ABSTRACT
Descriptive power of a Markov random field (MRF) of signals at the lattice sites is considerably higher than of any IRF because interdependencies, or interactions between different signals are taken into account. As was already detailed in Chapter 2; Equations 2.14 and 2.15, the probability or density P(g) of an image g sampled from a certain MRF, such that the positivity condition P(g) > 0 holds for all the images, is factored over a system C of selected subsets c of the lattice sites (cliques of an interaction graph, Γ). Thus it is represented in the exponential form as the GPD:
P(g) = 1 Z ∏c∈C
fc(g(r) : r ∈ c) ≡ 1Z exp ( − ∑
) defining the corresponding Markov-Gibbs random field (MGRF). The largest clique cardinality, k = maxc∈C |c|, specifies the order of the MGRF. One of basic difficulties of using and learning a vast majority of the MGRF image models is that their normalizer, or partition function Z = ∑g∈G exp (−∑c∈C Vc(g(r) : r ∈ c)) is computationally infeasible.
