Power-spectrum condition for energy-efficient watermarking

The paper presents a model for watermarking and some attacks on watermarks. Given the watermarked signal, the so-called Wiener attack performs minimum mean-squared error (MMSE) estimation of the watermark and subtracts the weighted MMSE estimate from the watermarked signal. Under the assumption of a fixed correlation detector, the attack is shown to minimize the expected correlation statistic for the same attack distortion among linear, shift-invariant filtering attacks. It also leads to the idea of energy-efficient watermarking: watermarking that resists MMSE estimation as much as possible, and provides a meaningful way to evaluate robustness. The paper shows that energy-efficient watermarks must satisfy a power-spectrum condition (PSC), which states that the watermark's power spectrum should be directly proportional to the original signal's. PSC-compliant watermarks are proven to be most robust. Experiments with signal models and natural images demonstrate that watermarks that do not closely fulfill the PSC are vulnerable to the Wiener attack, while PSC-compliant watermarks are highly resistant to it. These theoretical and experimental results justify prior heuristic arguments that, for maximum robustness, a watermark should be closely matched to the spectral content of the original signal. The results also discourage the use of watermarks that do not approximately satisfy the PSC.