6.3.2
This is an issue in MRF representation which affects parameter estimation. If an MRF can be represented using fewer parameters, then algorithms for it can be more efficient. It is known that Gibbs clique potential parameter representations for an MRF are not unique [Griffeath 1976,Kindermann and Snell 1980]. For the sake of economy, it is preferred to use a representation which requires the smallest possible number of nonzero clique parameters. The normalized clique potential parameterization described in Section 1.2.5 can be used for this purpose.
Given a countable set 
of labels and a set 
of all cliques,
one may choose an arbitrary label 
and let 
whenever 
for some 
. Then he can specify nonzero
clique parameters for the other configurations to define an MRF.
Figure 6.7: Normalized nonzero clique potentials estimated by using the
least squares method (top) and nonzero clique potentials chosen ad hoc
(bottom). Filled bars are edges and non-filled bars non-edges.
Figure 6.8: Step edge detection using various schemes. From left to right:
the input image; edges detected by using estimated canonical clique
potentials; edges obtained using the ad hoc
clique potentials; edges
detected using the Canny edge detector.
From (Nadabar and Jain 1992) with permission; © 1992 IEEE.
Nadabar and Jain (1991) use a normalized clique potential
representation for MRF-based edge
detection. Assuming the 4-neighborhood system on the image lattice,
there can be eight different types of edge cliques whose shapes are
shown in the top part of Fig.6.7. The label
at each edge site takes one of the two values, non-edge and edge, in the
label set 
={ NE, E}. These gives
thirty-five (35) {edge-label, clique-type} combinations. In the
normalized representation, clique potentials 
are set to zero
whenever 
takes a particular value, say, 
= NE for
some 
. In this way, only 8 clique potentials are nonzero; thus
the number of nonzero parameters are reduced from 35 to 8.
In [Zhang 1995], parameters reduction is performed for the compound Gauss-Markov model of [Jeng and Wood 1990,Jeng and Wood 1991]. The model contains line process variables on which clique potential parameters are dependent. By imposing reasonable symmetry constraints, the 80 interdependent parameters of the model is reduced to 7 independent ones. This not only reduces the parameter number but also guarantees the consistency of the model parameters.
After the normalized representation is chosen, the parameter estimation can be formulated using any method such as one of those described in the previous sections. In [Jain and Nadabar 1990] for example, the Derin-Elliott LS method is adopted to estimate line process parameters for edge detection. The estimated normalized clique potentials is shown in Fig.6.7 in comparison with an ad hoc choice made in their previous work [Jain and Nadabar 1990]. The corresponding results of edges detected using various schemes are shown in Fig.6.8.
After a set of parameters are estimated, one would evaluate the
estimate. The evaluation is to test the goodness of fit: How well does
the sample distribution fit the estimated MRF model with parameter
. This is a statistical problem. Some validation schemes are
discussed in [Cross and Jain 1983,Chen 1988] for testing the fit
between the observed distribution and the assumed distribution with the
estimated parameters. One available technique is the well-known 
test for measuring the correspondence between two distributions.