Power Spectrum Condition for Energy-Efficient Watermarking

A simple but powerful theoretical model for watermarking and attacks is presented. The attack consists of Wiener estimation of the watermark, followed by weighted feedback of the estimate to reduce the detector's correlation value. The Wiener attack is provably optimal, and it leads to the notion of energy-e cient watermarking, the generation of watermarks that resist Wiener estimation. It also leads to a meaningful way to measure watermark robustness. Designing watermarks to resist the attack produces a power-spectrum condition (PSC): the watermark's power spectrum should be a scaled version of the original document's. Experiments with signal models demonstrate the importance of the PSC and show that watermarks that fail to meet the PSC may be vulnerable to attack. These analytic predictions are con rmed with experiments on natural images. The paper provides a theoretical basis for the popular heuristic argument that a watermark should be embedded in the \perceptually signi cant frequency components" of the original document and discourages the use of watermarks whose power spectra are not closely matched to the original document's power spectrum.