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
is the likelihood energy. Now the posterior probability is
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.