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Prelims-P373624.tex 7/8/2006 13: 3 Page ix Preface Since the book, Discrete Cosine Transform by K. R. Rao and P. Yip (Academic Press, Boston) was published in 1990, the discrete cosine transform (DCT) has increasingly attracted the attention of scientiﬁc, engineering and research communities. The DCT is used in many applications and in data compression in particular. This is due to the fact that the DCT has excellent energy-packing capability and also approaches the statistically optimal Karhunen–Loéve transform (KLT) in decorrrelating a signal. The development of various fast algorithms for the efﬁcient implementation of the DCT involving real arithmetic only, further contributed to its popularity. In the last several years there have been signiﬁcant advances and developments in both theory and applications relating to transform processing of signals. In particular, digital processing motivated the investigation of other forms of DCTs for their integer approximations. International standards organizations (ISO/IEC and ITU-T) have adopted the use of various forms of the integer DCT. At the same time, the investigation of other forms of discrete sine transforms (DSTs) has made a similar impact. There is therefore a need to extend the coverage to include these techniques. This book is aimed at doing just that. The authors have retained much of the basic theory of transforms and transform processing, since the basic mathematics remains valid and valuable. The theory and fast algorithms of the DCTs, as well as those for the DSTs, are dealt with in great detail. There is also an appendix covering some of the fundamental mathematical aspects underlying the theory of transforms. It is no exaggeration to say that applications using DCT are numerous and it is with this in mind that the authors have decided not to include applications explicitly. Readers of this book will either have practical problems requiring the use of DCT, or want to examine the more general theory and techniques for future applications. There is no practical way of comprehensively dealing with all possible applications. However, it must be emphasized that implementation of the various transforms is considered an integral part of our presentation. It is the authors’hope that readers will not only gain some understanding of the various transforms, but also take this knowledge to apply to whatever processing problems they may encounter. The book Discrete Cosine and Sine Transforms: General properties, Fast algorithms and Integer Approximations is aimed at both the novice and the expert. The fervent hopes and aspirations of the authors are that the latest developments in the general DCT/DST ﬁeld ix

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Prelims-P373624.tex 7/8/2006 13: 3 Page x x Preface further lead into additional applications and also provide the incentive and inspiration to further modify/customize these transforms with the overall motivation to improve their efﬁciencies while retaining the simplicity in implementations. V. Britanak P. C. Yip K. R. Rao February 2006

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Prelims-P373624.tex 7/8/2006 13: 3 Page xi Acknowledgments Many many hours and efforts have been spent behind the computer in preparing and typing electronic form of the manuscript, in analytical derivation of various matrix factorizations, drawing the ﬁgures of signal ﬂow graphs and verifying by computer programs. This book is the result of long-term association of the three authors, V. Britanak, P.Yip and K. R. Rao. Special thanks go to their respective families for their support, perseverance and under- standing. We appreciate also the continued encouragement and many helpful suggestions of our colleagues and friends in the departments of the institute and universities. Especially, the leading author wishes to thank his love, Gulinka, for her patience, understanding and encouragement during the years of preparation of this book. Finally, it is also appropriate to acknowledge here the ﬁnancial support provided by Slovak Scientiﬁc Agency VEGA, Project No. 2/4149/24. The authors have been honored to have worked with Academic Press Inc. Elsevier Science on this project. The encouragement, support and understanding for the delayed completion of the book provided by the publishing editorial staff at Materials Science Department, and in particular, Christopher Greenwell and Dr. Jonathan Agbenyega, have been greatly appreciated. xi

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Prelims-P373624.tex 7/8/2006 13: 3 Page xiii List of Acronyms 1-D One-Dimensional 2-D Two-dimensional 3-D Three-dimensional BinDCT Binary arithmetic DCT BinDST Binary arithmetic DST CMT C-Matrix Transform CPU Central Processor Unit DCT Discrete Cosine Transform DFT Discrete Fourier Transform DHT Discrete Hartley Transform DLU DLU matrix factorization or DLU computational structure DPCM Differential Pulse Code Modulation DST Discrete Sine Transform DTT Discrete Trigonometric Transform DUL DUL matrix factorization or DUL computational structure DWT Discrete W Transform EOT Even/Odd Transform FCT Fourier Cosine Transform FFCT Fast Fourier Cosine Transform FFT Fast Fourier Transform FRDCT Fractional Discrete Cosine Transform FRDFT Fractional Discrete Fourier Transform FRDST Fractional Discrete Sine Transform FST Fourier Sine Transform GCMT Generalized C-Matrix Transform GCT Generalized Chen Transform GDFT Generalized Discrete Fourier Transform GDHT Generalized Discrete Hartley Transform HA Half sample Antisymmetric HS Half sample Symmetric ICT Integer Cosine Transform IEC International Electrotechnical Commission IEEE Institute of Electrical and Electronics Engineers IntDCT Integer DCT xiii

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Prelims-P373624.tex 7/8/2006 13: 3 Page xiv xiv List of Acronyms INTDCT Integer DCT IntDST Integer DST ISO International Organization for Standardization IST Integer Sine Transform ITU-T International Telecommunication Standardization Sector JPEG2000 Joint Photographic Experts Group KLT Karhunen–Loéve Transform LDCT Lossless DCT LDST Lossless DST LDU LDU matrix factorization LU LU matrix factorization LUL LUL matrix factorization or LUL computational structure MDCT Modiﬁed Discrete Cosine Transform MDL Multi-Dimensional computational structure MDST Modiﬁed Discrete Sine Transform MLT Modulated Lapped Transform MPEG Moving Picture Experts Group MSE Mean Square Error PCA Principal Component Analysis PLUS PLUS matrix factorization POS Points Of Symmetry QR QR matrix factorization RDCT Reversible DCT RDST Reversible DST SCT Symmetric Cosine Transform SignDCT Signed DCT SMT S-Matrix Transform SOPOT Sum Of Powers Of Two SST Symmetric Sine Transform SVD Singular Value Decomposition ULD ULD matrix factorization or ULD computational structure ULU ULU matrix factorization or ULU computational structure VLSI Very Large-Scale Integration WA Whole sample Antisymmetric WHT Walsh–Hadamard Transform WS Whole sample Symmetric

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Ch01-P373624.tex 7/8/2006 12: 52 Page 1 CHAPTER 1 Discrete Cosine and Sine Transforms 1.1 Introduction Since the publication of original book [1] more than 15 years ago many new con- tributions/extensions/modiﬁcations/updates/improvements to the origin, theoretical and practical aspects of the discrete cosine transforms (DCTs) and discrete sine transforms (DSTs) have been developed. Although the original book [1] has focused almost exclu- sively on the fast algorithms and applications of the DCT of type II (DCT-II) which has become the heart of many established international image/video coding standards [2], since then other forms of the DCT and DST have been investigated in detail. The complete set of DCTs and DSTs, called the discrete trigonometric transforms, has found a number of digital signal processing applications. Among them, for example, the DCT/DST of type IV (DCT- IV/DST-IV) and DCT-II/DST-II are used for the efﬁcient implementation of lapped orthog- onal transforms [6] and perfect reconstruction cosine/sine modulated ﬁlter banks (known as modiﬁed discrete cosine/sine transforms (MDCTs/MDSTs) or equivalently modulated lapped transforms (MLTs) [6]) for high-quality transform/subband audio coding. The complete set of DCTs and DSTs constituting the entire class of discrete sinusoidal uni- tary transforms is presented including their deﬁnitions, general mathematical properties, relations to the Karhunen–Loève transform (KLT), with the emphasis on fast algorithms and integer approximations for their efﬁcient implementations in the integer domain. The DCTs and DSTs are real-valued transforms that map integer-valued signals to ﬂoating- point coefﬁcients. One of the important issues for the applicability of DCTs and DSTs is the existence of fast algorithms that allow their efﬁcient computation. Although the fast algorithms reduce the computational complexity signiﬁcantly, they still need ﬂoating-point operations. To eliminate the ﬂoating-point operations, methods of integer approximations have been proposed to construct and ﬂexibly generate a family of integer transforms with arbitrary accuracy and performance. The integer transforms currently represent the mod- ern transform technologies for lossless transform-based coding. 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. 1

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Ch01-P373624.tex 7/8/2006 12: 52 Page 2 2 Discrete Cosine and Sine Transforms The book covers various latest developments in DCTs and DSTs in a uniﬁed way, and it is essentially a detailed excursion on orthogonal/orthonormal DCT and DST matrices, their matrix factorizations and integer approximations. It is hoped that the book will serve as an excellent reference in developing integer DCTs and DSTs as well as an inspiration for further advanced research. 1.2 Organization of the book The book is organized in terms of chapters starting with this introductory chapter; each chapter has its own list of general references and appendices. Chapter 2 covers deﬁnitions and general properties of classical integral transforms, Fourier cosine transform and Fourier sine transform. The general properties of these continuous transforms such as inversion, linearity, shift in time/frequency, differentiation in time/ frequency, asymptotic behavior, integration in time/frequency and convolution in time together with examples of integral transforms for selected continuous functions are pre- sented in Sections 2.2–2.5. All the DCTs and DSTs are not simply discretized versions of the corresponding integral continuous transforms rather, the discretized cosine and sine functions form the basis functions for an entire family of DCTs and DSTs, and are actually eigenfunctions (or eigenvectors) of certain tridiagonal matrix forms. This issue is addressed in Sections 2.6 and 2.7. DCTs and DSTs possess nice mathematical properties such as uni- tarity, linearity, scaling and shift in time, and, in particular, convolution properties which are discussed in detail in Sections 2.8 and 2.9. KLT an optimal transform from a statistical viewpoint is deﬁned in Chapter 3 (Section 3.2) along with the demonstration of the asymptotic equivalence of DCT-I and DCT-II to KLT in Section 3.3. Section 3.4 addresses the asymptotic equivalence of different types of correlation matrices and their orthonormal representations leading to a general procedure for generating certain discrete unitary transforms for a given class of signal correlation matrices. For the DCT and DST to be viable, feasible and practical, the fast algorithms for their efﬁcient implementation in terms of reduced memory, implementation complexity and recursivity are essential. The fast algorithms for both one- and two-dimensional (1-D, 2-D, respectively) DCTs/DSTs are the main thrust in Chapter 4. In Section 4.2, the deﬁni- tions, properties of and relations between DCTs and DSTs are ﬁrst presented, followed by presentation of the explicit forms of orthonormal DCT and DST matrices for N = 2, 4 and 8 in Section 4.3. The fast 1-D rotation-based algorithms for the computation of DCTs and DSTs based on the (recursive) sparse matrix factorizations of the corresponding DCT and DST matrices and represented by the generalized signal ﬂow graphs are discussed in Section 4.4. The matrix factorizations reveal various interrelations between different versions of the DCT and DST. These selected fast algorithms are very convenient in con- structing integer approximations of DCTs and DSTs. Section 4.5 analyzes existing 2-D fast DCT/DST algorithms and suggests a simple method for generating 2-D direct DCT/DST algorithms from the corresponding 1-D ones. As integer versions of the DCT/DST have attracted the attention of researchers resulting in substantial simpliﬁcation in their implementation while still maintaining performance

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Ch01-P373624.tex 7/8/2006 12: 52 Page 3 Discrete Cosine and Sine Transforms 3 nearly equal to their earlier versions, it is only logical that this arena be focused in much detail and depth in Chapter 5. Section 5.2 presents the basic material from linear alge- bra, theory of matrices and matrix computations which is fundamental for understanding the approximation methods. In order to evaluate the approximation error between the approximated and original transform matrix and to measure the performance of resulting approximated transform used in data compression, some theoretical criteria are deﬁned in Section 5.3. Finally, various developed methods and design approaches to integer approx- imation of the DCT and DST are detailed in Section 5.4. More recent developments in designing lossless DCTs, invertible integer DCTs and reversible DCTs including the latest developments are discussed in Sections 5.5 and 5.6. All chapters end with a summary, problems/exercises and references. Problems/exercises reﬂect the contents of the corresponding chapters and are intended for the reader in terms of refresh/review/reinforce their contents. Extensive deﬁnitions, principles, properties, signal ﬂow graphs, derivations, proofs and examples are provided throughout the book for proper understanding of the strengths and shortcomings of the spectrum of cosine/sine transforms and their application in diverse disciplines. 1.3 Appendices Appendices A.1 through A.3 review the important basic concepts of linear algebra such as vector spaces (Appendix A.1), matrix eigenvalue problem (Appendix A.2) and matrix decompositions (Appendix A.3) in the form of deﬁnitions and theorems with exercises/ problems at the end. Deterministic as well as random signals, their classiﬁcation and repre- sentations are discussed in Appendix A.4. A number of examples are listed in Appendices to illustrate the use of basic concepts in practical applications. 1.4 References To retain the connectivity among the chapters of the book as much as possible, each chapter in the book includes its own list of references related to the discussed sub- ject. Therefore, some references may appear in the lists of references of chapters more than once. 1.5 Additional references An extensive list of additional references have been appended to this chapter. No claim for completeness of this list is made. Additional references, although not cited in subsequent chapters, reﬂect the various recent/latest developments in the efﬁcient implementations of DCTs and DSTs, mainly 1-D, 2-D, 3-D and in general, multi-dimensional fast DCT/DST algorithms for the time period from 1989/1990 up to now. They supplement the compre- hensive list of references related to DCTs and DSTs in the original books [1, 2]. Thus, this book and books [1, 2] cover completely the theoretical developments, algorithmic history of DCTs and DSTs including the recent active research topics.

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Ch01-P373624.tex 7/8/2006 12: 52 Page 4 4 Discrete Cosine and Sine Transforms For clarity, the additional references are classiﬁed into the following categories with guidelines: • Other books discussing DCTs, DSTs and KLT [3–9] The recent published books discuss both the theoretical and practical aspects of DCTs and DSTs including the KLT. • Fast 1-D radix-2 DCT/DST algorithms [10–67] This category is further subdivided into three parts: fast algorithms for computation of DCT-I, -II, -III, -IV and corresponding DST-I, -II, -III, -IV [10–32], fast DCT algorithms only [33–59] and fast DST algorithms only [60–67]. • Fast direct 2-D DCT/DST algorithms [68–87] This category includes the direct 2-D radix-2 DCT/DST algorithms, and direct even/prime-length 2-D algorithms based on cyclic convolutions and circular or skew- circular correlations. Since 2-D DCT/DST kernels are separable, the 2-D DCT/DST computation can simply be realized by the so-called row–column method which sequentially uses any fast 1-D DCT/DST algorithm on rows and columns of the input data matrix. In general, many 1-D DCT/DST algorithms can be extended to the direct 2-D case using a 2-D decomposition process. • Fast direct 3-D and multi-dimensional DCT/DST algorithms [88–97] The higher-dimensional DCT/DST algorithms can be obtained by the similar methods as those of 2-D DCT/DST ones. • Fast even/odd/composite-length, prime-factor, radix-q and mixed-radix DCT/ DST algorithms [98–126] The limitation common to most fast DCT/DST algorithms is that N must be a power of 2 (radix-2 DCT/DST algorithms). In practice, various sequence lengths other than a power of 2 may occur. To deal with such sequence lengths, new fast even/odd-length (N is an even/odd integer), composite-length (N = p · q, where p and q are relatively n primes), prime-factor, radix-q (N = q , where q is an odd integer) and mixed-radix n (N = 2 · q, where q = 3, 5, 6, 7, 9, . . .) DCT/DST algorithms have been proposed. Even/odd-length and prime-factor DCT/DST algorithms can be directly mapped into the corresponding even/odd-length and prime-factor complex-valued or real-valued FFT modules, or they are based on shorter cyclic/skew-cyclic convolutions and skew- n circular correlations. The algorithms for sequence lengths other than 2 need quite different methods for their derivation, and generally they have a higher computational complexity and have more complex structure. • Fast pruning DCT algorithms [127–134] The standard DCT (radix-2) algorithms inherently assume that the lengths of input and output data sequences are equal. However, in many applications such as data compression, the most important information about the signal is kept by the low- frequency DCT coefﬁcients. Therefore, from N coefﬁcients (N being the length of data sequence) only N1 (N1 < N) lowest-frequency coefﬁcients need to be computed. Such a method where only a subset of the output coefﬁcients is utilized to accelerate the computation is referred to as “pruning”. Therefore the algorithms, called fast pruning DCT algorithms, have been developed just for this purpose. In general, the fast DCT algorithm to be pruned must be deﬁned by a simple structured recursive matrix factorization of the transform matrix and represented by the regular signal ﬂow graph.

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Ch01-P373624.tex 7/8/2006 12: 52 Page 5 Discrete Cosine and Sine Transforms 5 • DCT/DST computation by recursive ﬁlter structures [135–149] A class of algorithms for arbitrary length forward and inverse DCT/DST computations are recursive algorithms where DCT/DST kernels are converted to regular regres- sive structures based on sinusoidal recursive formulae, or recurrence formulae for Chebyshev polynomials (of the second and third kind), or Clenshaw’s recurrence for- mula. Although these recursive algorithms are not efﬁcient in terms of computational complexity, regressive structures provide simple and efﬁcient schemes for the parallel VLSI implementation of the variable length DCTs/DSTs. • Fractional DCTs and DSTs [150–152] Recently, the fractional DCTs (FRDCTs) and fractional DSTs (FRDSTs) for DCT-II, symmetric cosine and symmetric sine transforms have been introduced. The deﬁ- nitions of FRDCTs and FRDSTs are based on eigen decompositions (eigenvalues and eigenvectors) of the corresponding DCT and DST matrices; or simply by other words, FRDCTs and FRDSTs are deﬁned through the “fractional” real powers of DCT and DST matrices. It is the same idea as that of the fractional discrete Fourier transform (FRDFT). The investigation of FRDCT and FRDST, their general proper- ties are recently an active and interesting research topic. Open problems involve the rigorous deﬁnitions of FRDCTs and FRDSTs for other forms of DCT and DST, study of their general properties, matrix representations and, in particular, fast algorithms for their practical implementations [152]. • Fast quantum algorithms for DCTs and DSTs [153] Quantum computing has recently become an exciting area of emerging digital signal processing applications. A classical computer does not allow to calculate N-point n DCTs or DSTs, where N = 2 , in less than linear time. This trivial lower bound is no longer valid for a quantum computer. In fact, it is possible to realize N-point DCTs 2 and DSTs with as little as O(log N) operations on a quantum computer, whereas 2 the all known fast DCT/DST algorithms realized on a classical computer require O(N log N) operations. Based on existing efﬁcient quantum circuits for the DFT, the 2 (extremely) fast quantum DCT/DST algorithms can be derived and implemented on a number of quantum computing technologies. We believe that the additional references, although not used in the book, will be a valuable and useful source for the reader in her/his further study or advanced research or in solving speciﬁc problems in the area of DCT/DST applications. References [1] K. R. Rao and P. Yip, Discrete Cosine Transform: Algorithms, Advantages, Applications, Academic Press, Boston MA, 1990. [2] K. R. Rao and J. J. Hwang, Techniques and Standards for Digital Image/Video/Audio Coding, Prentice-Hall, Upper Saddle River, NJ, 1996. Other books discussing DCTs, DSTs and KLT [3] A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall, Englewood Cliffs, NJ, 1989.

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