6.1.3
Figure 6.2: Codings for the 4-neighborhood (left) and
8-neighborhood (right) systems. Pixels marked number k
belong to 
.
Using the coding method [Besag 1974], the
likelihood formula of the form (6.14) can be made a genuine one.
The idea is the following: Partition the set 
into several disjoint
sets 
, called codings, such that no two sites in one 
are
neighbors. Fig.6.2 illustrates the codings for the 4- and
8- neighborhood systems. For the 4-neighborhood system, two sets 
and 
are needed, one composed of the black sites and the other of
the white sites in a checkerboard. Four codings are needed for the
8-neighborhood system.
The fact that the sites within each 
are not neighboring to each
other provides computational advantages. Under the Markovian assumption,
the variables associated with the sites in an 
, given the labels
at all other sites, are mutually independent. This gives the following
simple product for the likelihood
in which the need to evaluate Z is also dispensed with. Unlike the
pseudo-likelihood, the above definition is a genuine expression for the
likelihood function. Maximizing (6.18) gives
coding estimates 
.
A disadvantage is the inefficiency: Each of the ML estimates 
is estimated based on information on only one coding. It is not clear
how to combine the results optimally. The simplest scheme of the
arithmetic average
is usually used.
Consider the homogeneous and isotropic auto-logistic MRF model described by (6.7) and (6.8). Its log coding likelihood take a similar form to that for the pseudo-likelihood
where for k=1,2. The ML estimates 
(k=1,2)
can be obtained by solving
Taking the arithmetic averages, we obtain the final estimate as
