On the Optimum Multiplicative Watermark Detection in the Transform Domain

This paper presents an investigation on the optimum detection of multiplicative watermarks based on statistical behaviour of image contents in the transform domain. In recent works, the problem of watermark detection is viewed as a binary decision where the observation is the possibly watermarked transformed coefficients. Indeed, the detector verifies whether the watermark presented to its input is actually embedded in the input image (hypothesis HI) or not (hypothesis HQ). Such a detection scheme relies on the Neyman-Pearson criterion to derive a decision threshold by minimising the probability of missed detection with respect to a given probability of false alarm. Previous works approximate the probability density function (pdf) of the observation when hypothesis HQ holds by the pdf when hypothesis of having no watermark embedded in the input image is in force by assuming that the watermark strength is weak to some extent. However, the weakness of the watermark does fulfil the requirement on the robustness. Moreover, from the viewpoint of the decision theory, the smaller the embedding depth, the worse the watermark detection. This paper describes the drawback behind this approximation in the general case and proposes an efficient solution closer to the theoretical derivation. To validate the proposed technique, we consider a special case in which the Laplace statistical model is used.