On Successive Refinement for the Kaspi/Heegard-Berger Problem

Abstract

Consider a source, {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> , Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i=1</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">infin</sup> , producing independent copies of a triplet of jointly distributed random variables (RVs). The {X <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } part of the process is observed at the encoder, and is supposed to be reproduced at two decoders, decoder 1 and decoder 2 , where the {Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } and the {Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } parts of the process are observed, respectively, in either a causal or a non-causal manner. The communication between the encoder and the decoders is carried in two successive communication stages. In the first stage, the transmission is available to both decoders and the decoders reconstruct the source according to the received stream and individual side information ({Z <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> } or {Y <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> }). In the second stage, additional information is sent to both decoders and the decoders refine the reconstruction of the source according to the side information available to it and the transmissions in both stages. It is desired to find the necessary and sufficient conditions on communication between the encoder and decoders, so that the distortions incurred (at each stage) will not exceed given thresholds. For such a multi-decoder coding setting with successive refinement and non-causal degraded side information, we derive inner and outer bounds to the achievable rate-distortion region. Then, for the case of general causal side information at the decoders, we derive a single-letter characterization of the achievable region for a multi-decoder source-coding problem with successive refinement.