Asymptotically Optimum Universal One-Bit Watermarking for Gaussian Covertexts and Gaussian Attacks

The problem of optimum watermark embedding and detection was addressed in a recent paper by Merhav and Sabbag, where the optimality criterion was the maximum false–negative error exponent subject to a guaranteed false– positive error exponent. In particular, Merhav and Sabbag derived universal asymptotically optimum embedding and detection rules under the assumption that the detector relies solely on second order joint empirical statistics of the received signal and the watermark. In the case of a Gaussian host signal and a Gaussian attack, however, closed–form expressions for the optimum embedding strategy and the false–negative error exponent were not obtained in that work. In this paper, we derive such expressions, again, under the universality assumption that neither the host variance nor the attack power are known to either the embedder or the detector. The optimum embedding rule turns out to be very simple and with an intuitively–appealing geometrical interpretation. The improvement with respect to existing sub–optimum schemes is demonstrated by displaying the optimum false–negative error exponent as a function of the guaranteed false–positive error exponent. Index Terms Watermarking, watermark embedding, watermark detection, hypothesis testing, Neyman–Pearson. P. Comesaña is with the Signal Theory and Communications Department, University of Vigo, Campus Lagoas-Marcosende, Vigo 36310, Spain, (phone: +34 986 812683, fax: +34 986 812116, e-mail: [email protected]), N. Merhav is with the Department of Electrical Engineering, Technion – I.I.T., Haifa 32000, Israel, (phone/fax: +972-4-8294737, e-mail: [email protected]), M. Barni is with the Department of Information Engineering, University of Siena, Via Roma 56, Siena 53100, Italy, (phone: +39 0577 234624 / +39 0577 234621, fax: +39 0577 233630, e-mail: [email protected]). This work was partially supported by the Italian Ministry of Research and Education under FIRB project no. RBIN04AC9W. December 21, 2013 DRAFT