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<p><it>Abstract</it>—We describe a family of approximations, denoted by “cluster approximations,” for the computation of the mean of a Markov random field (MRF). This is a key computation in image processing when applied to the a <it>posteriori</it> MRF. The approximation is to account exactly for only spatially local interactions. Application of the approximation requires the solution of a nonlinear multivariable fixed-point equation for which we prove several existence, uniqueness, and convergence-of-algorithm results. Four numerical examples are presented, including comparison with Monte Carlo calculations.</p>
Markov random fields, image restoration, Bayesian estimation, thresholded posterior mean estimator.

P. C. Doerschuk and C. Wu, "Cluster Expansions for the Deterministic Computation of Bayesian Estimators Based on Markov Random Fields," in IEEE Transactions on Pattern Analysis & Machine Intelligence, vol. 17, no. , pp. 275-293, 1995.
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