11 research outputs found
Type-II/III DCT/DST algorithms with reduced number of arithmetic operations
We present algorithms for the discrete cosine transform (DCT) and discrete
sine transform (DST), of types II and III, that achieve a lower count of real
multiplications and additions than previously published algorithms, without
sacrificing numerical accuracy. Asymptotically, the operation count is reduced
from ~ 2N log_2 N to ~ (17/9) N log_2 N for a power-of-two transform size N.
Furthermore, we show that a further N multiplications may be saved by a certain
rescaling of the inputs or outputs, generalizing a well-known technique for N=8
by Arai et al. These results are derived by considering the DCT to be a special
case of a DFT of length 4N, with certain symmetries, and then pruning redundant
operations from a recent improved fast Fourier transform algorithm (based on a
recursive rescaling of the conjugate-pair split radix algorithm). The improved
algorithms for DCT-III, DST-II, and DST-III follow immediately from the
improved count for the DCT-II.Comment: 9 page
On the realization of discrete cosine transform using the distributed arithmetic
Version of RecordPublishe
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for DCTs and DSTs
This paper presents a systematic methodology based on the algebraic theory of
signal processing to classify and derive fast algorithms for linear transforms.
Instead of manipulating the entries of transform matrices, our approach derives
the algorithms by stepwise decomposition of the associated signal models, or
polynomial algebras. This decomposition is based on two generic methods or
algebraic principles that generalize the well-known Cooley-Tukey FFT and make
the algorithms' derivations concise and transparent. Application to the 16
discrete cosine and sine transforms yields a large class of fast algorithms,
many of which have not been found before.Comment: 31 pages, more information at http://www.ece.cmu.edu/~smar
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Distributed arithmetic architecture for the discrete cosine transform
The Discrete Cosine Transform is used in many image and video compression
standards. Many methods have been developed for efficiently computing the Discrete
Cosine Transform including flowgraph algorithms, distributed arithmetic and
two-dimensional decompositions.
A new architecture based on distributed arithmetic is presented for computing
the Discrete Cosine Transform and it's inverse. The main objective of the design is
to minimize the area of the VLSI implementation while maintaining the throughput
necessary for video and image compression standards such as MPEG and JPEG.
Several improvements have been made compared to previously published distributed
arithmetic architectures. These include elimination of four lookup tables and implementation
of the lookup tables using logic instead of ROM.
A model of the proposed architecture was written in C. The model was used to
verify the accuracy of the architecture and to do JPEG compression on a series of
test images. Behavioral simulations were performed with a hardware model written
in the Verilog hardware description language. These behavioral simulations verify
that the hardware implementation matches the C model. The model was synthesized
using the Synopsis synthesis tool. The gate count and clock rate of the design were
estimated using the synthesis results
Three dimensional DCT based video compression.
by Chan Kwong Wing Raymond.Thesis (M.Phil.)--Chinese University of Hong Kong, 1997.Includes bibliographical references (leaves 115-123).Acknowledgments --- p.iTable of Contents --- p.ii-vList of Tables --- p.viList of Figures --- p.viiAbstract --- p.1Chapter Chapter 1 : --- IntroductionChapter 1.1 --- An Introduction to Video Compression --- p.3Chapter 1.2 --- Overview of Problems --- p.4Chapter 1.2.1 --- Analog Video and Digital Problems --- p.4Chapter 1.2.2 --- Low Bit Rate Application Problems --- p.4Chapter 1.2.3 --- Real Time Video Compression Problems --- p.5Chapter 1.2.4 --- Source Coding and Channel Coding Problems --- p.6Chapter 1.2.5 --- Bit-rate and Quality Problems --- p.7Chapter 1.3 --- Organization of the Thesis --- p.7Chapter Chapter 2 : --- Background and Related WorkChapter 2.1 --- Introduction --- p.9Chapter 2.1.1 --- Analog Video --- p.9Chapter 2.1.2 --- Digital Video --- p.10Chapter 2.1.3 --- Color Theory --- p.10Chapter 2.2 --- Video Coding --- p.12Chapter 2.2.1 --- Predictive Coding --- p.12Chapter 2.2.2 --- Vector Quantization --- p.12Chapter 2.2.3 --- Subband Coding --- p.13Chapter 2.2.4 --- Transform Coding --- p.14Chapter 2.2.5 --- Hybrid Coding --- p.14Chapter 2.3 --- Transform Coding --- p.15Chapter 2.3.1 --- Discrete Cosine Transform --- p.16Chapter 2.3.1.1 --- 1-D Fast Algorithms --- p.16Chapter 2.3.1.2 --- 2-D Fast Algorithms --- p.17Chapter 2.3.1.3 --- Multidimensional DCT Algorithms --- p.17Chapter 2.3.2 --- Quantization --- p.18Chapter 2.3.3 --- Entropy Coding --- p.18Chapter 2.3.3.1 --- Huffman Coding --- p.19Chapter 2.3.3.2 --- Arithmetic Coding --- p.19Chapter Chapter 3 : --- Existing Compression SchemeChapter 3.1 --- Introduction --- p.20Chapter 3.2 --- Motion JPEG --- p.20Chapter 3.3 --- MPEG --- p.20Chapter 3.4 --- H.261 --- p.22Chapter 3.5 --- Other Techniques --- p.23Chapter 3.5.1 --- Fractals --- p.23Chapter 3.5.2 --- Wavelets --- p.23Chapter 3.6 --- Proposed Solution --- p.24Chapter 3.7 --- Summary --- p.25Chapter Chapter 4 : --- Fast 3D-DCT AlgorithmsChapter 4.1 --- Introduction --- p.27Chapter 4.1.1 --- Motivation --- p.27Chapter 4.1.2 --- Potentials of 3D DCT --- p.28Chapter 4.2 --- Three Dimensional Discrete Cosine Transform (3D-DCT) --- p.29Chapter 4.2.1 --- Inverse 3D-DCT --- p.29Chapter 4.2.2 --- Forward 3D-DCT --- p.30Chapter 4.3 --- 3-D FCT (3-D Fast Cosine Transform Algorithm --- p.30Chapter 4.3.1 --- Partitioning and Rearrangement of Data Cube --- p.30Chapter 4.3.1.1 --- Spatio-temporal Data Cube --- p.30Chapter 4.3.1.2 --- Spatio-temporal Transform Domain Cube --- p.31Chapter 4.3.1.3 --- Coefficient Matrices --- p.31Chapter 4.3.2 --- 3-D Inverse Fast Cosine Transform (3-D IFCT) --- p.32Chapter 4.3.2.1 --- Matrix Representations --- p.32Chapter 4.3.2.2 --- Simplification of the calculation steps --- p.33Chapter 4.3.3 --- 3-D Forward Fast Cosine Transform (3-D FCT) --- p.35Chapter 4.3.3.1 --- Decomposition --- p.35Chapter 4.3.3.2 --- Reconstruction --- p.36Chapter 4.4 --- The Fast Algorithm --- p.36Chapter 4.5 --- Example using 4x4x4 IFCT --- p.38Chapter 4.6 --- Complexity Comparison --- p.43Chapter 4.6.1 --- Complexity of Multiplications --- p.43Chapter 4.6.2 --- Complexity of Additions --- p.43Chapter 4.7 --- Implementation Issues --- p.44Chapter 4.8 --- Summary --- p.46Chapter Chapter 5 : --- QuantizationChapter 5.1 --- Introduction --- p.49Chapter 5.2 --- Dynamic Ranges of 3D-DCT Coefficients --- p.49Chapter 5.3 --- Distribution of 3D-DCT AC Coefficients --- p.54Chapter 5.4 --- Quantization Volume --- p.55Chapter 5.4.1 --- Shifted Complement Hyperboloid --- p.55Chapter 5.4.2 --- Quantization Volume --- p.58Chapter 5.5 --- Scan Order for Quantized 3D-DCT Coefficients --- p.59Chapter 5.6 --- Finding Parameter Values --- p.60Chapter 5.7 --- Experimental Results from Using the Proposed Quantization Values --- p.65Chapter 5.8 --- Summary --- p.66Chapter Chapter 6 : --- Entropy CodingChapter 6.1 --- Introduction --- p.69Chapter 6.1.1 --- Huffman Coding --- p.69Chapter 6.1.2 --- Arithmetic Coding --- p.71Chapter 6.2 --- Zero Run-Length Encoding --- p.73Chapter 6.2.1 --- Variable Length Coding in JPEG --- p.74Chapter 6.2.1.1 --- Coding of the DC Coefficients --- p.74Chapter 6.2.1.2 --- Coding of the DC Coefficients --- p.75Chapter 6.2.2 --- Run-Level Encoding of the Quantized 3D-DCT Coefficients --- p.76Chapter 6.3 --- Frequency Analysis of the Run-Length Patterns --- p.76Chapter 6.3.1 --- The Frequency Distributions of the DC Coefficients --- p.77Chapter 6.3.2 --- The Frequency Distributions of the DC Coefficients --- p.77Chapter 6.4 --- Huffman Table Design --- p.84Chapter 6.4.1 --- DC Huffman Table --- p.84Chapter 6.4.2 --- AC Huffman Table --- p.85Chapter 6.5 --- Implementation Issue --- p.85Chapter 6.5.1 --- Get Category --- p.85Chapter 6.5.2 --- Huffman Encode --- p.86Chapter 6.5.3 --- Huffman Decode --- p.86Chapter 6.5.4 --- PutBits --- p.88Chapter 6.5.5 --- GetBits --- p.90Chapter Chapter 7 : --- "Contributions, Concluding Remarks and Future Work"Chapter 7.1 --- Contributions --- p.92Chapter 7.2 --- Concluding Remarks --- p.93Chapter 7.2.1 --- The Advantages of 3D DCT codec --- p.94Chapter 7.2.2 --- Experimental Results --- p.95Chapter 7.1 --- Future Work --- p.95Chapter 7.2.1 --- Integer Discrete Cosine Transform Algorithms --- p.95Chapter 7.2.2 --- Adaptive Quantization Volume --- p.96Chapter 7.2.3 --- Adaptive Huffman Tables --- p.96Appendices:Appendix A : The detailed steps in the simplification of Equation 4.29 --- p.98Appendix B : The program Listing of the Fast DCT Algorithms --- p.101Appendix C : Tables to Illustrate the Reording of the Quantized Coefficients --- p.110Appendix D : Sample Values of the Quantization Volume --- p.111Appendix E : A 16-bit VLC table for AC Run-Level Pairs --- p.113References --- p.11