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

TL;DR

This paper derives the optimal universal one-bit watermark embedding and detection strategies for Gaussian signals under Gaussian attacks, providing explicit formulas and demonstrating improved error exponents over sub-optimal methods.

Contribution

It provides closed-form expressions for the asymptotically optimal embedding and detection rules in Gaussian watermarking with unknown parameters, extending previous work.

Findings

Explicit formulas for optimal embedding and detection in Gaussian watermarking.

Demonstration of improved false-negative error exponents.

Simple geometrical interpretation of the optimal embedding rule.

Abstract

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.