Optimal prefix codes for sources with two-sided geometric distributions

A complete characterization of optimal prefix codes for off-centered, two-sided geometric distributions of the integers is presented. These distributions are often encountered in lossless image compression applications, as probabilistic models for image prediction residuals. The family of optimal codes described is an extension of the Golomb codes, which are optimal for one-sided geometric distributions. The new family of codes allows for encoding of prediction residuals at a complexity similar to that of Golomb codes, without recourse to the heuristic approximations frequently used when modifying a code designed for nonnegative integers so as to apply to the encoding of any integer. Optimal decision rules for choosing among a lower complexity subset of the optimal codes, given the distribution parameters, are also investigated, and the relative redundancy of the subset with respect to the full family of optimal codes is bounded.