Side-information Scalable Source Coding

arXiv:0707.4597v1 [cs.IT] 31 Jul 2007 Side-information Scalable Source Coding Chao Tian, Member, IEEE, Suhas N. Diggavi, Member, IEEE Abstract The problem of side-information scalable (SI-scalable) source coding is considered in this work, where the encoder constructs a progressive description, such that the receiver with high quality side information will be able to truncate the bitstream and reconstruct in the rate distortion sense, while the receiver with low quality side information will have to receive further data in order to decode. We provide inner and outer bounds for general discrete memoryless sources. The achievable region is shown to be tight for the case that either of the decoders requires a lossless reconstruction, as well as the case with degraded deterministic distortion measures. Furthermore we show that the gap between the achievable region and the outer bounds can be bounded by a constant when square error distortion measure is used. The notion of perfectly scalable coding is introduced as both the stages operate on the Wyner-Ziv bound, and necessary and sufficient conditions are given for sources satisfying a mild support condition. Using SI-scalable coding and successive refinement Wyner-Ziv coding as basic building blocks, a complete characterization is provided for the important quadratic Gaussian source with multiple jointly Gaussian side-informations, where the side information quality does not have to be monotonic along the scalable coding order. Partial result is provided for the doubly symmetric binary source with Hamming distortion when the worse side information is a constant, for which one of the outer bound is strictly tighter than the other one. I. INTRODUCTION Consider the following scenario where a server is to broadcast multimedia data to multiple users with different side informations, however the side informations are not available at the server. A user may have such strong side information that only minimal additional information is required from the server to satisfy a fidelity criterion, or a user may have barely any side information and expect the server to provide virtually everything to satisfy a (possibly different) fidelity criterion. A naive strategy is to form a single description and broadcast it to all the users, who can decode only after receiving it completely regardless of the quality of their individual side informations. However, for the users with good-quality side information (who will simply be referred to as the good users), most of the information received is redundant, which introduces a delay caused simply by the existence of users with poor-quality side informations (referred to as the bad users) in the network. It is natural to ask whether an opportunistic method exists, i.e., whether it is possible to construct a two-layer description, such that the good users can decode with only the first layer, and the bad users receive both the first and the second layer to reconstruct. Moreover, it is of importance to investigate whether such a coding order introduces any performance loss. We call this coding strategy side-information scalable (SI-scalable) source coding, since the scalable coding direction is from the 1 Encoder Decoder X 1R 2R 1Y 2 Y 1ˆX 2ˆX Encoder Decoder 1 1R 2R 1Y 2 Y 1ˆX 2ˆX Decoder 2 X Encoder Decoder 1 1R 2R 1Y 2 Y 1ˆX 2ˆX Decoder 2 X 1 2 Y Y X l l 2 1 Y Y X l l Fig. 1. The SR-WZ system vs. the SI-scalable system. good users to the bad users. In this work, we consider mostly two-layer systems, except the quadratic Gaussian source for which the solution to the general multi-layer problem is given. This work is related to the successive refinement problem, where a source is to be encoded in a scalable manner to satisfy different distortion requirement at each individual stage. This problem was studied by Koshelev [1], and by Equitz and Cover [2]; a complete characterization of the rate-distortion region can be found in [3]. Another related problem is the rate-distortion for source coding with side information at the decoder [4], for which Wyner and Ziv provided conclusive result (now widely known as the Wyner-Ziv problem). Steinberg and Merhav [5] recently extended the successive refinement problem in the Wyner-Ziv setting (SR-WZ), when the second stage side information Y2 is better than that of the first stage Y1, in the sense that X ↔Y2 ↔Y1 forms a Markov string. The extension to multistage systems with degraded side informations in such a direction was recently completed in [6]. Also relevant is the work by Heegard and Berger [7] (see also [8]), where the problem of source coding when side information may be present at the decoder was considered; the result was extended to the multistage case when the side informations are degraded. This is quite similar to the problem being considered here and in [5][6], however without the scalable coding requirement. Both the SR-WZ [5][6] and SI-scalable problems can be thought as special cases of

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