Fundamental performance limits of power-spectrum condition-compliant watermarks

Our previous work introduced a stochastic framework for studying watermarking. When the receiver used direct correlation detection and the attack employed linear shift-invariant (LSI) ltering, the optimal attack consisted of Wiener ltering and adding noise. Resisting the Wiener attack led to a power-spectrum condition (PSC): the watermark and original power spectra should be directly proportional. PSC-compliant watermarks are most diÆcult to estimate from the watermarked document in the minimum mean-squared error sense. This paper investigates the fundamental limits of PSC-compliant watermarks; it has two main parts. First, expressions are derived for the detection-error probabilities after the Wiener attack with PSC-compliant Gaussian noise. An explicit relationship between attacked-document distortion and capacity results. The second part studies optimal LSIltering attacks and receivers. Because direct correlation detection is only optimal for white Gaussian noise (WGN), the Wiener attack and PSC may actually be suboptimal. It is shown that the PSC holds for a suboptimal, e ective white-noise attack. The optimum attack is derived, but there does not appear to be an analytical solution for the optimum watermark power spectrum. Numerical experiments suggest that, as a rule of thumb, white watermarks perform well at lower distortions, while PSC-compliant watermarks perform better at higher distortions.