Asymptotically Optimum Universal Watermark Embedding and Detection in the High-SNR Regime
The problem of optimum watermark embedding and detection was addressed in a recent paper by Merhav and Sabbag, where the optimality criterion was the maximum false-negative error exponent subject to a guaranteed false-positive error exponent. In particular, Merhav and Sabbag derived universal asymptotically optimum embedding and detection rules under the assumption that the detector relies solely on second-order joint empirical statistics of the received signal and the watermark. In the case of a Gaussian host signal and a Gaussian attack, however, closed-form expressions for the optimum embedding strategy and the false-negative error exponent were not obtained in that work. In this paper, we derive the false-negative error exponent for any given embedding strategy and use such a result to show that in general the optimum embedding rule depends on the variance of the host sequence and the variance of the attack noise. We then focus on high signal-to-noise ratio (SNR) regime, deriving the optimum embedding strategy for such a setup. In this case, a universally optimum embedding rule turns out to exist and to be very simple with an intuitively appealing geometrical interpretation. The effectiveness of the newly proposed embedding strategy is evaluated numerically.