This article focuses on the compressed representations of the pictures

Lossy coding of speech, high-quality audio, still images, and video is commonplace today. However, in 1948, few lossy compression systems were in service. Shannon introduced and developed the theory of source coding with a fidelity criterion, also called rate-distortion theory. For the first 25 years of its existence, rate-distortion theory had relatively little impact on the methods and systems actually used to compress real sources. Today, however, rate-distortion theoretic concepts are an important component of many lossy compression techniques and standards. We chronicle the development of rate-distortion theory and provide an overview of its influence on the practice of lossy source coding. Index Terms---Data compression, image coding, speech coding, rate distortion theory, signal coding, source coding with a fidelity criterion, video coding. I.

During recent years there has been strong interest in a certain source coding problem, which some authors call the "problem of multiple descriptions". Old and new wringing techniques enable us to establish a single--letter characterization of the rate--distrotion region in the case of no excess rate for the joint description. 1 The Result Since the origin of the problem of multiple descriptiona and motivations for its study have already been described in an extensive literature [1]--[9], we present our result immediately. It goes considerably beyond those of [17], where the reader also will find a detailed discussion of previously known results. We are given the following. 1) A sequence (X t ) 1 t=1 of independent and identically distributed random variables with values in a finite set X , that is, a discrete memoryless source (DMS). 2) Three finite reconstruction spaces X 0 , X 1 , and X 2 , together with associated per-- letter distortion measures d i : X \Theta X i ! R ...

Abstract—We consider transmitting a source across a pair of independent, nonergodic channels with random states (e.g., slow-fading channels) so as to minimize the average distortion. The general problem is unsolved. Hence, we focus on comparing two commonly used source and channel encoding systems which correspond to exploiting diversity either at the physical layer through parallel channel coding or at the application layer through multiple description (MD) source coding. For on–off channel models, source coding diversity offers better performance. For channels with a continuous range of reception quality, we show the reverse is true. Specifically, we introduce a new figure of merit called the distortion exponent which measures how fast the average distortion decays with signal-to-noise ratio. For continuous-state models such as additive white Gaussian noise (AWGN) channels with multiplicative Rayleigh fading, optimal channel coding diversity at the physical layer is more efficient than source coding diversity at the application layer in that the former achieves a better distortion exponent. Finally, we consider a third decoding architecture: MD encoding with joint source–channel decoding. We show that this architecture achieves the same distortion exponent as systems with optimal channel coding diversity for continuous-state channels, and maintains the advantages of MD systems for on–off channels. Thus, the MD system with joint decoding achieves the best performance from among the three architectures considered, on both continuous-state and on–off channels. Index Terms—Application layer diversity, diversity, joint source–channel coding, multiple description coding, parallel channels. I.

Abstract—An achievable region for the-channel multiple description coding problem is presented. This region generalizes twochannel results of El Gamal and Cover and of Zhang and Berger. It further generalizes three-channel results of Gray and Wyner and of Zhang and Berger. A source that is successively refinable on chains is shown to be successively refinable on trees. A new outer bound on the rate-distortion (RD) region for memoryless Gaussian sources with mean squared error distortion is also derived. The achievable region meets this outer bound for certain symmetric cases. Index Terms—Multiple descriptions, rate-distortion (RD) theory, source coding.

A high rate analysis is presented for multiple description quantizers introduced in [1], for rth power distortions and general source densities. Both, fixed length and variable length encoding of the quantizer indices are considered. Optimal companding functions are shown to be the same as for single channel quantizers. As compared to the bound in [2], a gap of 8.69 dB and 3.07 dB exists between the entropy-constrained and level-constrained cases, respectively, for a memoryless Gaussian source and r = 2, Index terms: Source Coding, Quantization, Data Compression, Multiple Descriptions, Diversity Systems. This work was was presented in part at the 1994 IEEE International Symposium on Information Theory, Trondheim, Norway, and was supported by grants NSF NCR-9104566 and NSF NCR9314221. This work was performed while the first author was at Texas A&M University. 1 I Introduction We present an asymptotic analysis of multiple description scalar quantizers (MDSQ's), introduced in [1] fo...

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. We consider the problem of communication over multiple channels in which channels can fail and the information sent over those channels is lost. The goal is to recover as much of the original information as possible while having as little redundancy as possible. The information is encoded into an ordered tuple of indices, each component of which is sent over a separate channel. We analyze the problem of designing an encoding that minimizes the decoding error in case of channel failure, give tight lower bounds and optimal algorithms in the case of two channels, and generalize the results to arbitrary number of channels. Key words. Multichannel communication, diversity systems, quantization, source coding, multiple descriptions, index assignment, algorithm design. AMS(MOS) subject classifications. 94A05, 94A24, 94A40, 68Q25 1 Introduction Consider sending lg m bits of information over k independent channels. We want to partition the lg m bits into k subsets so that if all k subsets...

Witsenhausen's hyperbola bound for the multiple description problem without excess rate in case of a binary source is not right for exact joint reproductions. However, this bound is tight for almost--exact joint reproductions (Theorem 1, conjectured by Witsenhausen). The proof is based on an approximative form of the team guessing lemma for sequences of random variables. (This result may be of interest also for team guessing.) The hyperbola bound is also tight for exact joint reproductions and arbitrarily small, but possible, excess rate (Theorem 2). The proof of this result uses our covering lemma. 1 The Problem of Multiple Descriptions During the last years a strong interest has developed in a certain source--coding problem called the "problem of multiple descriptions". Since the origin of this problem and the motivations for its study have already been extensively described (see [1]--[9]), we begin immediately with the formal setup. Let (X t ) 1 t=1 be a sequence of independent a...