We describe a Bayesian restoration and segmentation algorithm based on a pixel-line Markov random field and using an efficient approximation based on locality of interactions. A medical tomography example is given.

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 posteriori 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.>

Methods for approximately computing the marginal probability mass functions and means of a Markov random field (MRF) by approximating the lattice by a tree are described. Applied to the a posteriori MRF these methods solve Bayesian spatial pattern classification and image restoration problems. The methods are described, several theoretical results concerning fixed-point problems are proven, and four numerical examples are presented, including comparison with optimal estimators and the iterated conditional mode estimator and including two agricultural optical remote sensing problems.>

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.>

Describes a family of approximations, denoted "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 posteriori MRF. The approximation is a form of mean field theory and accounts exactly for only spatially local interactions. The implementation of the approximation requires the solution of a nonlinear multivariable fixed-point equation. Unlike many forms of mean field theory, the approximation is easy to apply even to nonquadratic Hamiltonians (e.g., it requires no analytical calculations), the structure of the gray level values in the original problem is retained, the resulting approximate mean field can be proven to lie in the same set which contains the true mean of the MRF, and there are existence, uniqueness, and convergence-of-algorithm results for the fixed-point equation. Two numerical examples are presented which emphasize nonlinear observation processes.>