  On the basis of statistical mechanics of the Q-Ising model, we formulate the
Bayesian inference to the problem of inverse halftoning, which is the inverse
process of representing gray-scales in images by means of black and white dots.
Using Monte Carlo simulations, we investigate statistical properties of the
inverse process, especially, we reveal the condition of the Bayes-optimal
solution for which the mean-square error takes its minimum. The numerical
result is qualitatively confirmed by analysis of the infinite-range model. As
demonstrations of our approach, we apply the method to retrieve a grayscale
image, such as standard image `Lenna', from the halftoned version. We find that
the Bayes-optimal solution gives a fine restored grayscale image which is very
close to the original.
