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Chapter 6
Modelling context using Markov random fields

Bayesian theory has had a long and profound influence on statistical modelling. Two key elements make up a Bayesian classification formula, namely, the prior and conditional probability density functions (p.d.f.). By combining these functions, a classification can be expressed in terms of maximum a posteriori (MAP) criteria (Chapter 2). In practice, there are difficulties in using MAP estimates. One of the problems is that prior information or information concerning the data distribution may not always be available. As a result, alternative criteria must be used in place of MAP. For example, if knowledge of data distributions is available, but not prior information about the data being dealt with, then the maximum likelihood (ML) criterion may be used. Conversely, if one has prior information but no knowledge about the data distribution, then the maximum entropy criterion can be employed (Jaynes, 1982).

The maximum likelihood criterion has been widely adopted in remotely sensed image classification, since in most classification experiments we generally use the normal (Gaussian) distribution to model the class-conditional p.d.f., while the prior p.d.f. is generally not used. It is likely that the classification result would be improved if: (1) a reasonable assumption could be made in order to model the prior p.d.f.; and (2) the class-conditional p.d.f. could be incorporated in order to establish a MAP estimate. One assumption for modelling prior probability is context.

There has been an increasing interest in use of contextual information for modelling the prior p.d.f. (Derin and Elliott, 1987; Dubes and Jain, 1989; Jhung and Swain, 1996; Schistad et al., 1996). The concept is generally called the ‘smoothness prior’, because the aim is to generate a smooth image classification pattern.

In the interpretation of visual information, context is very important. It may be derived from spectral, spatial, or even temporal attributes. The suitable use of context allows the elimination of possible ambiguities, the recovery of missing information, and the correction of errors (Li, 1995a). The use of context to model the prior p.d.f. in order to help in the interpretation of remotely sensed imagery is considered as a reasonable procedure,

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