Describes a family of approximations, denoted "Bethe tree approximations", for the computation of the marginal probability mass functions (pmfs) of a Markov random field (MRF). This is a key computation in spatial pattern classification when applied to the a posteriori MRF. The approximation modifies the graph on which the MRF is defined: the original lattice is modified into a tree. Then the marginal pmfs on the tree can be computed exactly by fast recursive algorithms. A key issue is how to terminate the tree at its leaves and 4 solutions are explored of which 3 result in the solution of nonlinear multivariable fixed-point equations for which some existence, uniqueness, and convergence-of-algorithm results can be proven. The algorithm has given excellent performance on a variety of segmentation problems (1994) and a 9-class agricultural remote-sensing example is described.
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