The zero-rate spread-spectrum watermarking game

This paper develops a game-theoretic methodology to design and embed M messages in signals and images in the presence of an adversary. Here, M is assumed to be subexponential in the signal's sample size (zero-rate transmission), and the embedding is done using spread-spectrum watermarking. The detector performs statistical hypothesis testing. The system is designed to minimize probability of error under the worst-case attack in a prescribed class of attacks. The variables in this game are probability distributions for the watermarker and attacker. Analytical solutions are derived under the assumption of Gaussian host vectors, watermarks and attacks, and squared-error distortion constraints for the watermarker and the attacker. The Karhunen-Loeve transform (KLT) plays a central role in this study. The optimal distributions for the watermarker and the attacker are Gaussian test channels applied to the KLT coefficients; the game is then reduced to a maxmin power-allocation problem between the channels. As a byproduct of this analysis, we can determine the optimal tradeoff between using the most efficient (in terms of detection performance) signal components for transmission and spreading the transmission across many components (to fool the attacker's attempts to eliminate the watermark). We also conclude that in this framework, additive watermarks are suboptimal; they are, however, nearly optimal in a small-distortion regime. The theory is applied to watermarking of autoregressive processes and to wavelet-based image watermarking. The optimal watermark design outperforms conventional designs based on heuristic power allocations and/or simple correlation detectors.