1.5.2
In the MAP-MRF labeling, 
is the posterior distribution of
an MRF. An important step in Bayes labeling of MRFs is to derive this
distribution. Here we use a simple formulation of MRF restoration as an
example to illustrate MAP-MRF labeling. Assuming the underlying surface is
flat, then the joint prior distribution is
where 
is the prior energy given in
(1.57). Assuming that the observation is the
truth plus the independent Gaussian noise, 
, where 
, then the likelihood density is
where
is the likelihood energy. Now the posterior probability is
where
is the posterior energy. The MAP estimate is equivalently found by minimizing the posterior energy
When the parameters 
are given so that 
is
fully specified, the MAP-MRF labeling solution is completely
defined.
