Statistical mechanical evaluation of a spread-spectrum watermarking model with image restoration II AT stability of a hybrid system with message decoding and image

RESUMO

In the previous paper ( arxiv.org/abs/1209.4772 ), we proposed a spread-spectrum watermarking model with image restoration based on Bayes estimation assuming several prior probabilities and adopting the Gaussian channel model to represent attacks from unauthorized users. When an image is generated from the infinite range Ising model, we analyzed the model using the statistical mechanical method, the replica method, and derived the replica symmetric (RS) solution and performed Markov chain Monte Carlo simulations. The theoretical results of the RS solution and the simulation results were in good agreement except for some range of parameters. We treated the informed case where only the original image is known and the blind case where both the original message and the original image are unknown and found that the difference between these cases was small as long as the embedding and attack rates were small. In this paper, we treat the blind case and investigate the de Almeida-Thouless (AT) stability of the RS solution and reveal that the AT stability is broken in the range of parameters where theoretical and simulation results do not agree.