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Publication# Approximating the Gaussian Multiple Description Rate Region Under Symmetric Distortion

Abstract

We consider multiple description (MD) coding for the Gaussian source with K descriptions under the symmetric mean-squared error (MSE) distortion constraints, and provide an approximate characterization of the rate region. We show that the rate region can be sandwiched between two polytopes, between which the gap can be upper-bounded by constants dependent on the number of descriptions, but independent of the distortion constraints. Underlying this result is an exact characterization of the lossless multilevel diversity source coding problem: a lossless counterpart of the NW problem. This connection provides a polytopic template for the inner and outer bounds to the rate region. In order to establish the outer bound, we generalize Ozarow's technique to introduce a strategic expansion of the original probability space by more than one random variable. For the symmetric rate case with any number of descriptions, we show that the gap between the upper bound and the lower bound for the individual description rate-distortion function is no larger than 0.92 bit. The results developed in this work also suggest that the "separation" approach of combining successive refinement quantization and lossless multilevel diversity coding is a competitive one, since its performance is only a constant away from the optimum. The results are further extended to general sources under the MSE distortion measure, where a similar but looser bound on the gap holds.

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Related concepts (10)

Gaussian function

In mathematics, a Gaussian function, often simply referred to as a Gaussian, is a function of the base form
f(x) = \exp (-x^2)
and with parametric extension

Distortion

In signal processing, distortion is the alteration of the original shape (or other characteristic) of a signal. In communications and electronics it means the alteration of the waveform of an infor

Multivariate normal distribution

In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) no

Successive refinement is a technique to encode a source iteratively, improving the distortion at each step. If the code rates of all steps combined can be as efficient as coding only for the final distortion in one go, then the source is said to be successively refinable. We turn our attention to vector sources and we propose to replace the high-distortion first stage by a projection to low-dimensional space. The goal remains the same: code the original, high-dimensional source afterwards by only a small addendum. We look at Gaussian sources subject to an MSE criterion as a first step in this direction. A coding scheme for this problem is developed by turning the projection into an estimator of the source. It is shown that Gaussian vector sources coded via a projection can be successively refinable. We derive a parametric description of distortion pairs that can achieve this state; this region exists for any projection matrix, but its size may vary.

2014We consider the multiuser successive refinement (MSR) problem, where the users are connected to a central server via links with different noiseless capacities, and each user wishes to reconstruct in a successive-refinement fashion. An achievable region is given for the two-user two-layer case and it provides the complete rate-distortion region for the Gaussian source under the MSE distortion measure. The key observation is that this problem includes multiple description (MD) coding as a subsystem, and the techniques useful in the MD problem can be extended to this case. We show that the coding scheme based on the universality of random binning is sub-optimal, because multiple Gaussian side informations only at the decoders do incur performance loss, in contrast to the case of single side information at the decoder. We further show that unlike the single user case, when there are multiple users, the loss of performance by a multistage coding approach can be unbounded for the Gaussian source. The result suggests that in such a setting, the benefit of using successive refinement is not likely to justify the accompanying performance loss. The MSR problem is also related to the source coding problem where each decoder has its individual side information, while the encoder has the complete set of the side informations. The MSR problem further includes several variations of the MD problem, for which the specialization of the general result is investigated and the implication is discussed.

2007We consider multiple description coding for the Gaussian source with K descriptions under the symmetric mean squared error distortion constraints. One of the main contributions is a novel lower bound on the sum rate, derived by generalizing Ozarow’s well-known technique in treating the Gaussian two-description problem, with the new ingredient of expanding the probability space by more than one auxiliary random variables. Two achievability schemes are investigated in details, the first of which is based on successive refinement coding coupled with multilevel diversity coding (SR-MLD), and the second is a generalization of the multi-layer coding scheme proposed by Puri et al.. Moreover, to make the inner and outer bounds compatible, method similar to the enhancement technique in establishing the Gaussian multiple-input multiple-output broadcast channel capacity is used. Comparison between the lower and upper bounds for the Gaussian symmetric-rate individual-description R-D function shows that they are less than a constant away from each other. The bounding constant depends only on the number of descriptions, but not the distortion constraints. Moreover, regardless of the number of descriptions, this gap between the lower bound and the upper bound using the SR-MLD coding scheme is less than 1.5 bits, and for the other case, the gap is less than 1 bit. The results are further extended to general sources, where a similar but looser bound on the gap holds. Generalizing the proof technique for lower bounding the sum rate provides an outer bound for the R-D region. Since the two achievable regions, which have a geometry structure exactly the same as the rate region of the symmetric lossless multilevel diversity coding, are specially tailored for easy comparison with the outer bound, we are able to show the following results. For the case K = 3, the inner and outer bounds can both be represented by ten planes with matching normal directions, and the pairwise difference between them is small. This result is further generalized to K-description case by incorporating the α-resolution method, which yields upper and lower bounds on any bounding plane of the R-D region, and thus provides an approximate characterization of the R-D region.

2008