Bethe approximation to inverse halftoning using multiple halftone images

We formulate the problem of inverse halftoning using multiple dithered images utilizing the Bayesian inference via the maximizer of the posterior marginal (MPM) estimate on the basis of statistical mechanics of the Q-Ising model. From the theoretical point of view, the Monte Carlo simulation for a set of snapshots of the Q-Ising model clarifies that the performance is improved introducing the prior information on original images into the MPM estimate and that the optimal performance is realized around the Bayes-optimal condition within statistical uncertainty. Then, these properties are qualitatively confirmed by the analytical estimate via the infinite-range model. Next, we try the Bethe approximation established in statistical mechanics for this problem. Numerical simulations clarify that the Bethe approximation works as well as the MPM estimate via the Monte Carlo simulation for 256-level standard images, if we set parameters of the model prior appropriately.