On the rate-distortion region for multiple descriptions

In this paper, we study the problem of source coding with multiple descriptions, which is described as follows. Let X be a discrete memoryless source. There are two encoders, Encoders 1 and 2, and three decoders, Decoders 0, 1, and 2. Encoders 1 and 2 describe the source X at respective rates R-1 and R-2. Decoder 1 receives the output of Encoder 1 only, and it can recover X with distortion D-1. Decoder 2 receives the output of Encoder 2 only, and it can recover X with distortion D-2. Decoder 0 receives the outputs of both Encoders 1 and 2, and it can recover X with distortion D-0. We show that if Decoder 2 (or Decoder 1) is required to recover a function of the source X perfectly in the usual Shannon sense, the El Gamal-Cover inner bound on the rate distortion region is tight. This finding subsumes the Rimoldi rate-distortion region for successive refinement of information, the Kaspi rate-distortion function when side information may be present at the decoder, and the El Gamal-Cover achievable rate region for multiple descriptions with deterministic distortion measures. We have also obtained a new outer bound on the rate-distortion region which enhances the outer bound due to Witsenhausen and Wyner. This new outer bound implies some interesting facts regarding the achievable rate-distortion vectors. Finally, we pose a multilevel diversity source coding problem for further study.