5.2.1

Posterior Probability and Energy

In all cases, an of the neighborhood system on can consist of all the other sites . This is a trivial case for MRF. In RS matching, it can consist of all the other sites which are related to i by any relations in d. When the scene is very large, includes only those of the other sites which are within a spatial distance r from i  

which was given in (1.1.4). The distance threshold r may be reasonably related to the size of the model object. The set of first order cliques is

The set of second order cliques is

Here, only cliques of up to order two are considered.

The single-site potential is defined as  

where is a constant. If is the NULL label, it incurs a penalty of ; or the nil penalty is imposed otherwise. The pair-site potential is defined as

where is a constant. If either or is the NULL , it incurs a penalty of or nil penalty otherwise. The above clique potentials define the prior energy . The prior energy is then

The definitions of the above prior potentials are a generalization of that penalizing line process variables [Geman and Geman 1984,Marroquin 1985]. The potentials may also be defined in terms of stochastic geometry [Baddeley and van Lieshout 1992].

The conditional p.d.f., , of the observed data d, also called the likelihood function  ) when viewed as a function of f given d fixed, has the following characteristics:

1. It is conditioned on pure non- NULL matches ,
2. It is independent of the neighborhood system , and
3. It depends on how the model object is observed in the scene which in turn depends on the underlying transformations and noise.

where e is additive independent zero mean Gaussian noise. The assumptions of the independent and Gaussian noise may not be accurate but offers an approximation when an accurate observation model is not available.

Then the likelihood function is a Gibbs distribution with the energy

where the constraints, and , restrict the summations to take over the non- NULL matches. The likelihood potentials are

and

where ( and n=1,2) are the variances of the corresponding noise components. The vectors and are the ``mean vectors'', conditioned on and , for the random vectors and , respectively, conditioned on the configuration f.

Using , we obtain the posterior energy

There are several parameters involved in the posterior energy: the noise variances and the prior penalties . Only the relative, not absolute, values of and are important because the solution remains the same after the energy E is multiplied by a factor. The in the MRF prior potential functions can be specified to achieve the desired system behavior. The higher the prior penalties , the fewer features in the scene will be matched to the NULL for the minimal energy solution.

Normally, symbolic relations are represented internally by a number. The variances for those relations are zero. One may set corresponding to (a very small positive number), which is consistent with the concept of discrete distributions. Setting causes the corresponding distance to be infinitely large when the compared symbolic relations are not the same. This inhibits symbolically incompatible matches, if an optimal solution is sought, and thus imposes the desired symbolic constraint. A method for learning parameters from examples will be presented in Chapter 7.