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 MD 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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机译：我们考虑在对称均方误差（MSE）失真约束下使用$ K $描述对高斯源进行多描述（MD）编码，并提供速率区域的近似特征。我们表明速率区域可以夹在两个多面体之间，间隙之间可以由取决于描述数量但不受变形约束约束的常数上限限制。该结果的基础是无损多级分集源编码问题的精确表征：与MD问题无损对应。此连接为速率区域的内部和外部边界提供了一个多主题模板。为了建立外边界，我们推广了Ozarow的技术，以通过多个随机变量对原始概率空间进行战略扩展。对于具有任意数量描述的对称速率情况，我们表明，各个描述速率-失真函数的上限和下限之间的差距不大于0.92位。这项工作中得出的结果还表明，将连续精细化量化和无损多级分集编码相结合的“分离”方法是一种竞争性方法，因为它的性能与最佳值之间只有一个常数。结果在MSE失真度量下进一步扩展到一般来源，其中在间隙上存在相似但宽松的约束。
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