On joint information embedding and lossy compression

We consider the problem of optimum joint information embedding and lossy compression with respect to a fidelity criterion. The goal is to find the minimum achievable compression (composite) rate R_c as a function of the embedding rate R_e and the average distortion level Δ allowed, such that the average probability of error in decoding of the embedded message can be made arbitrarily small for sufficiently large block length. We characterize the minimum achievable composite rate for both the public and the private versions of the problem and demonstrate how this minimum can be approached in principle. We also provide an alternative single-letter expression of the maximum achievable embedding rate (embedding capacity) as a function of R_c and Δ, above which there exist no reliable embedding schemes.