Computing the Probability of False Watermark Detection

Several methods of watermark detection involve computing a v ector from some input media, computing the normalized correlation between that vector and a predeened watermark vector, and comparing the result against a threshold. We show that, if the probability density function of vectors that arise from random, unwatermarked media is a zero-mean, spherical Gaussian, then the probability that such a detector will give a false detection is given exactly by a simple ratio of two deenite integrals. This expression depends only on the detection threshold and the dimensionality of the watermark vector.