On the relationship between embedding costs and steganographic capacity

Contemporary steganography in digital media is dominated by the framework of additive distortion minimization: every possible change is given a cost, and the embedder minimizes total cost using some variant of the Syndrome-Trellis Code algorithm. One can derive the relationship between the cost of each change c_i and the probability that it should be made pi_i, but the literature has not examined the relationship between the costs and the total capacity (secure payload size) of the cover. In this paper we attempt to uncover such a relationship, asymptotically, for a simple independent pixel model of covers. We consider a 'knowing' detector who is aware of the embedding costs, in which case sum pi_i^2 c_i should be optimized. It is shown that the total of the inverse costs, sum c_i^-1, along with the embedder's desired security against an optimal opponent, determines the asymptotic capacity. This result also recovers a Square Root Law. Some simple simulations confirm the relationship between costs and capacity in this ideal model