Second Edition

Many new DCT-like transforms have been proposed since the first edition of this book.

Author: Humberto Ochoa-Dominguez

Publisher: CRC Press

ISBN: 9781351396479

Category: Technology & Engineering

Page: 358

View: 936

Many new DCT-like transforms have been proposed since the first edition of this book. For example, the integer DCT that yields integer transform coefficients, the directional DCT to take advantage of several directions of the image and the steerable DCT. The advent of higher dimensional frames such as UHDTV and 4K-TV demand for small and large transform blocks to encode small or large similar areas respectively in an efficient way. Therefore, a new updated book on DCT, adapted to the modern days, considering the new advances in this area and targeted for students, researchers and the industry is a necessity.
Categories: Technology & Engineering

Discrete Cosine Sine Transform

Discrete Cosine   Sine Transform

The goals of this thesis are the following: to provide a tutorial in the field of discrete cosine transforms and discrete sine transforms in one, two, and three dimensions; to show how these transforms can be used in the field of image ...

Author: Jaskeerat S. Baweja

Publisher:

ISBN: OCLC:44494914

Category: Algorithms

Page: 266

View: 962

The goals of this thesis are the following: to provide a tutorial in the field of discrete cosine transforms and discrete sine transforms in one, two, and three dimensions; to show how these transforms can be used in the field of image processing; and finally to compare their image coding capabilities.
Categories: Algorithms

Hardware Implementation Design Synthesis and Analysis of Odd Discrete Cosine Transform and Odd Discrete Sine Transform Using Verilog HDL

Hardware Implementation  Design Synthesis and Analysis of Odd Discrete Cosine Transform and Odd Discrete Sine Transform Using Verilog HDL

This objective of this work is to address the hardware implementation of Odd Discrete Cosine Transform (ODCT) and Odd Discrete Sine Transform (ODST) by utilizing the ODCT/ODST coefficients obtained from the conventional algorithms.

Author: Sanskruti Mainkar

Publisher:

ISBN: OCLC:841574073

Category:

Page: 170

View: 437

This objective of this work is to address the hardware implementation of Odd Discrete Cosine Transform (ODCT) and Odd Discrete Sine Transform (ODST) by utilizing the ODCT/ODST coefficients obtained from the conventional algorithms. The Discrete Cosine Transform (DCT) is a Fourier-related transform similar to the Discrete Fourier Transform (DFT), but using a purely real representation. The DCT Algorithm is widely used due to its autocorrelation properties, bandwidth reduction and energy compaction efficiency and is used extensively in the field of data and image compression, audio-visual, communication and speech enhancement techniques. The Discrete Sine Transform (DST) is equivalent to a DFT of real and odd functions. It is equivalent to the imaginary part of a DFT of roughly twice the length, operating on real data with odd symmetry (since the Fourier transform of a real and odd function is imaginary and odd), where in some variants the input and/or output data are shifted by half a sample. In this thesis, the digital hardware implementations of these two transforms are achieved using a hybrid design process by modifying the traditional butterfly flow-graph. This is achieved by combining the butterfly flow-graph and the matrix-vector multiplication method. A set of 4 data points is used to implement the hybrid design, but the study can be extended to larger data sets in case of more data points with minor modifications, due to the symmetry in the basic butterfly.
Categories:

Computational Frameworks for the Fast Fourier Transform

Computational Frameworks for the Fast Fourier Transform

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: Charles Van Loan

Publisher: SIAM

ISBN: 1611970997

Category: Fourier transformations

Page: 273

View: 837

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.
Categories: Fourier transformations

Linear Difference Equations with Discrete Transform Methods

Linear Difference Equations with Discrete Transform Methods

This book covers the basic elements of difference equations and the tools of difference and sum calculus necessary for studying and solv ing, primarily, ordinary linear difference equations.

Author: A.J. Jerri

Publisher: Springer Science & Business Media

ISBN: 9781475756579

Category: Mathematics

Page: 442

View: 504

This book covers the basic elements of difference equations and the tools of difference and sum calculus necessary for studying and solv ing, primarily, ordinary linear difference equations. Examples from various fields are presented clearly in the first chapter, then discussed along with their detailed solutions in Chapters 2-7. The book is in tended mainly as a text for the beginning undergraduate course in difference equations, where the "operational sum calculus" of the di rect use of the discrete Fourier transforms for solving boundary value problems associated with difference equations represents an added new feature compared to other existing books on the subject at this introductory level. This means that in addition to the familiar meth ods of solving difference equations that are covered in Chapter 3, this book emphasizes the use of discrete transforms. It is an attempt to introduce the methods and mechanics of discrete transforms for solv ing ordinary difference equations. The treatment closely parallels what many students have already learned about using the opera tional (integral) calculus of Laplace and Fourier transforms to solve differential equations. As in the continuous case, discrete operational methods may not solve problems that are intractable by other meth ods, but they can facilitate the solution of a large class of discrete initial and boundary value problems. Such operational methods, or what we shall term "operational sum calculus," may be extended eas ily to solve partial difference equations associated with initial and/or boundary value problems.
Categories: Mathematics

Transforms and Applications Handbook

Transforms and Applications Handbook

3. Sine. and. Cosine. Transforms. 3-1 3-1 3-11 3-16 3.1 3.2 The Fourier Cosine
Transform (FCT) . ... Fourier Transforms • Basic Properties and Operational Rules
• Selected Fourier Sine Transforms 3.4 The Discrete Sine and Cosine ...

Author: Alexander D. Poularikas

Publisher: CRC Press

ISBN: 9781420066531

Category: Mathematics

Page: 911

View: 484

Updating the original, Transforms and Applications Handbook, Third Edition solidifies its place as the complete resource on those mathematical transforms most frequently used by engineers, scientists, and mathematicians. Highlighting the use of transforms and their properties, this latest edition of the bestseller begins with a solid introduction to signals and systems, including properties of the delta function and some classical orthogonal functions. It then goes on to detail different transforms, including lapped, Mellin, wavelet, and Hartley varieties. Written by top experts, each chapter provides numerous examples and applications that clearly demonstrate the unique purpose and properties of each type. The material is presented in a way that makes it easy for readers from different backgrounds to familiarize themselves with the wide range of transform applications. Revisiting transforms previously covered, this book adds information on other important ones, including: Finite Hankel, Legendre, Jacobi, Gengenbauer, Laguerre, and Hermite Fraction Fourier Zak Continuous and discrete Chirp-Fourier Multidimensional discrete unitary Hilbert-Huang Most comparable books cover only a few of the transforms addressed here, making this text by far the most useful for anyone involved in signal processing—including electrical and communication engineers, mathematicians, and any other scientist working in this field.
Categories: Mathematics

A Hardware Implementation and Analysis of the Discrete Cosine Transform Update Algorithms

A Hardware Implementation and Analysis of the Discrete Cosine Transform Update Algorithms

This work can be divided into three main objectives.

Author: Hazem Bassam Alassaly

Publisher:

ISBN: OCLC:841573135

Category: Computer algorithms

Page: 480

View: 770

This work can be divided into three main objectives. The first objective is to implement hardware circuits that are capable of computing the Discrete Cosine Transform (DCT) coefficients and the Discrete Sine Transform (DST) coefficients in an efficient and optimized way (conventional mode of operation). Those circuits are built using a hybrid design that combines the butterfly flow-graph and the matrix-vector multiplication method. The hybrid design helps solving the problem of global communications that are necessary to implement the traditional butterfly flow-graph. An optimum size of 8 data points is used to implement the hybrid design. The second objective is to incorporate hardware circuits that are capable of updating both the DCT coefficients and the DST coefficients simultaneously using moving window of size 8 and step size of 1 and 4 (update mode of operation). This mode of operation aids in recalculating the DCT and DST coefficients in their entirety (i.e. updated) based on receiving a number of data points that is smaller than the size of the window. By having both modes of operation implemented together, the capability of switching between the two modes becomes available which can be exercised in some specific cases. Finally, the third objective is to investigate the possibilities of speeding-up the architecture along with reducing the power requirements.
Categories: Computer algorithms

Communication Theory and Signal Processing for Transform Coding

Communication Theory and Signal Processing for Transform Coding

This book is tailored to fulfil the requirements in the area of the signal processing in communication systems.

Author: Khamies El-Shennawy

Publisher: Bentham Science Publishers

ISBN: 9781608058303

Category: Technology & Engineering

Page: 302

View: 928

This book is tailored to fulfil the requirements in the area of the signal processing in communication systems. The book contains numerous examples, solved problems and exercises to explain the methodology of Fourier Series, Fourier Analysis, Fourier Transform and properties, Fast Fourier Transform FFT, Discrete Fourier Transform DFT and properties, Discrete Cosine Transform DCT, Discrete Wavelet Transform DWT and Contourlet Transform CT. The book is characterized by three directions, the communication theory and signal processing point of view, the mathematical point of view and utility computer programs. The contents of this book include chapters in communication system and signals, Fourier Series and Power Spectra, Fourier Transform and Energy Spectra, Fourier Transform and Power Spectra, Correlation Function and Spectral Density, Signal Transmission and Systems, Hilbert Transform, Narrow Band-Pass Signals and Systems and Numerical Computation of Transform Coding. This book is intended for undergraduate students in institutes, colleges, universities and academies who want to specialize in the field of communication systems and signal processing. The book will also be very useful to engineers of graduate and post graduate studies as well as researchers in research centers since it contains a great number of mathematical operations that are considered important in research results.
Categories: Technology & Engineering

Fourier Analysis

Fourier Analysis

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online.

Author: Source Wikipedia

Publisher: University-Press.org

ISBN: 1230550585

Category:

Page: 194

View: 892

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. Pages: 192. Chapters: Nyquist-Shannon sampling theorem, Discrete cosine transform, Discrete Fourier transform, Bessel function, Dirac delta function, Autocorrelation, Laplace's equation, Convolution, Topological group, Banach algebra, List of Fourier-related transforms, Frequency spectrum, Haar measure, Laplace transform, Fourier transform spectroscopy, Convolution theorem, Harmonic function, Basis function, Periodic function, Whittaker-Shannon interpolation formula, Fundamental frequency, Laplace operator, Modified discrete cosine transform, Fourier optics, Spherical harmonics, LTI system theory, Window function, Pontryagin duality, Multiplier, Discrete-time Fourier transform, Modulus of continuity, Sobolev space, Almost periodic function, Carleson's theorem, Ewald summation, Relations between Fourier transforms and Fourier series, Poisson summation formula, Analytic signal, Reciprocal lattice, Fractional Fourier transform, Solid harmonics, Spin-weighted spherical harmonics, Short-time Fourier transform, Uncertainty principle for the short-time Fourier transform, Riesz-Thorin theorem, Discrete Hartley transform, Linear canonical transformation, SigSpec, Discrete sine transform, Chirplet transform, Homogeneous distribution, Peter-Weyl theorem, Bilinear time-frequency distribution, Metaplectic group, Bloch wave - MoM method, Marcinkiewicz interpolation theorem, Spectral leakage, Orthonormal basis, Paley-Wiener theorem, Overlap-add method, FBI transform, Unit circle, Fourier inversion theorem, DFT matrix, Poisson kernel, Interpolation space, Littlewood-Paley theory, Motions in the time-frequency distribution, List of cycles, Parametrix, A derivation of the discrete Fourier transform, Parseval's theorem, Number-theoretic transform, Compact group, Overlap-save method, Set of uniqueness, Multitaper, Convolution power, ..
Categories:

Fast Transform Methods in Digital Signal Processing

Fast Transform Methods in Digital Signal Processing

"This ebook covers fast transform algorithms, analyses, and applications in a single volume.

Author: Leonid Yaroslavsky

Publisher: Bentham Science Publishers

ISBN: 9781608052301

Category: Technology & Engineering

Page: 119

View: 208

"This ebook covers fast transform algorithms, analyses, and applications in a single volume. It is the result of the collaboration by the author with others in the world wide university community and has been accumulated over the author's working lifetime "
Categories: Technology & Engineering

Discrete Transforms and Their Applications

Discrete Transforms and Their Applications

2 ) Compute the discrete cosine transform on the sequence f ( n ) . The FDCT
developed by Chen et al . [ 4 ) is a good implementation of this step . 3 ) By
reversing the sequence order of data which were produced by step 2 ) , the
discrete sine ...

Author: Kamisetty Ramamohan Rao

Publisher: Krieger Publishing Company

ISBN: IND:30000000944078

Category: Mathematics

Page: 334

View: 837

Categories: Mathematics

Multiple Transforms for Video Coding

Multiple Transforms for Video Coding

Several designs are presented, allowing for different complexity and bitrate savings trade-offs. These systems reveal the interest of using multiple transforms for video coding.

Author: Adrià Arrufat Batalla

Publisher:

ISBN: OCLC:959222416

Category:

Page: 115

View: 900

State of the art video codecs use transforms to ensure a compact signal representation. The transform stage is where compression takes place, however, little variety is observed in the type of transforms used for standardised video coding schemes: often, a single transform is considered, usually a Discrete Cosine Transform (DCT). Recently, other transforms have started being considered in addition to the DCT. For instance, in the latest video coding standard, High Efficiency Video Coding (HEVC), the 4x4 sized blocks can make use of the Discrete Sine Transform (DST) and, in addition, it also possible not to transform them. This fact reveals an increasing interest to consider a plurality of transforms to achieve higher compression rates. This thesis focuses on extending HEVC through the use of multiple transforms. After a general introduction to video compression and transform coding, two transform designs are studied in detail: the Karhunen Loève Transform (KLT) and a Rate-Distortion Optimised Transform are considered. These two methods are compared against each other by replacing the transforms in HEVC. This experiment validates the appropriateness of the design. A coding scheme that incorporates and boosts the use of multiple transforms is introduced: several transforms are made available to the encoder, which chooses the one that provides the best rate-distortion trade-off. Consequently, a design method for building systems using multiple transforms is also described. With this coding scheme, significant amounts of bit-rate savings are achieved over HEVC, especially when using many complex transforms. However, these improvements come at the expense of increased complexity in terms of coding, decoding and storage requirements. As a result, simplifications are considered while limiting the impact on bit-rate savings. A first approach is introduced, in which incomplete transforms are used. This kind of transforms use one single base vector and are conceived to work as companions of the HEVC transforms. This technique is evaluated and provides significant complexity reductions over the previous system, although the bit-rate savings are modest. A systematic method, which specifically determines the best trade-offs between the number of transforms and bit-rate savings, is designed. This method uses two different types of transforms based separable orthogonal transforms and Discrete Trigonometric Transforms (DTTs) in particular. Several designs are presented, allowing for different complexity and bitrate savings trade-offs. These systems reveal the interest of using multiple transforms for video coding.
Categories:

The continuous boundary local trigonometric transform

The continuous boundary local trigonometric transform

DCT2 - IV The 2D Discrete Cosine Transform ( type IV boundary conditions ) .
DCT / DST- LCT / LST Real - valued brushlet transform based on the global
Discrete Cosine ( Sine ) Transform followed by the Local Cosine ( Sine )
Transform .

Author: Brons Michael Larson

Publisher:

ISBN: UCAL:X64598

Category:

Page: 434

View: 916

Categories:

Indian Science Abstracts

Indian Science Abstracts

This result in quicker compulation of the discrete cosine and sine transform
coefficients . Also looks at the relations among the family of discrete cosine and
sine transforms and presents the mapping relationships for various discrete
cosine and ...

Author:

Publisher:

ISBN: UCBK:C086062859

Category: Science

Page:

View: 666

Categories: Science

Cosine Sine Modulated Filter Banks

Cosine  Sine Modulated Filter Banks

This book covers various algorithmic developments in the perfect reconstruction cosine/sine-modulated filter banks (TDAC-MDCT/MDST or MLT, MCLT, low delay MDCT, complex exponential/cosine/sine-modulated QMF filter banks), and near-perfect ...

Author: Vladimir Britanak

Publisher: Springer

ISBN: 9783319610801

Category: Technology & Engineering

Page: 645

View: 533

This book covers various algorithmic developments in the perfect reconstruction cosine/sine-modulated filter banks (TDAC-MDCT/MDST or MLT, MCLT, low delay MDCT, complex exponential/cosine/sine-modulated QMF filter banks), and near-perfect reconstruction QMF banks (pseudo-QMF banks) in detail, including their general mathematical properties, matrix representations, fast algorithms and various methods to integer approximations being recently a new transform technology for lossless audio coding. Each chapter will contain a number of examples and will conclude with problems and exercises. The book reflects the research efforts/activities and achieved results of the authors in the time period over the last 20 years.
Categories: Technology & Engineering