Additive successive refinement

Rate-distortion bounds for scalable coding, and conditions under which they coincide with nonscalable bounds, have been extensively studied. These bounds have been derived for the general tree-structured refinement scheme, where reproduction at each layer is an arbitrarily complex function of all encoding indexes up to that layer. However, in most practical applications (e.g., speech coding) "additive" refinement structures such as the multistage vector quantizer are preferred due to memory limitations. We derive an achievable region for the additive successive refinement problem, and show via a converse result that the rate-distortion bound of additive refinement is above that of tree-structured refinement. Necessary and sufficient conditions for the two bounds to coincide are derived. These results easily extend to abstract alphabet sources under the condition E{d(X,a)}<∞ for some letter a. For the special cases of square-error and absolute-error distortion measures, and subcritical distortion (where the Shannon lower bound (SLB) is tight), we show that successive refinement without rate loss is possible not only in the tree-structured sense, but also in the additive-coding sense. We also provide examples which are successively refinable without rate loss for all distortion values, but the optimal refinement is not additive.