Maximum likelihood amplitude scale estimation for quantization-based watermarking in the presence of dither

Quantization-based watermarking schemes comprise a class of watermarking schemes that achieves the channel capacity in terms of additive noise attacks. The existence of good high dimensional lattices that can be efficiently implemented and incorporated into watermarking structures, made quantization-based watermarking schemes of practical interest. Because of the structure of the lattices, watermarking schemes making use of them are vulnerable to non-additive operations, like amplitude scaling in combination with additive noise. In this paper, we propose a secure Maximum Likelihood (ML) estimation technique for amplitude scaling factors using subtractive dither. The dither has mainly security purposes and is assumed to be known to the watermark encoder and decoder. We derive the probability density function (PDF) models of the watermarked and attacked data in the presence of subtractive dither. The derivation of these models follows the lines of, where we derived the PDF models in the absence of dither. We derive conditions for the dither sequence statistics such that a given security level is achieved using the error probability of the watermarking system as objective function. Based on these conditions we are able to make approximations to the PDF models that are used in the ML estimation procedure. Finally, experiments are performed with real audio and speech signals showing the good performance of the proposed estimation technique under realistic conditions.