On Successive Refinement for the Kaspi/Heegard-Berger Problem
Consider a source that produces independent copies of a triplet of jointly distributed random variables {X(i), Y(i), Z(i)}(i=1)(infinity). The process {X(i)} is observed at the encoder, and is supposed to be reproduced at two decoders, decoder Y and decoder Z, where {Y(i)} and {Z(i)} are observed, respectively, in either a causal or noncausal manner. The communication between the encoder and the decoders is carried in two successive stages. In the first stage, the transmission is available to both decoders and they reconstruct the source according to the received bit-stream and the individual side information ({Z(i)} or {Y(i)}). In the second stage, additional information is sent to both decoders and they refine the reconstructions of the source according to the available side information and the transmissions at both stages. It is desired to find the necessary and sufficient conditions on the communication rates between the encoder and decoders, so that the distortions incurred (at each stage) will not exceed given thresholds. For the case of causal availability of side information at the decoders, an exact single-letter characterization of the achievable region is derived for the case of pure source-coding. Then, for the case of communication between the encoder and decoders carried over independent memoryless discrete channels with random states known causally/noncausally at the encoder and with causal side information about the source at the decoders, a single-letter characterization of all achievable distortion in both stages is provided and it is shown that the separation theorem holds. The results are derived without assuming any structural restrictions on side information, such as a Markov structure, etc. Finally, for noncausal degraded side information, inner and outer bounds to the achievable rate-distortion region are derived. These bounds are shown to be tight for certain cases of reconstruction requirements at the decoders. Due to the system setup, these results also shed some light on a problem of successive refinement with side information which is not degraded in the usual sense.
successive refinement separation theorem causal/noncausal side information channel capacity degraded side-information joint source-channel coding source coding