151.

[Cover] [Contents] [Index]

Page 233

neighbourhood order up to five is shown in Figure 6.1c. More specifically, when the sites form a regular rectangular lattice, as do pixels in a two-dimension image, the site r=(i, j) has four nearest (first-order) neighbours, denoted by N(i, j)={(i−1, j), (i+1, j), (i, j−1), (i, j+1)}.

6.1.2 Gibbs random fields

A Gibbs random field (GRF) provides a global model for an image by specifying a p.d.f. in the following form:

(6.1)

where w is defined in Section 6.1.1, U(w) is called an energy function, T is a constant termed temperature (which should be assumed to take the value 1 unless otherwise stated) and Z is called the partition function, which is expressed by:

(6.2)

The Gaussian distribution is a special member of this Gibbs distribution family. Since Z is the sum of all possible configurations of w (e.g. in the case of a 256×256 image with eight defined classes, the total configurations will be 8^256×256), it follows that Z is regarded as not computable except in extremely simple cases. The difficulty of computing Z considerably complicates sampling and estimation problems.

Based on Equation (6.1), it can be shown that maximising P(w) is equivalent to minimising the energy function U(w), given by Equation (6.3):

(6.3)

In this equation, C is known as a clique (defined below). …, is the collection of all possible cliques, and V_c(w) is called the potential function with respect to clique type C. A clique C is a subset in which all pairs of sites are mutual neighbours. It can be a single site, or a pair of neighbouring sites, or a triple of neighbouring sites, and so forth. Figure 6.2 shows clique types of the first and second order. The parameters associated with each clique type are also given. Figures 6.2a and b show those from the first-order neighbourhood system; Figures 6c, d and e are derived from the second-order neighbourhood system. Clearly, as the order of the neighbourhood system increases, the number of cliques grows rapidly, and

[Cover] [Contents] [Index]