157.

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we apply different symbols to denote potential parameters for different clique types. The potential parameters corresponding to clique types from C₂ to C₄ are illustrated in Figure 6.2 (e.g. α for C₁, β for C₂, γ for C₃, and ξ for C₄).

A special case of the MLL model is defined when only pairwise cliques are active, that is, only parameter β is non-zero (β>0). The potential can therefore be simplified to

(6.13)

When the model is anisotropic (i.e. rotation variant), the β coefficients may take on different values for different orientations as illustrated in Figure 6.2b and c from β₁ to β₄. Owing to its simplicity, the pairwise MLL model has been widely used for modelling regions and texture (Geman and Geman, 1984; Derin and Elliott, 1987; Won and Derin, 1992). It should be noted that, under the isotropic assumption (i.e. the same value β for all directions), blob-like regions are generated. As the value of β increases, regions that are more homogeneous will be favoured. However, if the isotropy limitation does not exist, the result will show patterns that are more texture-like. This topic is discussed further in the following sections.

6.1.5 Generation of texture patterns using MRF

In order to show how an image configuration is affected by its associated MRF parameters, an example specifying the effect of parameter selection is given below in which we adopted a MLL model with only pairwise cliques considered. The images are generated using Metropolis’s algorithm (1953), which is illustrated in Figure 6.4. This algorithm has been successfully tested by Dubes and Jain (1989) and results showed that fifty iterations (i.e. n=50 in Figure 6.4, step (1)) are sufficient to achieve convergence. In interpreting Figure 6.4, note that the higher the energy, the lower the probability.

Texture patterns derived using this algorithm are shown in Figure 6.5. Note that the resulting image patterns are not affected solely by the magnitude of parameter β. The choice of the second term in Equation (6.13), i.e. β or 0, will also contribute some variation. Figure 6.5a and c were constructed by using β in the case w_rw_r′, while in Figures 6.5b and d, β is replaced by 0. It is clear that the resulting texture patterns do reveal some level of difference. These texture patterns can be used for testing the robustness of texture quantisation methodology, as described in Chapter 5. These texture images can also be used for evaluating the approach to MRF model parameter estimation, as described in Section 6.4.

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