Deriving Posterior Energy

Let us derive the posterior energy using the four-step procedure summarized in Section 1.5.4:

1. Define a neighborhood system and the set of cliques for it. The neighborhood system is defined according to (1.13). The set of cliques for the 4-neighborhood system are shown in Fig.1.2(d)-(e). For the 8-neighborhood system, cliques in Fig.1.2(f)-(h) are also included.
2. Define the prior clique potential functions in the Gibbs prior distribution (1.24-1.26). For the MLL prior, it is (2.9).
3. Derive the likelihood energy from the observation model. Assume the i.i.d. additive Gaussian model. The likelihood function takes the form of (2.5) with taking a discrete value.
4. Add the prior energy and the likelihood energy to yield the posterior energy

    

is simply the number of neighboring sites whose label is different from .

In this problem, as well as in other labeling problems in this chapter, the parameters, such as and here, are assumed known. The more advanced topic of labeling with unknown noise and MRF parameters will be discussed in Section 6.2.1.