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6.1.1 Markov random field

Given the definition of a random field specified earlier, we define the configuration w for the set S as w={d₁=w₁, d₂=w_2,…, d_m=w_m}, where w_r L (1≤r ≤m). For convenience, we simplify the notation of w to w ={w₁, w₂, …, w_m}. A random field, with respect to a neighbourhood system, is a Markov random field if its probability density function satisfies the following three properties:

1 Positivity: P(w)>0 for all possible configurations of w,

2 Markovianity: P(w_r|w_s−r)=P(w_r|w_Nr), and

3 Homogeneity: P(w_r|w_Nr) is the same for all sites r.

S−r is the set difference (i.e. all pixels in the set S excluding r), w_s−r denotes the set of labels at the sites in S–r, and Nr denotes the neighbours (defined below) of site r.

The first property, that of positivity, can usually be satisfied in practice, and the joint probability P(w) can be uniquely determined by local conditional properties as long as the positivity property sustains (Besag, 1974). Markovianity indicates that labelling of a site r is only dependent on its neighbouring sites. The property of homogeneity specifies the conditional probability for the label of a site r, given the labels of the neighbouring pixels, regardless of the relative position of site r in S. An MRF may also incorporate other properties, such as isotropy. Isotropy is the property of direction-independence, that is, the neighbouring sites surrounding a site r have the same contributing effect to the labelling of site r.

The usual neighbourhood system used in image analysis defines the first-order neighbours of a pixel as the four pixels sharing a side with the given pixel, as shown in Figure 6.1a. Second-order neighbours are the four pixels having corner boundaries with the pixel of interest, as shown in Figure 6.1b. Higher-order neighbours can be extended in a similar manner. The

Figure 6.1 Neighbourhood ordering for a pixel r. (a), (b) and (c) depict the first, second and fifth order neighbourhood system, respectively.

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