DOCSLIB.ORG

Data Compression Using Adaptive Transform Coding

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Data Compression Using Adaptive Transform Coding Data Compression Using Adaptive Transform Coding by Martin C. Rost A DISSERTATION Presented to the Faculty of The Graduate College in the University of Nebraska In Partial Fulfillment of Requirements For the Degree of Doctor of Philosophy Major: Engineering (Electrical Engineering) Under the Supervision of Professor Khalid Sayood Lincoln, Nebraska October, 1988 (NASA-CR-189956) DATA COMPRESSION USING N92-19119 ADAPTIVE TRANSFORM CODING. APPENDIX 1: ITEM 1 Ph.D. Thesis (Nebraska Univ.) 194 p CSCL 098 Unclas G3/61 0072200 This work was supported by the NASA Goddard Space Flight Center under grant NAG-916. ACKNOWLEDGEMENTS I thank all the members of my committee for their participation and helpful comments. I am grateful to my readers, Dean Stanley Liberty and Dr. William Brogan, for their suggestions. Their help has improved the quality of this dissertation. I thank Dr. Jerry Gibson for his comments concerning the presentation of the text and for the time he took from his schedule to come and serve on my committee. Special thanks go to my advisor, Dr. Khalid Sayood. It was his patient guidance that kept this ship afloat. Data Compression Using Adaptive Transform Coding Martin Christopher Rost, Ph.D. University of Nebraska, 1988 Adviser: Khalid Sayood The data of natural images is not stationary, and the coding complexity of images varies from region to region. How well any particular source coding system works is dependent upon its design assumptions and how well these assumptions match the data. In this disser- tation adaptive low-rate source coders are developed. These coders adapt by adjusting the com- plexity of the coder to match the local coding difficulty of the image. This is accomplished by using a threshold driven maximum distortion criterion to select the specific coder used. The different coders are built using variable blocksized transform techniques, and the threshold criterion selects small transform blocks to code the more difficult regions and larger blocks to code the less complex regions. The different coders are interconnected using a quad tree structure. The algorithm that generates the tree and tests the thresholds is independent of the design of block coders. This allows the system designer to select different coders for the different types of images being coded without having to change the entire coding system. A progressive transmission scheme based upon these coders is also developed. Progres- sive coding provides a recognizable image to the viewer in a very short time. As this image is viewed, more data can be received to update the image and improve its visual quality. A set of example systems are constructed to test the feasibility of these systems of source coders. These systems use scalar quantized and vector quantized transform block coders. Some of the systems are extended to include a new modified block truncation coding scheme. They are used to code both monochromatic and color images with good results to rates as low as 0.3 bits/pel. A theoretical framework is constructed from which the study of these coders can be explored, and an algorithm for selecting the optimal bit allocation for the quantization of transform coefficients is developed. The bit allocation algorithm is more fully developed, and can be used to achieve more accurate bit assignments than the algorithms currently used in the literature. Some upper and lower bounds for the bit-allocation distortion-rate function are developed. An obtainable distortion-rate function is developed for a particular scalar quantizer mixing method that can be used to code transform coefficients at any rate is also presented. ACKNOWLEDGEMENTS I thank all the members of my committee for their participation and helpful comments. I am grateful to my readers, Dean Stanley Liberty and Dr. William Brogan, for their suggestions. Their help has improved the quality of this dissertation. I thank Dr. Jerry Gibson for his comments concerning the presentation of the text and for the time he took from his f schedule to come and serve on my committee. Special thanks go to my advisor, Dr. Khalid Sayood. It was his patient guidance that kept this ship afloat. This work was supported by the NASA Goddard Space Flight Center under grant NAG-916. Data Compression Using Adaptive Transform Coding Table of Contents Abstract i List of Figures viii List of Tables x List of Abbreviations xi Chapter 1. Introduction 1 Chapter 2. The Coding and Perception of Images 4 2.1 Modelling overview 2.2 Digital images 2.3 Perceptual considerations 2.3a Spatial, spectral and luminance considerations 2.3b Low-bandwidth temporal considerations Chapter 3. Image Compression Techniques 20 3.1 Measuring data compression and image quality 3.2 Quantization 3.2a Scalar quantization 3.2b Vector quantization 3.3 Transform coding 3.3a KLT and DCT coding 3.4 Block truncation coding Chapter 4. Progressive Transmission of Images 38 4.1 Progressive image coding examples for the literature 4.la Fixed blocksize methods 4.1b Variable blocksize methods Chapter 5. Mixture Block Coding and Mixture Block Coding with Progressive Transmission . 44 JM Design considerations 5. la The largest blocksize 5.1b The smallest blocksize 5.1c Quad tree structure and the MBC coding rate 5.Id The MBC/PT coder and its coding rate 5.2 The MBC and MBC/PT simulators 5.2a The DCT block coders 5.2b Using block truncation coding with MBC 5.2c The distortion measures 5.2d The quantizers 5.3 MBC and MBC/PT computer simulations Chapter 6. A Distortion-rate Function for Transform Coding 123 6.1 The distortion-rate problem 6.2 The transform coefficient distortion-rate problem 6.3 The simplest rate solutions 6.4 The non-negative rate solutions 6.5 Distortion function for constant performance factors 6.6 Distortion function for variable-rate performance factors Chapter 7. A Vector Quantization Distortion-rate Function for MBC and MBC/PT 156 7.1 Block partitioning for vector quantization 7.2 Rate as a function of mixture and thresholds 7.3 Threshold driven distortion-rate function 7.4 Distortion function for MBC 7.5 Distortion function for MBC/PT Chapter 8. Summary and Conclusions 168 Appendix 1. A Simple Bit Allocation Example 171 Appendix 2. Another Approach to the Bit Allocation Problem 173 References 176 List of Figures Figure Page 2.1 Mannos and Sakrison visual perception model. 19 5.1 Example 16x16 block quad tree. 81 5.2 Example 16x16 block for MBC and default sub-block numbering for MBC. 82 5.3 MBC and MBC/PT DCT transform coefficients. 83 5.4 Design and use of the MBC and MBC/PT vector quantizer. 84 5.5 Mixture fractions versus blocksize-constant distortion thresholds. 85 5.6 PSNR versus rate as a function of distortion threshold. 86 5.7 Distortion versus rate as a function of distortion threshold. 87 5.8 MBC mixture fractions versus threshold for BW woman/hat. 88 5.9 MBC PSNR and distortion versus rate for BW woman/hat. 89 5.10 MBC mixture fractions versus threshold for BW F-16. 90 5.11 MBC PSNR and distortion versus rate for BW F-16. 91 5.12 MBC/PT mixture fractions versus threshold for BW woman/hat and F-16. 92 5.13 MBC/PT PSNR versus rate for BW woman/hat and F-16. 93 5.14 MBC/PT distortion versus rate for BW woman/hat and F-16. 94 5.15 MBC mixture fractions versus threshold for YIQ woman/hat and F-16. 95 5.16 MBC PSNR versus rate for YIQ woman/hat and F-16. 96 5.17 MBC distortion versus rate for YIQ woman/hat and F-16. 97 5.18 MBC/PT mixture fractions versus threshold for YIQ woman/hat and F-16. 98 5.19 MBC/PT PSNR versus rate for YIQ woman/hat and F-16. 99 5.20 MBC/PT distortion versus rate for YIQ woman/hat and F-16. 100 5.21 MBC mixture fractions versus threshold for RGB woman/hat and F-16. 101 5.22 MBC PSNR versus rate for RGB woman/hat and F-16. 102 5.23 MBC distortion versus rate for RGB woman/hat and F-16. 103 5.24 MBC/PT mixture fractions versus threshold for RGB woman/hat and F-16. 104 5.25 MBC/PT PSNR versus rate for RGB womwi/hat and F-16. 105 5.26 MBC/PT distortion versus rate for RGB woman/hat and F-16. 106 5.27 256x256 central portion of 512x512 BW woman/hat. 107 5.28 256x256 central portion of 512x512 BW F-16. 108 5.29 256x256 BW USC girl training image. 109 5.30 256x256 BW small face training image. 110 5.31 256x256 BW photo face training image. Ill 5.32 256x256 central portion of MBC woman/hat. 112 5.33 256x256 central portion MBC woman/hat block profile. 113 5.34 256x256 central portion MBC/PT woman/hat block profile. 114 5.35 256x256 central portion of MBC F-16. 115 5.36 256x256 central portion of MBC/BTC F-16. 116 5.37 256x256 central portion of MBC F-16 block profile. 117 5.38 256x256 central portion of MBC/BTC F-16 block profile. 118 5.39 256x256 central portion of first pass MBC/PT woman/hat coded with 16x16 blocks. 119 5.40 256x256 central portion of second pass MBC/PT woman/hat adding 8x8 blocks. 120 5.41 256x256 central portion of third pass MBC/PT woman/hat adding 4x4 blocks. 121 5.42 256x256 central portion of final pass MBC/PT woman/hat adding 2x2 blocks. 122 6.1 Scalar quantizer performance factors.
Load more

Recommended publications
  • 16.1 Digital “Modes”
    Contents 16.1 Digital “Modes” 16.5 Networking Modes 16.1.1 Symbols, Baud, Bits and Bandwidth 16.5.1 OSI Networking Model 16.1.2 Error Detection and Correction 16.5.2 Connected and Connectionless 16.1.3 Data Representations Protocols 16.1.4 Compression Techniques 16.5.3 The Terminal Node Controller (TNC) 16.1.5 Compression vs. Encryption 16.5.4 PACTOR-I 16.2 Unstructured Digital Modes 16.5.5 PACTOR-II 16.2.1 Radioteletype (RTTY) 16.5.6 PACTOR-III 16.2.2 PSK31 16.5.7 G-TOR 16.2.3 MFSK16 16.5.8 CLOVER-II 16.2.4 DominoEX 16.5.9 CLOVER-2000 16.2.5 THROB 16.5.10 WINMOR 16.2.6 MT63 16.5.11 Packet Radio 16.2.7 Olivia 16.5.12 APRS 16.3 Fuzzy Modes 16.5.13 Winlink 2000 16.3.1 Facsimile (fax) 16.5.14 D-STAR 16.3.2 Slow-Scan TV (SSTV) 16.5.15 P25 16.3.3 Hellschreiber, Feld-Hell or Hell 16.6 Digital Mode Table 16.4 Structured Digital Modes 16.7 Glossary 16.4.1 FSK441 16.8 References and Bibliography 16.4.2 JT6M 16.4.3 JT65 16.4.4 WSPR 16.4.5 HF Digital Voice 16.4.6 ALE Chapter 16 — CD-ROM Content Supplemental Files • Table of digital mode characteristics (section 16.6) • ASCII and ITA2 code tables • Varicode tables for PSK31, MFSK16 and DominoEX • Tips for using FreeDV HF digital voice software by Mel Whitten, KØPFX Chapter 16 Digital Modes There is a broad array of digital modes to service various needs with more coming.
    [Show full text]
  • Shannon's Coding Theorems
    Saint Mary's College of California Department of Mathematics Senior Thesis Shannon's Coding Theorems Faculty Advisors: Author: Professor Michael Nathanson Camille Santos Professor Andrew Conner Professor Kathy Porter May 16, 2016 1 Introduction A common problem in communications is how to send information reliably over a noisy communication channel. With his groundbreaking paper titled A Mathematical Theory of Communication, published in 1948, Claude Elwood Shannon asked this question and provided all the answers as well. Shannon realized that at the heart of all forms of communication, e.g. radio, television, etc., the one thing they all have in common is information. Rather than amplifying the information, as was being done in telephone lines at that time, information could be converted into sequences of 1s and 0s, and then sent through a communication channel with minimal error. Furthermore, Shannon established fundamental limits on what is possible or could be acheived by a communication system. Thus, this paper led to the creation of a new school of thought called Information Theory. 2 Communication Systems In general, a communication system is a collection of processes that sends information from one place to another. Similarly, a storage system is a system that is used for storage and later retrieval of information. Thus, in a sense, a storage system may also be thought of as a communication system that sends information from one place (now, or the present) to another (then, or the future) [3]. In a communication system, information always begins at a source (e.g. a book, music, or video) and is then sent and processed by an encoder to a format that is suitable for transmission through a physical communications medium, called a channel.
    [Show full text]
  • Information Theory and Coding
    Information Theory and Coding Introduction – What is it all about? References: C.E.Shannon, “A Mathematical Theory of Communication”, The Bell System Technical Journal, Vol. 27, pp. 379–423, 623–656, July, October, 1948. C.E.Shannon “Communication in the presence of Noise”, Proc. Inst of Radio Engineers, Vol 37 No 1, pp 10 – 21, January 1949. James L Massey, “Information Theory: The Copernican System of Communications”, IEEE Communications Magazine, Vol. 22 No 12, pp 26 – 28, December 1984 Mischa Schwartz, “Information Transmission, Modulation and Noise”, McGraw Hill 1990, Chapter 1 The purpose of telecommunications is the efficient and reliable transmission of real data – text, speech, image, taste, smell etc over a real transmission channel – cable, fibre, wireless, satellite. In achieving this there are two main issues. The first is the source – the information. How do we code the real life usually analogue information, into digital symbols in a way to convey and reproduce precisely without loss the transmitted text, picture etc.? This entails proper analysis of the source to work out the best reliable but minimum amount of information symbols to reduce the source data rate. The second is the channel, and how to pass through it using appropriate information symbols the source. In doing this there is the problem of noise in all the multifarious ways that it appears – random, shot, burst- and the problem of one symbol interfering with the previous or next symbol especially as the symbol rate increases. At the receiver the symbols have to be guessed, based on classical statistical communications theory. The guessing, unfortunately, is heavily influenced by the channel noise, especially at high rates.
    [Show full text]
  • Source Coding: Part I of Fundamentals of Source and Video Coding Full Text Available At
    Full text available at: http://dx.doi.org/10.1561/2000000010 Source Coding: Part I of Fundamentals of Source and Video Coding Full text available at: http://dx.doi.org/10.1561/2000000010 Source Coding: Part I of Fundamentals of Source and Video Coding Thomas Wiegand Berlin Institute of Technology and Fraunhofer Institute for Telecommunications | Heinrich Hertz Institute Germany [email protected] Heiko Schwarz Fraunhofer Institute for Telecommunications | Heinrich Hertz Institute Germany [email protected] Boston { Delft Full text available at: http://dx.doi.org/10.1561/2000000010 Foundations and Trends R in Signal Processing Published, sold and distributed by: now Publishers Inc. PO Box 1024 Hanover, MA 02339 USA Tel. +1-781-985-4510 www.nowpublishers.com [email protected] Outside North America: now Publishers Inc. PO Box 179 2600 AD Delft The Netherlands Tel. +31-6-51115274 The preferred citation for this publication is T. Wiegand and H. Schwarz, Source Coding: Part I of Fundamentals of Source and Video Coding, Foundations and Trends R in Signal Processing, vol 4, nos 1{2, pp 1{222, 2010 ISBN: 978-1-60198-408-1 c 2011 T. Wiegand and H. Schwarz All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, mechanical, photocopying, recording or otherwise, without prior written permission of the publishers. Photocopying. In the USA: This journal is registered at the Copyright Clearance Cen- ter, Inc., 222 Rosewood Drive, Danvers, MA 01923. Authorization to photocopy items for internal or personal use, or the internal or personal use of specific clients, is granted by now Publishers Inc for users registered with the Copyright Clearance Center (CCC).
    [Show full text]
  • Fundamental Limits of Video Coding: a Closed-Form Characterization of Rate Distortion Region from First Principles
    1 Fundamental Limits of Video Coding: A Closed-form Characterization of Rate Distortion Region from First Principles Kamesh Namuduri and Gayatri Mehta Department of Electrical Engineering University of North Texas Abstract—Classical motion-compensated video coding minimum amount of rate required to encode video. On methods have been standardized by MPEG over the the other hand, a communication technology may put years and video codecs have become integral parts of a limit on the maximum data transfer rate that the media entertainment applications. Despite the ubiquitous system can handle. This limit, in turn, places a bound use of video coding techniques, it is interesting to note on the resulting video quality. Therefore, there is a that a closed form rate-distortion characterization for video coding is not available in the literature. In this need to investigate the tradeoffs between the bit-rate (R) paper, we develop a simple, yet, fundamental character- required to encode video and the resulting distortion ization of rate-distortion region in video coding based (D) in the reconstructed video. Rate-distortion (R-D) on information-theoretic first principles. The concept of analysis deals with lossy video coding and it establishes conditional motion estimation is used to derive the closed- a relationship between the two parameters by means of form expression for rate-distortion region without losing a Rate-distortion R(D) function [1], [6], [10]. Since R- its generality. Conditional motion estimation offers an D analysis is based on information-theoretic concepts, elegant means to analyze the rate-distortion trade-offs it places absolute bounds on achievable rates and thus and demonstrates the viability of achieving the bounds derives its significance.
    [Show full text]
  • Beyond Traditional Transform Coding
    Beyond Traditional Transform Coding by Vivek K Goyal B.S. (University of Iowa) 1993 B.S.E. (University of Iowa) 1993 M.S. (University of California, Berkeley) 1995 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering|Electrical Engineering and Computer Sciences in the GRADUATE DIVISION of the UNIVERSITY OF CALIFORNIA, BERKELEY Committee in charge: Professor Martin Vetterli, Chair Professor Venkat Anantharam Professor Bin Yu Fall 1998 Beyond Traditional Transform Coding Copyright 1998 by Vivek K Goyal Printed with minor corrections 1999. 1 Abstract Beyond Traditional Transform Coding by Vivek K Goyal Doctor of Philosophy in Engineering|Electrical Engineering and Computer Sciences University of California, Berkeley Professor Martin Vetterli, Chair Since its inception in 1956, transform coding has become the most successful and pervasive technique for lossy compression of audio, images, and video. In conventional transform coding, the original signal is mapped to an intermediary by a linear transform; the final compressed form is produced by scalar quantization of the intermediary and entropy coding. The transform is much more than a conceptual aid; it makes computations with large amounts of data feasible. This thesis extends the traditional theory toward several goals: improved compression of nonstationary or non-Gaussian signals with unknown parameters; robust joint source–channel coding for erasure channels; and computational complexity reduction or optimization. The first contribution of the thesis is an exploration of the use of frames, which are overcomplete sets of vectors in Hilbert spaces, to form efficient signal representations. Linear transforms based on frames give representations with robustness to random additive noise and quantization, but with poor rate–distortion characteristics.
    [Show full text]
  • 19It501 Information Theory and Coding Techniques
    19IT501 INFORMATION THEORY AND CODING L T P C TECHNIQUES 3 0 0 3 PREAMBLE This course is used to design an application with error control. It is used to help to identify the multimedia concepts. It is one of the most important and direct applications of information theory and also make use of compression and decompression techniques. PREREQUISITE NIL COURSE OUTCOMES At the end of the course learners will be able to CO1 Construct Huffman coding applications Understand CO2 Recall Differential pulse code modulation Apply CO3 Design an application with error–control Apply CO4 Apply the concepts of multimedia communication Analyze CO5 Make Use of compression and decompression techniques Evaluate MAPPING OF COs WITH POs AND PSOs PROGRAMME COURSE PROGRAMME OUTCOMES SPECIFIC OUTCOMES OUTCOMES PO1 PO2 PO3 PO4 PO5 PO6 PO7 PO8 PO9 PO10 PO11 PO12 PSO1 PSO2 CO1 - 2 3 2 3 1 - - - - 3 - 3 - CO2 1 1 3 2 2 1 - - - - - - - - CO3 1 1 3 2 2 1 - - - - 1 - - 1 CO4 1 3 3 1 1 1 - - - - 1 - 3 - CO5 - 2 3 1 3 - - - - - - - 3 - CO6 - 2 3 1 3 - - - - - 2 - - 1 1. LOW 2. MODERATE 3. SUBSTANTIAL CONCEPT MAP Information Theory and Coding Techniques Information Data and voice Error control Compression coding TheoryFundame coding Techniques Modulation ntals Codes Text Differential pulse Syndrome Image Source coding code Decoding Audio Huffman Adaptive Video cyclic codes coding Differential pulse Polynomial & Shannon coding code Parity check Channel coding Delta Polynomial Channel capacity Encoder 1. Source coding Development of Multimedia application SYLLABUS UNIT I INFORMATION ENTROPY FUNDAMENTALS 9 Uncertainty, Information and Entropy – Source coding Theorem – Huffman coding –Shannon Fanocoding – Discrete Memory less channels – channel capacity – channel coding Theorem – Channel capacity Theorem.
    [Show full text]
  • 01 2007 DCC.Pmd
    TPSK31: Getting the Trellis Coded Modulation Advantage A Tutorial on Information Theory, Coding, and Us with an homage to Claude Shannon and Ungerboeck’s genius applied to PSK31. By: Bob McGwier, N4HY As just about everyone who is alive and awake and not suffering from some awful brain disease knows, PSK31 is an extremely popular mode for doing keyboard to keyboard communications. Introduced in the late 1990’s (1,2) it quickly swept the amateur radio world. Its major characteristics are a 31.25 baud differentially encoded PSK signal with keyboard characters encoded into an compression algorithm called varicode by Peter (1) before being modulated with the differentially encoded psk. This differential psk has some serious advantages on HF, including insensitivity to sideband selection, no need for a training sequence to allowing break in decoding to occur and myriad others discussed elsewhere. The slow data rate allows it to use our awful HF channels on 40 and 20 meters effectively and it is more than fast enough to have a good conversation just as we have always done with RTTY. Further, it is very spectrally efficient. If we are going to improve on it, we better not mess with these characteristics. We will not. Peter will be the first person to tell you that he is not a trained communications theorist, engineer, or coding expert. Yet he is one of the great native intelligences in all of amateur radio. His inspirational muse (lend me some!) is one of the best and his perspiration coefficient are among the highest ever measured in amateur radio (this means he is smart and works hard, an extremely good combination).
    [Show full text]
  • Quantum Rate Distortion, Reverse Shannon Theorems, and Source-Channel Separation Nilanjana Datta, Min-Hsiu Hsieh, and Mark M
    1 Quantum rate distortion, reverse Shannon theorems, and source-channel separation Nilanjana Datta, Min-Hsiu Hsieh, and Mark M. Wilde Abstract—We derive quantum counterparts of two key the- of the original data, then the compression is said to be orems of classical information theory, namely, the rate distor- lossless. The simplest example of an information source is tion theorem and the source-channel separation theorem. The a memoryless one. Such a source can be characterized by a rate-distortion theorem gives the ultimate limits on lossy data random variable U with probability distribution pU (u) and compression, and the source-channel separation theorem implies f g that a two-stage protocol consisting of compression and channel each use of the source results in a letter u being emitted with coding is optimal for transmitting a memoryless source over probability pU (u). Shannon’s noiseless coding theorem states a memoryless channel. In spite of their importance in the P that the entropy H (U) u pU (u) log2 pU (u) of such classical domain, there has been surprisingly little work in these an information source is≡ the − minimum rate at which we can areas for quantum information theory. In the present paper, we prove that the quantum rate distortion function is given in compress signals emitted by it [49], [21]. terms of the regularized entanglement of purification. We also The requirement of a data compression scheme being loss- determine a single-letter expression for the entanglement-assisted less is often too stringent a condition, in particular for the quantum rate distortion function, and we prove that it serves case of multimedia data, i.e., audio, video and still images as a lower bound on the unassisted quantum rate distortion or in scenarios where insufficient storage space is available.
    [Show full text]
  • Entropy Coding and Different Coding Techniques
    Journal of Network Communications and Emerging Technologies (JNCET) www.jncet.org Volume 6, Issue 5, May (2016) Entropy Coding and Different Coding Techniques Sandeep Kaur Asst. Prof. in CSE Dept, CGC Landran, Mohali Sukhjeet Singh Asst. Prof. in CS Dept, TSGGSK College, Amritsar. Abstract – In today’s digital world information exchange is been 2. CODING TECHNIQUES held electronically. So there arises a need for secure transmission of the data. Besides security there are several other factors such 1. Huffman Coding- as transfer speed, cost, errors transmission etc. that plays a vital In computer science and information theory, a Huffman code role in the transmission process. The need for an efficient technique for compression of Images ever increasing because the is a particular type of optimal prefix code that is commonly raw images need large amounts of disk space seems to be a big used for lossless data compression. disadvantage during transmission & storage. This paper provide 1.1 Basic principles of Huffman Coding- a basic introduction about entropy encoding and different coding techniques. It also give comparison between various coding Huffman coding is a popular lossless Variable Length Coding techniques. (VLC) scheme, based on the following principles: (a) Shorter Index Terms – Huffman coding, DEFLATE, VLC, UTF-8, code words are assigned to more probable symbols and longer Golomb coding. code words are assigned to less probable symbols. 1. INTRODUCTION (b) No code word of a symbol is a prefix of another code word. This makes Huffman coding uniquely decodable. 1.1 Entropy Encoding (c) Every source symbol must have a unique code word In information theory an entropy encoding is a lossless data assigned to it.
    [Show full text]
  • MIT6 451S05 Fulllecnotes.Pdf
    Chapter 1 Introduction The advent of cheap high-speed global communications ranks as one of the most important developments of human civilization in the second half of the twentieth century. In 1950, an international telephone call was a remarkable event, and black-and-white television was just beginning to become widely available. By 2000, in contrast, an intercontinental phone call could often cost less than a postcard, and downloading large files instantaneously from anywhere in the world had become routine. The effects of this revolution are felt in daily life from Boston to Berlin to Bangalore. Underlying this development has been the replacement of analog by digital communications. Before 1948, digital communications had hardly been imagined. Indeed, Shannon’s 1948 paper 1 [7] may have been the first to use the word “bit.” Even as late as 1988, the authors of an important text on digital communications [5] could write in their first paragraph: Why would [voice and images] be transmitted digitally? Doesn’t digital transmission squander bandwidth? Doesn’t it require more expensive hardware? After all, a voice- band data modem (for digital transmission over a telephone channel) costs ten times as much as a telephone and (in today’s technology) is incapable of transmitting voice signals with quality comparable to an ordinary telephone [authors’ emphasis]. This sounds like a serious indictment of digital transmission for analog signals, but for most applications, the advantages outweigh the disadvantages . But by their second edition in 1994 [6], they were obliged to revise this passage as follows: Not so long ago, digital transmission of voice and video was considered wasteful of bandwidth, and the cost .
    [Show full text]
  • Lossless Source Coding
    Lossless Source Coding Gadiel Seroussi Marcelo Weinberger Information Theory Research Hewlett-Packard Laboratories Palo Alto, California Seroussi/Weinberger – Lossless source coding 26-May-04 1 Notation • A = discrete (usually finite) alphabet • a = | A| = size of A (when finite) n n • x1 = x = x1x2 x3 Kxn = finite sequence over A ¥ ¥ • x1 = x = x1x2 x3 Kxt K = infinite sequence over A j • xi = xi xi+1Kx j = sub-sequence (i sometimes omitted if = 1) • pX(x) = Prob(X=x) = probability mass function (PMF) on A (subscript X and argument x dropped if clear from context) • X ~ p(x) : X obeys PMF p(x) • Ep [F] = expectation of F w.r.t. PMF p (subscript and [ ] may be dropped) n • pˆ n (x) = empirical distribution obtained from x1 x1 • log x = logarithm to base 2 of x, unless base otherwise specified • ln x = natural logarithm of x • H(X), H(p) = entropy of a random variable X or PMF p, in bits; also • H(p) = - p log p - (1-p) log (1-p), 0 £ p £ 1 : binary entropy function • D(p||q) = relative entropy (information divergence) between PMFs p and q Seroussi/Weinberger – Lossless source coding 26-May-04 2 LosslessLossless SourceSource CodingCoding 3. Universal Coding Seroussi/Weinberger – Lossless source coding 26-May-04 3 Universal Modeling and Coding q So far, it was assumed that a model of the data is available, and we aimed at compressing the data optimally w.r.t. the model q By Kraft’s inequality, the models we use can be expressed in terms of probability distributions: - L( s ) For every UD code with length function L(s), we have å2 £ 1 n
    [Show full text]