149.

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since a pixel classified as ‘ocean’ is likely to be surrounded by pixels of the same class and is unlikely to have neighbours from categories such as ‘pasture’ or ‘forest’. In other words, using the concept of context, pixels are not treated in isolation, but are considered to have a relationship with their neighbours. Thus, the relationship between the pixel of interest and its neighbours is treated as being statistically dependent.

The Markov random field (MRF) is a useful tool for characterising contextual information and has been widely used in image segmentation and image restoration (Besag, 1974; Geman and Geman, 1984; Derin and Elliott, 1987; Dubes and Jain, 1989; Geman and Reynolds, 1992). The use of MRF models for linear feature detection has also achieved satisfactory results (Tupin et al., 1998). The practical use of the MRF relies on its relationship to the Gibbs random field (GRF), which provides a tractable way to apply the MRF to deal with context. Moreover, owing to the MRF’s local property, the algorithm can be implemented in a highly parallel manner, which makes the MRF more attractive.

This chapter illustrates how MRF theory can be used to model the prior p.d.f. of contextual-dependent patterns. Using both the prior p.d.f and class-conditional p.d.f., one can establish a MAP estimate. The fundamental concept of MRF is explained in Section 6.1. The construction of the energy function relating to remote sensing image classification and restoration is considered in Section 6.2. Section 6.3 illustrates how to use MRF to achieve a robust maximum likelihood estimate, in order to avoid noise effects, and to obtain an efficient estimate of the sample mean. Parameter setting is another important issue in the application of MRF. A model is not complete if both the functional form of the model and its parameters are not fully specified. Methods for estimating model-associated parameters are therefore dealt with in Section 6.4. Classification algorithms based on the MAP-MRF framework and the experimental results derived from these algorithms are presented in the final sections of this chapter.

6.1 Markov random fields and Gibbs random fields

Let a set of random variables d={d₁, d₂,…, d_m} be defined on the set S containing m number of sites in which each random variable y_i (1≤i≤ m) takes a label from label set L. The family d is called a random field. The set S is equivalent to an image containing m pixels; d is a set of pixel DN values; and the label set L depends upon the application. The label set L is equivalent to a set of the user-defined information classes, e.g. L= {water, forest, pasture, or residential areas}, while in case of boundary detection, the label set L={boundary, non-boundary}. There are many kinds of random field models describing ways of labelling the random variables. Here, we are concerned with two special types of random fields, the Markov random field and the Gibbs random field.

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