A hierarchical two-level Gibbs model has been proposed to represent both noise-contaminated and textured images [Derin and Cole 1986; Derin and Elliott 1987]. The higher level Gibbs distribution uses an isotropic random field, e.g. MLL, to characterize the blob-like region formation process. A lower level Gibbs distribution describes the filling-in in each region. The filling-in may be independent noise or a type of texture, both of which can be characterized by Gibbs distributions. This provides a convenient approach for MAP-MRF modeling. In segmentation of noisy and textured image [Derin and Cole 1986; Derin and Elliott 1987; Lakshmanan and Derin 1989; Hu and Fahmy 1987; Won and Derin 1992], for example, the higher level determines the prior of f for the region process while the lower level Gibbs contributes to the conditional probability of the data given f. Note that different levels of MRFs in the hierarchy can have different neighborhood systems.
Various hierarchical Gibbs models result according to what are chosen for the regions and for the filling-in's, respectively. For example, each region may be filled in by an auto-normal texture [Simchony and Chellappa 1988; Won and Derin 1992] or an auto-binomial texture [Hu and Fahmy 1987]; the MLL for the region formation may be substituted by another appropriate MRF. The hierarchical MRF model for textured regions will be further discussed in Section 2.4.1.
A drawback of the hierarchical model is that the conditional probability for regions given by can not always be written exactly. For example, when the lower level MRF is a texture modeled as an auto-normal field, its joint distribution over an irregularly shaped region is not known. This difficulty may be overcome by using approximate schemes such as pseudo-likelihood to be introduced in Section 6.1 or by using the eigen-analysis method [Wu and Leahy 1993].