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Library of Congress Cataloging-in-Publication Data Dragotti, Pier Luigi. Distributed source coding :theory,algorithms, and applications / Pier Luigi Dragotti, Michael Gastpar. p. cm. Includes index. ISBN 978-0-12-374485-2 (hardcover :alk. paper) 1. Data compression (Telecommunication) 2. Multisensor data fusion. 3. Coding theory. 4. Electronic data processing–Distributed processing. I. Gastpar,Michael. II. Title. TK5102.92.D57 2009 Ј 621.382 16–dc22 2008044569
British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library
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Typeset by: diacriTech, India. List of Contributors
Chapter 1. Foundations of Distributed Source Coding Krishnan Eswaran Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720
Michael Gastpar Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720
Chapter 2. Distributed Transform Coding Varit Chaisinthop Department of Electrical and Electronic Engineering Imperial College, London SW7 2AZ London, UK
Pier Luigi Dragotti Department of Electrical and Electronic Engineering Imperial College, London SW7 2AZ London, UK
Chapter 3. Quantization for Distributed Source Coding David Rebollo-Monedero Department of Telematics Engineering Universitat Politècnica de Catalunya 08034 Barcelona, Spain
Bernd Girod Department of Electrical Engineering Stanford University Palo Alto, CA 94305-9515
xiii xiv List of Contributors
Chapter 4. Zero-error Distributed Source Coding Ertem Tuncel Department of Electrical Engineering University of California, Riverside Riverside, CA 92521
Jayanth Nayak Mayachitra, Inc. Santa Barbara, CA 93111
Prashant Koulgi Department of Electrical and Computer Engineering University of California, Santa Barbara Santa Barbara, CA 93106
Kenneth Rose Department of Electrical and Computer Engineering University of California, Santa Barbara Santa Barbara, CA 93106
Chapter 5. Distributed Coding of Sparse Signals Vivek K Goyal Department of Electrical Engineering and Computer Science Massachusetts Institute of Technology Cambridge, MA 02139
Alyson K. Fletcher Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720
Sundeep Rangan Qualcomm Flarion Technologies Bridgewater, NJ 08807-2856
Chapter 6. Toward Constructive Slepian–Wolf Coding Schemes Christine Guillemot INRIA Rennes-Bretagne Atlantique Campus Universitaire de Beaulieu 35042 Rennes Cédex, France List of Contributors xv
Aline Roumy INRIA Rennes-Bretagne Atlantique Campus Universitaire de Beaulieu 35042 Rennes Cédex, France
Chapter 7. Distributed Compression in Microphone Arrays Olivier Roy Audiovisual Communications Laboratory School of Computer and Communication Sciences Ecole Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland
Thibaut Ajdler Audiovisual Communications Laboratory School of Computer and Communication Sciences Ecole Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland
Robert L. Konsbruck Audiovisual Communications Laboratory School of Computer and Communication Sciences Ecole Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland
Martin Vetterli Audiovisual Communications Laboratory School of Computer and Communication Sciences Ecole Polytechnique Fédérale de Lausanne CH-1015 Lausanne, Switzerland and Department of Electrical Engineering and Computer Sciences University of California, Berkeley Berkeley, CA 94720
Chapter 8. Distributed Video Coding: Basics, Codecs, and Performance Fernando Pereira Instituto Superior Técnico—Instituto de Telecomunicações 1049-001 Lisbon, Portugal xvi List of Contributors
Catarina Brites Instituto Superior Técnico—Instituto de Telecomunicações 1049-001 Lisbon, Portugal
João Ascenso Instituto Superior Técnico—Instituto de Telecomunicações 1049-001 Lisbon, Portugal
Chapter 9. Model-based Multiview Video Compression Using Distributed Source Coding Principles Jayanth Nayak Mayachitra, Inc. Santa Barbara, CA 93111
Bi Song Department of Electrical Engineering University of California, Riverside Riverside, CA 92521
Ertem Tuncel Department of Electrical Engineering University of California, Riverside Riverside, CA 92521
Amit K. Roy-Chowdhury Department of Electrical Engineering University of California, Riverside Riverside, CA 92521
Chapter 10. Distributed Compression of Hyperspectral Imagery Ngai-Man Cheung Signal and Image Processing Institute Department of Electrical Engineering University of Southern California Los Angeles, CA 90089-2564
Antonio Ortega Signal and Image Processing Institute Department of Electrical Engineering University of Southern California Los Angeles, CA 90089-2564 List of Contributors xvii
Chapter 11. Securing Biometric Data Anthony Vetro Mitsubishi Electric Research Laboratories Cambridge, MA 02139
Shantanu Rane Mitsubishi Electric Research Laboratories Cambridge, MA 02139
Jonathan S. Yedidia Mitsubishi Electric Research Laboratories Cambridge, MA 02139
Stark C. Draper Department of Electrical and Computer Engineering University of Wisconsin, Madison Madison, WI 53706 Introduction
In conventional source coding,a single encoder exploits the redundancy of the source in order to perform compression. Applications such as wireless sensor and camera networks, however, involve multiple sources often separated in space that need to be compressed independently. In such applications, it is not usually feasible to first transport all the data to a central location and compress (or further process) it there. The resulting source coding problem is often referred to as distributed source coding (DSC). Its foundations were laid in the 1970s, but it is only in the current decade that practical techniques have been developed,along with advances in the theoretical underpinnings.The practical advances were,in part,due to the rediscovery of the close connection between distributed source codes and (standard) error-correction codes for noisy channels. The latter area underwent a dramatic shift in the 1990s, following the discovery of turbo and low-density parity-check (LDPC) codes. Both constructions have been used to obtain good distributed source codes. In a related effort,ideas from distributed coding have also had considerable impact on video compression, which is basically a centralized compression problem. In this scenario, one can consider a compression technique under which each video frame must be compressed separately,thus mimicking a distributed coding problem. The resulting algorithms are among the best-performing and have many additional features,including,for example,a shift of complexity from the encoder to the decoder. This book summarizes the main contributions of the current decade.The chapters are subdivided into two parts. The first part is devoted to the theoretical foundations, and the second part to algorithms and applications. Chapter 1, by Eswaran and Gastpar, summarizes the state of the art of the theory of distributed source coding, starting with classical results. It emphasizes an impor- tant distinction between direct source coding and indirect (or noisy) source coding: In the distributed setting, these two are fundamentally different. This difference is best appreciated by considering the scaling laws in the number of encoders: In the indirect case, those scaling laws are dramatically different. Historically, compression is tightly linked to transforms and thus to transform coding. It is therefore natural to investigate extensions of the traditional centralized transform coding paradigm to the distributed case. This is done by Chaisinthop and Dragotti in Chapter 2, which presents an overview of existing distributed transform coders. Rebollo-Monedero and Girod, in Chapter 3, address the important question of quantization in a distributed setting. A new set of tools is necessary to optimize quantizers, and the chapter gives a partial account of the results available to date. In the standard perspective, efficient distributed source coding always involves an error probability,even though it vanishes as the coding block length is increased. In Chapter 4,Tuncel, Nayak, Koulgi, and Rose take a more restrictive view: The error probability must be exactly zero.This is shown
Distributed Source Coding: Theory, Algorithms, and Applications Copyright © 2008 by Academic Press, Inc. All rights of reproduction in any form reserved. xix xx Introduction
to lead to a strict rate penalty for many instances. Chapter 5, by Goyal, Fletcher, and Rangan, connects ideas from distributed source coding with the sparse signal models that have recently received considerable attention under the heading of compressed (or compressive) sensing. The second part of the book focuses on algorithms and applications, where the developments of the past decades have been even more pronounced than in the the- oretical foundations.The first chapter,by Guillemot and Roumy,presents an overview of practical DSC techniques based on turbo and LDPC codes,along with ample exper- imental illustration. Chapter 7, by Roy, Ajdler, Konsbruck, and Vetterli, specializes and applies DSC techniques to a system of multiple microphones, using an explicit spatial model to derive sampling conditions and source correlation structures. Chap- ter 8, by Pereira, Brites, and Ascenso, overviews the application of ideas from DSC to video coding: A single video stream is encoded, frame by frame, and the encoder treats past and future frames as side information when encoding the current frame.The chapter starts with an overview of the original distributed video coders from Berkeley (PRISM) and Stanford,and provides a detailed description of an enhanced video coder developed by the authors (and referred to as DISCOVER). The case of the multiple multiview video stream is considered by Nayak, Song, Tuncel, and Roy-Chowdhury in Chapter 9,where they show how DSC techniques can be applied to the problem of multiview video compression. Chapter 10, by Cheung and Ortega, applies DSC tech- niques to the problem of distributed compression of hyperspectral imagery. Finally, Chapter 11, by Vetro, Draper, Rane, and Yedidia, is an innovative application of DSC techniques to securing biometric data.The problem is that if a fingerprint,iris scan,or genetic code is used as a user password, then the password cannot be changed since users are stuck with their fingers (or irises,or genes).Therefore,biometric information should not be stored in the clear anywhere. This chapter discusses one approach to this problematic issue, using ideas from DSC. One of the main objectives of this book is to provide a comprehensive reference for engineers, researchers, and students interested in distributed source coding. Results on this topic have so far appeared in different journals and conferences. We hope that the book will finally provide an integrated view of this active and ever evolving research area. Edited books would not exist without the enthusiasm and hard work of the contributors. It has been a great pleasure for us to interact with some of the very best researchers in this area who have enthusiastically embarked in this project and have contributed these wonderful chapters. We have learned a lot from them. We would also like to thank the reviewers of the chapters for their time and for their constructive comments. Finally we would like to thank the staff at Academic Press—in particular Tim Pitts, Senior Commissioning Editor,and Melanie Benson—for their continuous help.
Pier Luigi Dragotti, London, UK Michael Gastpar, Berkeley, California, USA CHAPTER Foundations of Distributed Source Coding 1
Krishnan Eswaran and Michael Gastpar Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA
CHAPTER CONTENTS Introduction ...... 3 Centralized Source Coding ...... 4 Lossless Source Coding ...... 4 Lossy Source Coding ...... 5 Lossy Source Coding for Sources with Memory ...... 8 Some Notes on Practical Considerations ...... 9 Distributed Source Coding ...... 9 Lossless Source Coding ...... 10 Lossy Source Coding ...... 11 Interaction ...... 15 Remote Source Coding...... 16 Centralized ...... 16 Distributed: The CEO Problem ...... 19 Joint Source-channel Coding ...... 22 Acknowledgments...... 23 APPENDIX A: Formal Definitions and Notations ...... 23 Notation...... 23 Centralized Source Coding ...... 25 Distributed Source Coding ...... 26 Remote Source Coding ...... 27 References ...... 28
Distributed Source Coding: Theory, Algorithms, and Applications Copyright © 2008 by Academic Press, Inc. All rights of reproduction in any form reserved. 3 4 CHAPTER 1 Foundations of Distributed Source Coding
1.1 INTRODUCTION Data compression is one of the oldest and most important signal processing ques- tions. A famous historical example is the Morse code, created in 1838, which gives shorter codes to letters that appear more frequently in English (such as “e” and “t”). A powerful abstraction was introduced by Shannon in 1948 [1]. In his framework, the original source information is represented by a sequence of bits (or, equivalently, by one out of a countable set of prespecified messages). Classically, all the informa- tion to be compressed was available in one place, leading to centralized encoding problems. However, with the advent of multimedia, sensor, and ad-hoc networks, the most important compression problems are now distributed: the source information appears at several separate encoding terminals. Starting with the pioneering work of Slepian and Wolf in 1973, this chapter provides an overview of the main advances of the last three and a half decades as they pertain to the fundamental performance bounds in distributed source coding. A first important distinction is lossless versus lossy compression, and the chapter provides closed-form formulas wherever possi- ble. A second important distinction is direct versus remote compression; in the direct compression problem, the encoders have direct access to the information that is of interest to the decoder,while in the remote compression problem, the encoders only access that information indirectly through a noisy observation process (a famous exam- ple being the so-called CEO problem). An interesting insight discussed in this chapter concerns the sometimes dramatic (and perhaps somewhat unexpected) performance difference between direct and remote compression. The chapter concludes with a short discussion of the problem of communicating sources across noisy channels,and thus, Shannon’s separation theorem.
1.2 CENTRALIZED SOURCE CODING 1.2.1 Lossless Source Coding The most basic scenario of source coding is that of describing source output sequences with bit strings in such a way that the original source sequence can be recovered without loss from the corresponding bit string. One can think about this scenario in two ways. First, one can map source realizations to binary strings of different lengths and strive to minimize the expected length of these codewords. Compres- sion is attained whenever some source output sequences are more likely than others: the likelier sequences will receive shorter bit strings. For centralized source coding (see Figure 1.1), there is a rich theory of such codes (including Huffman codes, Lempel–Ziv codes, and arithmetic codes). However, for distributed source coding, this perspective has not yet been very fruitful.The second approach to lossless source coding is to map L samples of the source output sequence to the set of bit strings of a fixed length N, but to allow a “small” error in the reconstruction. Here,“small” means that the probability of reconstruction error goes to zero as the source sequence length 1.2 Centralized Source Coding 5
S R Sˆ ENC DEC
FIGURE 1.1 Centralized source coding. goes to infinity.The main insight is that it is sufficient to assign bit strings to “typical” source output sequences. One measures the performance of a lossless source code by considering the ratio N/L of the number of bits N of this bit string to the num- ber of source samples L.Anachievable rate RϭN/L is a ratio that allows for an asymptotically small reconstruction error. Formal definitions of a lossless code and an achievable rate can be found in Appendix A (Definitions A.6 and A.7). The central result of lossless source coding is the following:
Theorem 1.1. Given a discrete information source {S(n)}nϾ0, the rate R is achievable via lossless source coding if R ϾHϱ(S), where Hϱ(S) is the entropy (or entropy rate) of the source. Conversely, if RϽHϱ(S), R is not achievable via lossless source coding.
A proof of this theorem for the i.i.d. case and Markov sources is due to Shannon [1]. A proof of the general case can be found, for example, in [2,Theorem 3, p. 757].
1.2.2 Lossy Source Coding In many source coding problems, the available bit rate is not sufficient to describe the information source in a lossless fashion. Moreover, for real-valued sources, loss- less reconstruction is not possible for any finite bit rate. For instance, consider a source whose samples are i.i.d. and uniform on the interval [0, 1]. Consider the binary representation of each sample as the sequence. B1B2 ...; here, each binary digit is independent and identically distributed (i.i.d.) with probability 1/2 of being 0 or 1. Thus, the entropy of any sample is infinite, and Theorem 1.1 implies that no finite rate can lead to perfect reconstruction. Instead, we want to use the available rate to describe the source to within the smallest possible average distortion D, which in turn is determined by a distortion function d(·, ·), a mapping from the source and reconstruction alphabets to the non- negative reals. The precise shape of the distortion function d(·, ·) is determined by the application at hand. A widely studied choice is the mean-squared error, that is, d(s, sˆ)ϭ|s Ϫsˆ|2. It should be intuitively clear that the larger the available rate R, the smaller the incurred distortion D. In the context of lossy source coding, the goal is thus to study the achievable trade-offs between rate and distortion. Formal definitions of a lossless code and achievable rate can be found in Appendix A (Definitions A.8 and A.9). Per- haps somewhat surprisingly, the optimal trade-offs between rate and distortion can 6 CHAPTER 1 Foundations of Distributed Source Coding
be characterized compactly as a“single-letter”optimization problem usually called the rate-distortion function.More formally,we have the following theorem:
Theorem 1.2. Given a discrete memoryless source {S(n)}n Ͼ 0 and bounded distortion ˆ ϩ function d:S ϫS →R , a rate R is achievable with distortion D for R Ͼ RS(D), where ˆ RS(D)ϭ min I(S; S)(1.1) p(sˆ|s): E[d(S,Sˆ)]рD
is the rate distortion function. Conversely, for RϽRS(D), the rate R is not achievable with distortion D.
A proof of this theorem can be found in [3, pp. 349–356]. Interestingly,it can also be shown that when D Ͼ0, RϭRS(D) is achievable [4]. Unlike the situation in the lossless case, determining the rate-distortion func- tion requires one to solve an optimization problem. The Blahut–Arimoto algorithm [5, 6] and other techniques (e.g., [7]) have been proposed to make this computation efficient. WhileTheorem 1.2 is stated for discrete memoryless sources and a bounded distor- tion measure, it can be extended to continuous sources under appropriate technical conditions. Furthermore, one can show that these technical conditions are satisfied for memoryless Gaussian sources with a mean-squared error distortion. This is some- times called the quadratic Gaussian case.Thus,one can use Equation (1.1) inTheorem 1.2 to deduce the following.
Proposition 1.1. Given a memoryless Gaussian source {S(n)}nϾ0 with S(n)∼ N (0, 2) and distortion function d(s, sˆ)ϭ(s Ϫsˆ)2,
2 1 ϩ R (D)ϭ log .(1.2) S 2 D For general continuous sources,the rate-distortion function can be difficult to deter- mine. In lieu of computing the rate-distortion function exactly, an alternative is to find closed-form upper and lower bounds to it. The idea originates with Shannon’s work [8],and it has been shown that under appropriate assumptions,Shannon’s lower bound for difference distortions (d(s, sˆ)ϭf (s Ϫsˆ)) becomes tight in the high-rate regime [9]. { ( )} 2 For a quadratic distortion and memoryless source S n nϾ0 with variance S and entropy power QS, these upper and lower bounds can be expressed as [10, p. 101] 2 1 QS 1 S log рRS(D)р log , ( . ) 2 D 2 D 1 3 where the entropy power is given in Definition A.4. From Table 1.1, one can see that the bounds in (1.3) are tight for memoryless Gaussian sources. 1.2 Centralized Source Coding 7
Table 1.1 Variance and Entropy Power of Common Distributions Source Name Probability Density Function Variance Entropy Power 2 2 Gaussian f (x)ϭ √ 1 eϪ(xϪ) /2 2 2 2e2 ( )ϭ Ϫ|xϪ| 2 e · 2 Laplacian f x 2 e 2 2 1 , Ϫaрx Ϫрa 2a a2 6 a2 Uniform f (x)ϭ · 0, otherwise 3 e 3
S 1 R ENC
ˆ DEC S 1
S 2
FIGURE 1.2 Conditional rate distortion.
The conditional source coding problem (see Figure 1.2) considers the case in which a correlated source is available to the encoder and decoder to potentially decrease the encoding rate to achieve the same distortion. Definitions A.10 and A.11 formalize the problem.
Theorem 1.3. Given a memoryless source S1, memoryless source side information S2 available at the encoder and decoder with the property that S1(k),S2(k) are i.i.d. in k, and ˆ ϩ distortion function d:S1 ϫS1 → , the conditional rate-distortion function is ( )ϭ ( ; ˆ | ). ( . ) RS1|S2 D min I S1 S1 S2 1 4 p(sˆ1|s1,s2): ˆ E[d(S1,S1)]рD A proof of Theorem 1.3 can be found in [11,Theorem 6, p. 11]. Because the rate-distortion theorem gives an asymptotic result as the blocklength gets large, convergence to the rate-distortion function for any finite blocklength has also been investigated. Pilc [4] as well as Omura [12] considered some initial inves- tigations in this direction. Work by Marton established the notion of a source coding error exponent [13], in which she considered upper and lower bounds to the prob- ability that for memoryless sources, an optimal rate-distortion codebook exceeds distortion D as a function of the blocklength. 8 CHAPTER 1 Foundations of Distributed Source Coding
1.2.3 Lossy Source Coding for Sources with Memory We start with an example. Consider a Gaussian source S with S(i)∼N (0, 2) where pairs Y (k)ϭ(S(2kϪ1), S(2k)) have the covariance matrix 21 ⌺ϭ ,(1.5) 12
and Y (k) are i.i.d. over k. The discrete Fourier transform (DFT) of each pair can be written as 1 S˜(2kϪ1)ϭ √ (S(2kϪ1)ϩS(2k)) (1.6) 2 1 S˜(2k)ϭ √ (S(2kϪ1)ϪS(2k)),(1.7) 2 which has the covariance matrix ˜ 30 ⌺ϭ ,(1.8) 01
and thus the source S˜ is independent,with i.i.d. even and odd entries. For squared error distortion, if C is the codeword sent to the decoder, we can express the distortion as ϭ n [( ( )Ϫ [ ( )| ])2] nD iϭ1 E S i E S i C . By linearity of expectation, it is possible to rewrite this as n 2 nD ϭ E S˜(i)ϪE[S˜(i)|C] (1.9) iϭ1 n/2 n/2 2 2 ϭ E S˜(2kϪ1)ϪE[S˜(2kϪ1)|C] ϩ E S˜(2k)ϪE[S˜(2k)|C] . kϭ1 kϭ1 (1.10) Thus, this is a rate-distortion problem in which two independent Gaussian sources of different variances have a constraint on the sum of their mean-squared errors. Sometimes known as the parallel Gaussian source problem,it turns out there is a well- known solution to it called reverse water-filling [3, p. 348,Theorem 13.3.3], which in this case, evaluates to the following: 2 2 1 1 1 2 RS(D)ϭ log ϩ log (1.11) 2 D1 2 D2 , Ͻ2, D ϭ i (1.12) i 2, у2, i i 2 ϭ 2 ϭ ϩ ϭ where 1 3, 2 1, and is chosen so that D1 D2 D. This diagonalization approach allows one to state the following result for stationary ergodic Gaussian sources. 1.3 Distributed Source Coding 9
Proposition 1.2. Let S be a stationary ergodic Gaussian source with autocorrela- tion function E[SnSnϪk]ϭ(k) and power spectral density ϱ Ϫjk ⌽()ϭ (k)e .(1.13) kϭϪϱ Then the rate-distortion function for S under mean-squared error distortion is given by: 1 ⌽() R(D)ϭ max 0, log d (1.14) 4 Ϫ 1 D ϭ min{, ⌽()}d.(1.15) 2 Ϫ PROOF. See Berger [10, p. 112]. ■ While it can be difficult to evaluate this in general,upper and lower bounds can give a better sense for its behavior. For instance,let 2 be the variance of a stationary ergodic Gaussian source. Then a result of Wyner and Ziv [14] shows that the rate-distortion function can be bounded as follows: 1 2 1 2 log Ϫ⌬ рR (D)р log ,(1.16) 2 D S S 2 D where ⌬S is a constant that depends only on the power spectral density of S.
1.2.4 Some Notes on Practical Considerations The problem formulation considered in this chapter focuses on the existence of codes for cases in which the encoder has access to the entire source noncausally and knows its distribution. However,in many situations of practical interest,some of these assump- tions may not hold. For instance,several problems have considered the effects of delay and causal access to a source [15–17]. Some work has also considered cases in which no underlying probabilistic assumptions are made about the source [18–20]. Finally, the work of Gersho and Gray [21] explores how one might actually go about designing implementable vector quantizers.
1.3 DISTRIBUTED SOURCE CODING The problem of source coding becomes significantly more interesting and challenging in a network context. Several new scenarios arise:
■ Different parts of the source information may be available to separate encoding terminals that cannot cooperate. ■ Decoders may have access to additional side information about the source infor- mation; or they may only obtain a part of the description provided by the encoders. 10 CHAPTER 1 Foundations of Distributed Source Coding
We start our discussion by an example illustrating the classical problem of source coding with side information at the decoder.
Example 1.1 Let {S(n)}nϾ0 be a source where source samples S(n) are uniform over an alphabet of size 8, which we choose to think of as binary vectors of length 3. The decoder has access to a ˜ ˜ corrupted version of the source {S(n)}nϾ0 where each sample S(n) takes values in a set of ternary sequences {0, ∗,1}3 of length 3 with the property that ⎧ ⎪ 1 , if (c , c , c )ϭ(b , b , b ) ⎪ 4 1 2 3 1 2 3 ⎪ ⎨ 1 , if (c , c , c )ϭ(∗, b , b ) ˜ 4 1 2 3 2 3 Pr S(n)ϭ(c1, c2, c3) S(n)ϭ(b1, b2, b3) ϭ (1.17) ⎪ 1 , if ( , , )ϭ( , ∗, ) ⎪ 4 c1 c2 c3 b1 b3 ⎩⎪ 1 , ( , , )ϭ( , , ∗) 4 if c1 c2 c3 b1 b2 Thus, the decoder has access to at least two of every three bits per source symbol, but the encoder is unaware of which ones. Consider the partition of the alphabet S into
S1 ϭ (0, 0, 0), S2 ϭ (1, 1, 1),
(0, 1, 1), (1, 0, 0), .(1.18) (1, 1, 0), (0, 0, 1), (1, 0, 1) (0, 1, 0)
If the decoder knows which of these partitions a particular sample S(n)isin,S˜ (n) is sufficient to determine the exact value of S(n). Thus, for each source output, the encoder can use one bit to indicate to which of the two partitions the source sample belongs. Thus, at a rate of 1 bit per source sample, the decoder can perfectly reconstruct the output. However, in the ˜ absence of {S(n)}nϾ0 at the decoder, Theorem 1.1 implies the best possible rate is H(S1)ϭ3 bits per source sample.
Example 1.1 illustrates a simple version of a strategy known as binning. It turns out that binning can be applied more generally and is used to prove many of the results in distributed source coding.
1.3.1 Lossless Source Coding More generally,we now consider the scenario illustrated in Figure 1.3,where two sepa- rate encoding terminals each observe part of the data.That is,with respect to Figure 1.3, the source streams S1 and S2 are dependent on each other.The coding question now involves two separate source codes that appear at rates R1 and R2, respectively,and a receiver where the source codes are jointly decoded. Formal definitions are provided in Appendix A (Definitions A.12 and A.13). Since the sources are dependent, the rates R1 and R2 constrain one another. That is, if more bits are used to describe one of the sources, typically, the number of bits for the other can be reduced. Specifically, if we 1.3 Distributed Source Coding 11
R S 1 1 ENC
ˆˆ DEC S 1, S 2
R S 2 2 ENC
FIGURE 1.3 Distributed source coding problem. assume R2 Ͼlog |S2|, we can assume that the decoder knows S2 without error, and thus this problem also includes the special case of side information at the decoder. Theorem 1.4. Given discrete memoryless sources S and S , define R as 1 2
Rϭ (R1, R2):R1 ϩR2 уH(S1, S2),
R1 уH(S1|S2), R2 уH(S2|S1) .(1.19)
0 0 Furthermore, let R be the interior of R. Then (R1,R2)∈R are achievable for the two- terminal lossless source coding problem, and (R1,R2)/∈R are not. The proof of this result was shown by Slepian and Wolf [22]; to show achievability involves a random binning argument reminiscent of Example 1.1. However,by contrast to that example, the encoders now bin over the entire vector of source symbols, and they only get probabilistic guarantees that successful decoding is possible. Variations and extensions of the lossless coding problem have been considered by Ahlswede and Körner [23], who examine a similar setup case in which the decoder is only interested in S1;Wyner [24], who considers a setup in which one again wants to reconstruct sources S1 and S2, but there is now an additional encoding terminal with access to another correlated information source S3;and Gel’fand and Pinsker [25],who consider perfect reconstruction of a source to which each encoding terminal has only a corrupted version. In all of these cases, a random binning strategy can be used to establish optimality. However,one notable exception to this is a paper by Körner and Marton [26], which shows that when one wants to reconstruct the modulo-2 sum of correlated sources, there exists a strategy that performs better than binning.
1.3.2 Lossy Source Coding By analogy to the centralized compression problem, it is again natural to study the problem where instead of perfect recovery, the decoder is only required to provide estimates of the original source sequences to within some distortions. Reconsidering Figure 1.3,we now ask that the source S1 be recovered at distortion D1, when assessed with distortion measure d1(·, ·), and S2 at distortion D2, when assessed with distortion 12 CHAPTER 1 Foundations of Distributed Source Coding
measure d2(·, ·). The question is again to determine the necessary rates, R1 and R2, respectively, as well as the coding schemes that permit satisfying the distortion con- straints. Formal definitions are provided in Appendix A (Definitions A.14 and A.15). For this problem,a natural achievable strategy arises. One can first quantize the sources at each encoder as in the centralized lossy coding case and then bin the quantized val- ues as in the distributed lossless case. The work of Berger and Tung [27–29] provides an elegant way to combine these two techniques that leads to the following result. The result was independently discovered by Housewright [30].
Theorem 1.5: “Quantize-and-bin.” Given sources S1 and S2 with distortion functions dk : ϩ Sk ϫUk → , k∈{1, 2}, the achievable rate-distortion region includes the following set: Rϭ{(R, D):∃U1, U2 s.t. U1 ϪS1 ϪS2 ϪU2,
E[d1(S1, U1)]рD1, E[d2(S2, U2)]рD2,(1.20)
R1 ϾI(S1; U1|U2), R2 ϾI(S2; U2|U1),
R1 ϩR2 ϾI(S1, S2; U1U2)}. A proof for the setting of more than two users is given by Han and Kobayashi [31,Theorem 1, pp. 280–284]. A major open question stems from the optimality of the “quantize-and-bin” achievable strategy. While work by Servetto [32] suggests it may be tight for the two-user setting, the only case for which it is known is the quadratic Gaussian setting, which is based on an outer bound developed by Wagner and Anantharam [33, 34].
Theorem 1.6: Wagner, Tavildar, and Viswanath [35]. Given sources S1 and S2 that are jointly Gaussian and i.i.d. in time, that is, S1(k),S2(k) ∼N (0,⌺), with covariance matrix 2 ⌺ϭ 1 1 2 ( . ) 2 1 21 1 2 2
for D1, D2 Ͼ0 define R(D1,D2)as Ϫ ( Ϫ2 ϩ2 2R2 )2 1 ϩ 1 2 1 R(D1, D2)ϭ (R1, R2):R1 у log , 2 D1 2 2 Ϫ2R 2 1 ϩ (1Ϫ ϩ 2 1 ) R у log 2 , 2 2 D 2 ( Ϫ2)22( , ) 1 ϩ 1 1 2 D1 D2 R1 ϩR2 у log ,(1.22) 2 2D1D2 where 42D D (D , D )ϭ1ϩ 1ϩ 1 2 .(1.23) 1 2 ( Ϫ2)222 1 1 2 0 Furthermore, let R be the interior of R. Then for distortions D1, D2 Ͼ0, (R1, R2)∈ 0 R (D1, D2) are achievable for the two-terminal quadratic Gaussian source coding problem and (R1, R2)/∈R(D1, D2) are not. 1.3 Distributed Source Coding 13
R S 1 1 ENC
ˆ DEC S 1
S 2
FIGURE 1.4 The Wyner–Ziv source coding problem.
In some settings,the rates given by the“quantize-and-bin”achievable strategy can be shown to be optimal. For instance, consider the setting in which the second encoder has an unconstrained rate link to the decoder, as in Figure 1.4. This configuration is often referred to as the Wyner–Ziv source coding problem.
Theorem 1.7. Given a discrete memoryless source S1, discrete memoryless side informa- tion source S2 with the property that (S1(k),S2(k)) are i.i.d. over k, and bounded distortion function d : S ϫU →ϩ, a rate R is achievable with lossy source coding with side information at the decoder and with distortion D if R Ͼ RWZ (D). Here S1|S2 WZ ( )ϭ ( ; | )(. ) RS |S D min I S1 U S2 1 24 1 2 p(u|s): UϪS1ϪS2 E[d(S1,U)]рD is the rate distortion function for side information at the decoder. Conversely, for R Ͻ RWZ (D), the rate R is not achievable with distortion D. S1|S2 Theorem 1.7 was first proved by Wyner and Ziv [36]. An accessible summary of the proof is given in [3,Theorem 14.9.1, pp. 438–443]. The result can be extended to continuous sources and unbounded distortion mea- sures under appropriate regularity conditions [37]. It turns out that for the quadratic Gaussian case, that is, jointly Gaussian source and side information with a mean- squared error distortion function, these regularity conditions hold, and one can char- acterize the achievable rates as follows. Note the correspondence between this result and Theorem 1.6 as R2 →ϱ.
Proposition 1.3. Consider a source S1 and side information source S2 such that (S (k), S (k))∼N (0, ⌺) are i.i.d. in k, with 1 2 1 ⌺ϭ2 (1.25) 1 2 Then for distortion function d(s1, u)ϭ(s1 Ϫu) and for D Ͼ0, ( Ϫ2)2 WZ 1 ϩ 1 R | (D)ϭ log .(1.26) S1 S2 2 D 14 CHAPTER 1 Foundations of Distributed Source Coding
1.3.2.1 Rate Loss Interestingly,in this Gaussian case,even if the encoder in Figure 1.4 also had access to the source X2 as in the conditional rate-distortion problem,the rate-distortion function would still be given by Proposition 1.3. Generally, however, there is a penalty for the absence of the side information at the encoder. A result of Zamir [38] has shown that for memoryless sources with finite variance and a mean-squared error distortion, the 1 rate-distortion function provided in Theorem 1.7 can be no larger than 2 bit/source sample than the rate-distortion function given in Theorem 1.3. It turns out that this principle holds more generally. For centralized source coding of a length M zero-mean vector Gaussian source {S(n)}nϾ0 with covariance matrix ⌺ 2 (M),...,(M) S having diagonal entries S and eigenvalues 1 M with the property that (M) m у for some Ͼ0 and all m, and squared error distortion, the rate-distortion function is given by [10]