The auto-logistic model can be generalized to multi-level logistic (MLL) model [Elliott et al. 1984 ; Derin and Cole 1986 ; Derin and Elliott 1987], also called Strauss process [Strauss 1977] and generalized Ising model [Geman and Geman 1984]. There are M (>2) discrete labels in the label set, . A clique potential depends on the type c (size, shape and possibly orientation) of the clique and the local configuration . For cliques containing more than one site (), the MLL clique potentials are defined by
where is the potential for type-c cliques; for single site cliques, they depends on the label assigned to the site
where is the potential for label value I. Fig.1.4 shows the clique types and the associated parameters in the second order (8-neighbor) neighborhood system.
Figure 1.4: Clique types and associated potential parameters for the second order neighborhood system. Sites are shown in dots and neighboring relationships in joining lines.
Assume that an MLL model is of second-order as in (1.37), so that only (for single-site cliques) and (for pair-site cliques) parameters are non-zero. The potential function for pair-wise cliques is written as
where is the parameter for type-c cliques and is set of pair-site cliques. For the 4-neighborhood system, there are four types of pair-wise cliques ( cf.
Fig.1.4) and so there may be four different 's. When the model is isotropic all the four take the same value. Owing to its simplicity, the pair-wise MLL model (1.54) has been widely used for modeling regions and textures [Elliott et al. 1984 ; Geman and Geman 1984 ; Derin and Cole 1986 ; Derin and Elliott 1987 ; Murray and Buxton 1987 ; Lakshmanan and Derin 1989 ; Won and Derin 1992].
When the MLL model is isotropic, it depicts blob-like regions. In this case, the conditional probability can be expressed as follows [Strauss 1977]
where is the number of sites in which are labeled I. It reduces to (1.40) when there are only two labels, 0 and 1. In contrast, an anisotropic model tends to generates texture-like patterns. See examples in Section 2.4.