Scalar Quadratic-Gaussian Soft Watermarking Games

We introduce the zero-sum game problem of soft watermarking: The hidden information (watermark) comes from a continuum and has a perceptual value; the receiver generates an estimate of the embedded watermark to minimize the expected estimation error (unlike the conventional watermarking schemes where both the hidden information and the receiver output are from a discrete finite set). Applications include embedding a multimedia content into another. We consider in this paper the scalar Gaussian case and use expected mean-squared distortion. We formulate the resulting problem as a zero-sum game between the encoder & receiver pair and the attacker. We show that for the lin- ear encoder, the optimal attacker is Gaussian-affine, derive the optimal system parameters in that case, and discuss the corresponding system behavior. We also provide numerical results to gain further insight and understanding of the system behavior at optimality.