sA very fruitful approach to the solution of image segmentation andssurface reconstruction tasks is their formulation as estimationsproblems via the use of Markov random field models and Bayes theory.sHowever, the Maximuma Posteriori (MAP) estimate, which is the one mostsfrequently used, is suboptimal in these cases. We show that forssegmentation problems the optimal Bayesian estimator is the maximizersof the posterior marginals, while for reconstruction tasks, thesthreshold posterior mean has the best possible performance. We presentsefficient distributed algorithms for approximating these estimates insthe general case. Based on these results, we develop a maximumslikelihood that leads to a parameter-free distributed algorithm forsrestoring piecewise constant images. To illustrate these ideas, thesreconstruction of binary patterns is discussed in detail.