Stealing bits from a quantized source

We consider "bit stealing" scenarios where the rate of a source code must be reduced without prior planning. We first investigate the efficiency of source requantization to reduce rate, which we term successive degradation. We focus on finite-alphabet sources with arbitrary distortion measures as well as the Gaussian-quadratic and high-resolution scenarios. We show an achievable rate-distortion tradeoff and prove that this is the best guaranteeable tradeoff for any good source code. This tradeoff is in general different from the rate-distortion tradeoff with successive refinement, where there is prior planning. But, we show that with quadratic distortion measures, for all sources with finite differential entropy and at least one finite moment, the gap is at most 1/2 bit or 3 dB in the high-resolution limit. In the Gaussian-quadratic case, the gap is at most 1/2 bit for all resolutions. We further consider bit stealing in the form of information embedding, whereby an embedder acts on a quantized source and produces an output at the same rate and in the original source codebook. We develop achievable distortion-rate tradeoffs. Two cases are considered, corresponding to whether or not the source decoder is informed of the embedding rate. In the Gaussian-quadratic case, we show the informed decoder need only augment the regular decoder with simple post-reconstruction distortion compensation in the form of linear scaling for the resulting system to be as efficient as bit stealing via successive degradation. Finally, we show that the penalty for uninformed versus informed decoders is at most 3 dB or 0.21-bit in the Gaussian-quadratic case and that their performance also lies within the 1/2-bit gap to that of successive refinement.