Type-IV DCT, DST, and MDCT algorithms with reduced numbers of arithmetic operations

We present algorithms for the type-IV discrete cosine transform (DCT-IV) and discrete sine transform (DST-IV), as well as for the modified discrete cosine transform (MDCT) and its inverse, 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 ~2NlogN to ~(17/9)NlogN for a power-of-two transform size N, and the exact count is strictly lowered for all N > 4. These results are derived by considering the DCT to be a special case of a DFT of length 8N, 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 DST-IV and MDCT follow immediately from the improved count for the DCT-IV.Comment: 11 page

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

High efficiency block coding techniques for image data.

by Lo Kwok-tung.Thesis (Ph.D.)--Chinese University of Hong Kong, 1992.Includes bibliographical references.ABSTRACT --- p.iACKNOWLEDGEMENTS --- p.iiiLIST OF PRINCIPLE SYMBOLS AND ABBREVIATIONS --- p.ivLIST OF FIGURES --- p.viiLIST OF TABLES --- p.ixTABLE OF CONTENTS --- p.xChapter CHAPTER 1 --- IntroductionChapter 1.1 --- Background - The Need for Image Compression --- p.1-1Chapter 1.2 --- Image Compression - An Overview --- p.1-2Chapter 1.2.1 --- Predictive Coding - DPCM --- p.1-3Chapter 1.2.2 --- Sub-band Coding --- p.1-5Chapter 1.2.3 --- Transform Coding --- p.1-6Chapter 1.2.4 --- Vector Quantization --- p.1-8Chapter 1.2.5 --- Block Truncation Coding --- p.1-10Chapter 1.3 --- Block Based Image Coding Techniques --- p.1-11Chapter 1.4 --- Goal of the Work --- p.1-13Chapter 1.5 --- Organization of the Thesis --- p.1-14Chapter CHAPTER 2 --- Block-Based Image Coding TechniquesChapter 2.1 --- Statistical Model of Image --- p.2-1Chapter 2.1.1 --- One-Dimensional Model --- p.2-1Chapter 2.1.2 --- Two-Dimensional Model --- p.2-2Chapter 2.2 --- Image Fidelity Criteria --- p.2-3Chapter 2.2.1 --- Objective Fidelity --- p.2-3Chapter 2.2.2 --- Subjective Fidelity --- p.2-5Chapter 2.3 --- Transform Coding Theroy --- p.2-6Chapter 2.3.1 --- Transformation --- p.2-6Chapter 2.3.2 --- Quantization --- p.2-10Chapter 2.3.3 --- Coding --- p.2-12Chapter 2.3.4 --- JPEG International Standard --- p.2-14Chapter 2.4 --- Vector Quantization Theory --- p.2-18Chapter 2.4.1 --- Codebook Design and the LBG Clustering Algorithm --- p.2-20Chapter 2.5 --- Block Truncation Coding Theory --- p.2-22Chapter 2.5.1 --- Optimal MSE Block Truncation Coding --- p.2-24Chapter CHAPTER 3 --- Development of New Orthogonal TransformsChapter 3.1 --- Introduction --- p.3-1Chapter 3.2 --- Weighted Cosine Transform --- p.3-4Chapter 3.2.1 --- Development of the WCT --- p.3-6Chapter 3.2.2 --- Determination of a and β --- p.3-9Chapter 3.3 --- Simplified Cosine Transform --- p.3-10Chapter 3.3.1 --- Development of the SCT --- p.3-11Chapter 3.4 --- Fast Computational Algorithms --- p.3-14Chapter 3.4.1 --- Weighted Cosine Transform --- p.3-14Chapter 3.4.2 --- Simplified Cosine Transform --- p.3-18Chapter 3.4.3 --- Computational Requirement --- p.3-19Chapter 3.5 --- Performance Evaluation --- p.3-21Chapter 3.5.1 --- Evaluation using Statistical Model --- p.3-21Chapter 3.5.2 --- Evaluation using Real Images --- p.3-28Chapter 3.6 --- Concluding Remarks --- p.3-31Chapter 3.7 --- Note on Publications --- p.3-32Chapter CHAPTER 4 --- Pruning in Transform Coding of ImagesChapter 4.1 --- Introduction --- p.4-1Chapter 4.2 --- "Direct Fast Algorithms for DCT, WCT and SCT" --- p.4-3Chapter 4.2.1 --- Discrete Cosine Transform --- p.4-3Chapter 4.2.2 --- Weighted Cosine Transform --- p.4-7Chapter 4.2.3 --- Simplified Cosine Transform --- p.4-9Chapter 4.3 --- Pruning in Direct Fast Algorithms --- p.4-10Chapter 4.3.1 --- Discrete Cosine Transform --- p.4-10Chapter 4.3.2 --- Weighted Cosine Transform --- p.4-13Chapter 4.3.3 --- Simplified Cosine Transform --- p.4-15Chapter 4.4 --- Operations Saved by Using Pruning --- p.4-17Chapter 4.4.1 --- Discrete Cosine Transform --- p.4-17Chapter 4.4.2 --- Weighted Cosine Transform --- p.4-21Chapter 4.4.3 --- Simplified Cosine Transform --- p.4-23Chapter 4.4.4 --- Generalization Pruning Algorithm for DCT --- p.4-25Chapter 4.5 --- Concluding Remarks --- p.4-26Chapter 4.6 --- Note on Publications --- p.4-27Chapter CHAPTER 5 --- Efficient Encoding of DC Coefficient in Transform Coding SystemsChapter 5.1 --- Introduction --- p.5-1Chapter 5.2 --- Minimum Edge Difference (MED) Predictor --- p.5-3Chapter 5.3 --- Performance Evaluation --- p.5-6Chapter 5.4 --- Simulation Results --- p.5-9Chapter 5.5 --- Concluding Remarks --- p.5-14Chapter 5.6 --- Note on Publications --- p.5-14Chapter CHAPTER 6 --- Efficient Encoding Algorithms for Vector Quantization of ImagesChapter 6.1 --- Introduction --- p.6-1Chapter 6.2 --- Sub-Codebook Searching Algorithm (SCS) --- p.6-4Chapter 6.2.1 --- Formation of the Sub-codebook --- p.6-6Chapter 6.2.2 --- Premature Exit Conditions in the Searching Process --- p.6-8Chapter 6.2.3 --- Sub-Codebook Searching Algorithm --- p.6-11Chapter 6.3 --- Predictive Sub-Codebook Searching Algorithm (PSCS) --- p.6-13Chapter 6.4 --- Simulation Results --- p.6-17Chapter 6.5 --- Concluding Remarks --- p.5-20Chapter 6.6 --- Note on Publications --- p.6-21Chapter CHAPTER 7 --- Predictive Classified Address Vector Quantization of ImagesChapter 7.1 --- Introduction --- p.7-1Chapter 7.2 --- Optimal Three-Level Block Truncation Coding --- p.7-3Chapter 7.3 --- Predictive Classified Address Vector Quantization --- p.7-5Chapter 7.3.1 --- Classification of Images using Three-level BTC --- p.7-6Chapter 7.3.2 --- Predictive Mean Removal Technique --- p.7-8Chapter 7.3.3 --- Simplified Address VQ Technique --- p.7-9Chapter 7.3.4 --- Encoding Process of PCAVQ --- p.7-13Chapter 7.4 --- Simulation Results --- p.7-14Chapter 7.5 --- Concluding Remarks --- p.7-18Chapter 7.6 --- Note on Publications --- p.7-18Chapter CHAPTER 8 --- Recapitulation and Topics for Future InvestigationChapter 8.1 --- Recapitulation --- p.8-1Chapter 8.2 --- Topics for Future Investigation --- p.8-3REFERENCES --- p.R-1APPENDICESChapter A. --- Statistics of Monochrome Test Images --- p.A-lChapter B. --- Statistics of Color Test Images --- p.A-2Chapter C. --- Fortran Program Listing for the Pruned Fast DCT Algorithm --- p.A-3Chapter D. --- Training Set Images for Building the Codebook of Standard VQ Scheme --- p.A-5Chapter E. --- List of Publications --- p.A-

Best wavelet packet bases in a rate-distortion sense

A fast rate-distortion (R-D) optimal scheme for coding adaptive trees whose individual nodes spawn descendents forming a disjoint and complete basis cover for the space spanned by their parent nodes is presented. The scheme guarantees operation on the convex hull of the operational R-D curve and uses a fast dynamic programing pruning algorithm to markedly reduce computational complexity. Applications for this coding technique include R. Coefman et al.'s (Yale Univ., 1990) generalized multiresolution wavelet packet decomposition, iterative subband coders, and quadtree structures. Applications to image processing involving wavelet packets as well as discrete cosine transform (DCT) quadtrees are presente