MARKOV RANDOM-FIELDS WITH HIGHER-ORDER INTERACTIONS

Discrete-state Markov random fields on regular arrays have played a si gnificant role in spatial statistics and image analysis. For example, they are used to represent objects against background in computer visi on and pixel-based classification of a region into different crop type s in remote sensing. Convenience has generally favoured formulations t hat involve only pairwise interactions. Such models are in themselves unrealistic and, although they often perform surprisingly well in task s such as the restoration of degraded images, they are unsatisfactory for many other purposes. In this paper, we consider particular forms o f Markov random fields that involve higher-order interactions and ther efore are better able to represent the large-scale properties of typic al spatial scenes. Interpretations of the parameters are given and rea lizations from a variety of models are produced via Markov chain Monte Carlo. Potential applications are illustrated in two examples. The fi rst concerns Bayesian image analysis and confirms that pairwise-intera ction priors may perform very poorly for image functionals such as num ber of objects, even when restoration apparently works well. The secon d example describes a model for a geological dataset and obtains maxim um-likelihood parameter estimates using Markov chain Monte Carlo. Desp ite the complexity of the formulation, realizations of the estimated m odel suggest that the representation is quite realistic.