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.