152.

[Cover] [Contents] [Index]

Page 234

Figure 6.2 Patterns (a) to (e) show possible cliques with a neighbourhood system of the first and second order.

so does the computational complexity. To help clarify the concept of the neighbourhood system, Figure 6.3 shows cliques C₁, C₂, C₃, and C₄ relating to a pixel r of interest.

The energy function U(w) in Equation (6.3) is more easily understood when it is expressed in the following expanding form:

(6.4)

C₁, C₂, C₃ and C₄ each represent a single site clique (Figure 6.2a), pair-site clique (Figure 6.2b and c), triple-site clique (Figure 6.2d), and quadruple-site clique (Figure 6.2e), respectively.

Schröder et al. (2000) provide a comprehensive account of Gibbs random field models and their application in spatial data analysis.

6.1.3 MRF-GRF equivalence

An MRF is defined in terms of local properties (i.e. the classification label assigned to a pixel is affected only by its neighbours), whereas a GRF describes the global properties of an image (i.e. the label given to a specific pixel is affected by the labels given to all other pixels) in terms of the joint distribution of classes for all pixels. The Hammersley-Clifford theorem describes the equivalence of GRF and MRF properties. The theorem states

[Cover] [Contents] [Index]