Assume a two-dimensional random field is with size I by J. For any pixel at location (i, j), 1 < i < I, 1 < j < J, its neighborhood, N j, can be defined as

2. for any pixel (p, q) e N j, there is (i, j) e Npq

To illustrate the neighborhood, an example is shown in Fig. 8.1. The shaded region is a 3 by 3 neighborhood of pixel s. The size of neighborhood generally reflects how far a pixels surrounding region has an affect on it. This is a detail of the implementation of the algorithm and depends on the characteristics of the processed image.

Assume X is a two-dimensional random field and Πdenotes the set of all the possible samples of X. The definition of Markov random field is given as follow: if X is an MRF, then for every Xit j e X it must satisfy

(i) P(Xi, j |Xpq all (p, q ) = (i, j)) = P(Xi, j |Xpq all (p, q ) e N j) (8.1.a)

Condition (i) is called the Markovian property that describes the statistical dependency of any pixel in the random field on its neighboring pixels. Under this constraint, only a small number of pixels within X, j's neighborhood, N j, instead of the whole image needs to be considered. Thereby, it reduces the model's complexity significantly. Condition (ii), the positivity property, restricts all the samples of X to have a positive probability.

Although the Markov property is very useful for a prior model, the definition in Eq. (8.1) is not directly suitable to specify an MRF. Fortunately, the Hammersley-Clifford theorem [33] proves the equivalence between MRF and Gibbs random field (GRF), which states that a random field X is an MRF, if and only if, a prior probability P(X) follows Gibbs distribution:

where Z, partition function, is a normalizing constant. The parameter T is a constant used to control the peaking of the distribution. U(x) is Gibbs energy function (or Gibbs potential). In the GRF model, a very important concept is called clique. The definition of clique C within the neighborhood of pixel s, Ns, is given as a set of points neighboring to each other, which satisfy

1. C consists of a single pixel, or

The collection of all cliques is denoted by C = C(s, Ns). An illustration of the types of cliques associated with C1 and C2 is shown in Fig. 8.2.

Based on this, the term of the energy function U(x) can be expressed as

s CeNs

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