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Signal Processing, IEEE Transactions On IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 11, NOVEMBER 2002 2843 An Improvement to Multiple Description Transform Coding Yao Wang, Senior Member, IEEE, Amy R. Reibman, Senior Member, IEEE, Michael T. Orchard, Fellow, IEEE, and Hamid Jafarkhani, Senior Member, IEEE Abstract—A multiple description transform coding (MDTC) comprehensive review of the literature in both theoretical and method has been reported previously. The redundancy rate distor- algorithmic development, see the comprehensive review paper tion (RRD) performance of this coding scheme for the independent by Goyal [1]. and identically distributed (i.i.d.) two-dimensional (2-D) Gaussian source has been analyzed using mean squared error (MSE) as The performance of an MD coder can be evaluated by the re- the distortion measure. At the small redundancy region, the dundancy rate distortion (RRD) function, which measures how MDTC scheme can achieve excellent RRD performance because fast the side distortion ( ) decreases with increasing redun- a small increase in redundancy can reduce the single description dancy ( ) when the central distortion ( ) is fixed. As back- distortion at a rate faster than exponential, but the performance ground material, we first present a bound on the RRD curve for of MDTC becomes increasingly poor at larger redundancies. This paper describes a generalization of the MDTC (GMDTC) scheme, an i.i.d Gaussian source with the MSE as the distortion mea- which introduces redundancy both by transform and through sure. This bound was derived by Goyal and Kovacevic [2] and correcting the error resulting from a single description. Its RRD was translated from the achievable region for multiple descrip- performance is closer to the theoretical bound in the entire range tions, which was previously derived by Ozarow [3]. It can be of redundancy. Analysis both for a single pair of variables and for seen that decays at a super-exponential rate for small and multiple variables is presented. then gradually slows down to an exponential rate, similar to the Index Terms—Error resilience, multiple description coding, rate-distortion (RD) function for the Gaussian source. source coding. As part of the background material, we also review two MD coders developed previously and compare their RRD perfor- I. INTRODUCTION mance with the bound. The first coder, called multiple descrip- tion transform coding (MDTC), was first presented in [4]–[6]. ULTIPLE description (MD) coding addresses the Goyal et al. extended the MDTC idea to consider the genera- problem of encoding a source into two (or more) M tion of more than two descriptions, which they refer to as “gen- bitstreams such that a high-quality reconstruction is decodable eralized multiple description coding” [2], [7], [8]. The basic from the two bitstreams together, while a lower, but still idea of MDTC is to introduce a controlled amount of corre- acceptable, quality reconstruction is decodable if either of the lation between two originally uncorrelated variables, generally two bitstreams is lost. To accomplish this goal, each description with unequal variances, by using a pairwise correlating trans- alone must carry a sufficient amount of information about the form (PCT). The resulting variables are assigned to two sepa- original source. This necessarily means that there is a certain rate streams, which are each then quantized and coded to form amount of shared information and, hence, correlation between one description. The correlation between the two variables en- the two descriptions. This correlation will increase the bit rate ables the estimation of one from the other, but this correlation required to code the two descriptions beyond that required for a also leads to a loss in coding efficiency (i.e., redundancy), com- single bitstream optimized for coding efficiency. The extra bit pared with coding the original two uncorrelated variables. The rate is the redundancy introduced by the MD coder to reduce redundancy can be controlled precisely by a single transform pa- the single description distortion. MD coding was first studied rameter. We show that the RRD function of the MDTC scheme from the information theory point of view, where the goal matches the bound very closely at small redundancies, but at was to find the achievable rate-distortion region for a given the higher redundancy regime, decays slower than exponen- source using multiple descriptions. Since then, various practical tially for increasing and converges to a constant that is half of coders have been proposed to achieve the MD objective. For a the smaller of the variances of the two variables. Manuscript received August 13, 2001; revised June 24, 2002. The associate The second coder, called multiple description layered coding editor coordinating the review of this paper and approving it for publication was (MDLC), is built on top of layered coding [9], [15], [16]. It du- Prof. Sheila S. Hemami. Y. Wang is with Polytechnic University, Brooklyn, NY 11201 USA (e-mail: plicates the base layer bits from a layered coder in both descrip- [email protected]). tions and splits the enhancement layer bits between the two de- A. R. Reibman is with AT&TLabs—Research, Florham Park, NJ 07932-0971 scriptions. Obviously, the redundancy in the MDLC system is USA (e-mail: [email protected]). M. T. Orchard is with the Department of Electrical and Computer Engi- equal to the bit rate used for the base layer. Because redundancy neering, Rice University, Houston, TX 77005 USA (e-mail: [email protected]). bits are exclusively used to protect the more important part of H. Jafarkhani is with the Center for Pervasive Comunications, Department the signal, this scheme is intuitively appealing. We will see that of Electrical and Computer Engineering, University of California, Irvine, CA 92697 USA (e-mail: [email protected]). the RRD function of this approach has an exponential decay rate Digital Object Identifier 10.1109/TSP.2002.804062. in the entire range of the redundancy, such that it is worse than 1053-587X/02$17.00 © 2002 IEEE 2844 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 50, NO. 11, NOVEMBER 2002 MDTC at small redundancies but better at higher redundancies. II. RRD BOUND AND REVIEW OF PREVIOUS WORK At the higher redundancy regime, it is above the bound by a A. RRD Bound for Gaussian Variables factor of two. To circumvent the disadvantages associated with both MDTC First, we review the definition of the RRD function, which and MDLC, we have developed a generalization of the MDTC was introduced in [5]. For a given source, we call a coder that is (GMDTC) scheme,1 which essentially combines the ideas in optimized for coding efficiency a single description (SD) coder. MDTC and MDLC such that it matches the performance of Let represent the base rate needed by the SD coder to achieve MDTC at small redundancies and matches that of MDLC at a distortion . Let represent the rate required by an MD large redundancies. The poor performance of MDTC at high coder to achieve the same distortion (call the central dis- redundancies is because MDTC includes only one variable in tortion) when both descriptions are available. Further, let each description so that the estimation error for the original (called the side distortion) represent the corresponding average two variables cannot be reduced to the two-description distor- distortion when only a single description is available. To reduce tion even at very high redundancy. To overcome this problem, , redundancy must be introduced so that . The ex- with GMDTC, in each description, we include not only one of cess rate is defined as the redundancy. An RRD the transformed variables but some information about the es- function captures the relationship between and for a fixed timation error for the other variable as well. The bits used for or, equivalently, . In this subsection, we present the lower coding the estimation error contribute to a second mode (called bound on the side distortion for an i.i.d. Gaussian source with perp-mode2 ) of redundancy, in addition to the first mode (called the MSE as the distortion measure. We first consider a one-di- transform-mode) introduced by the correlating transform. The mensional (1-D) source and then extend it to a two-dimensional RRD performance of the GMDTC scheme is optimized by allo- (2-D) source. cating a given total redundancy between the two modes to mini- Let and represent the rate and distortion associated mize . An interesting result is that at a total redundancy lower with description 1, 2, and let represent the distortion than a critical point , all the given redundancy should be used from both descriptions. Ozarow [3] derived the necessary and in the transform mode. Beyond this point, the transform should sufficient conditions for a quintuple ( , , , , )tobe only introduce bits of redundancy, and the remaining bits achievable for an i.i.d Gaussian source with unit variance using should be exclusively applied for coding the estimation error. the MSE distortion criterion. Combining this result with the RD Thus, the GMDTC coder corresponds to a transform coder at function for the Gaussian source, one can derive the lower bound low redundancies and a “hybrid” coder at higher redundancies. on the side distortion for given and .For In Section III, we describe the GMDTC scheme for a single the balanced case of , , Goyal and Kovacevic pair of Gaussian variables, discuss how to allocate redundancies [2] derived such a lower bound. For a source with variance , between the two modes of operations, and derive the overall this bound can be written as (1), shown at the bottom of the page, RRD function. We will see that the overall RRD performance where 2 . of GMDTC is much closer to the theoretical RRD bound: It has In this paper, we are primarily concerned with the behavior of a super-exponential decay-rate at smaller redundancies but an the above bound when the central distortion is substantially exponential decay-rate at large redundancies.
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