Statistical-mechanical approaches to the problem of phase retrieval in adaptive optics in astronomy

We construct the statistical-mechanical formulation for the problem of phase retrieval in adaptive optics for astronomy using the Q-Ising model on the square lattice. This method is based on the maximizer of the posterior marginal (MPM) estimate which is classified into two kinds selectively used according to the sensitivity of measurement. Next, using the Markov-chain Monte Carlo simulation for a typical wave-front in adaptive optics, we obtains the result that our method works well for phase retrieval if we appropriately select the model of the true prior respective of the sensitivity of measurement. Further, these results are qualitatively confirmed by the replica theory for the infinite-range model.