In cases in which an original image is blind, a decoding method where both
the image and the messages can be estimated simultaneously is desirable. We
propose a spread spectrum watermarking model with image restoration based on
Bayes estimation. We therefore need to assume some prior probabilities. The
probability for estimating the messages is given by the uniform distribution,
and the ones for the image are given by the infinite range model and 2D Ising
model. Any attacks from unauthorized users can be represented by channel
models. We can obtain the estimated messages and image by maximizing the
posterior probability.
We analyzed the performance of the proposed method by the replica method in
the case of the infinite range model. We first calculated the theoretical
values of the bit error rate from obtained saddle point equations and then
verified them by computer simulations. For this purpose, we assumed that the
image is binary and is generated from a given prior probability. We also assume
that attacks can be represented by the Gaussian channel. The computer
simulation retults agreed with the theoretical values.
In the case of prior probability given by the 2D Ising model, in which each
pixel is statically connected with four-neighbors, we evaluated the decoding
performance by computer simulations, since the replica theory could not be
applied. Results using the 2D Ising model showed that the proposed method with
image restoration is as effective as the infinite range model for decoding
messages.
We compared the performances in a case in which the image was blind and one
in which it was informed. The difference between these cases was small as long
as the embedding and attack rates were small. This demonstrates that the
proposed method with simultaneous estimation is effective as a watermarking
decoder
