We have compared the results we obtain from the proposed variational model with
the ones obtained using a stochastic based on Markov Random Fields
theory (cf. [2] for instance).
Let 
be the solution of classification, i.e. an image of labels, and let 
represent the observed data. We are
looking for the estimation 
of
by
maximizing (Maximum A Posteriori) the probability P(/).
Using Bayes rule, and according to the properties of Markov Random
Fields, the solution is found thanks to :
and 
are respectively the label assigned to the
site s and the observation at site s. As for the
proposed variational model, the label value assigned to
the site s is one of the values 
,
i=1..M. The set C is the one of considered cliques. Parameter
represents the weight of the regularization term and
the operator 
is the Dirac distribution defined as
stands for the temperature parameter.
The minimization of E is operated through a simulated annealing (with a Metropolis dynamic).