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Figure 8.2: Illustration of all the possible cliques types associated with a 3 by 3 pixels neighborhood. (a) One pixel clique. (b) Two-pixel cliques. (c) Four-pixel clique. (d) Three-pixel cliques.

It is a summation of all the clique energy of each pixel along the whole image. VsC(x) is called clique energy function.

The assignment of clique energy is completely application dependent [33]. In our study, in order to obtain a precise model description, clique energy is calculated as a summation of two parts, pixel constraint and edge constraint [19]. The expression of clique energy function is written as

where the VsP(x) is the energy function derived by considering the spatial constraint of pixel s and its neighboring pixels. It is defined as

where xs and xh are the labels at location s and h. VsE(x) is the energy function with an edge constraint:

heN,

Assume h is an edge pixel within the neighborhood of pixel s (see Fig. 8.1),

+p2 if xs = xg, h e Ns, and s, g are on different sides of an edge, —¡32 if xs = xg, h e Ns, and s, g are on different (8.8)

sides of an edge. 0 otherwise,

The introduction of edge constraint on energy expression is very straightforward. For two nonedge pixels, s and g, they are unlikely to be in the same region if an edge pixel, h, is in between of them (see Fig. 8.1). Compared to the energy function in traditional MRF models [20, 22], this additional edge constraint can provide more strict definition for energy function, so that the label-updating process can be more sensitive at boundaries of regions. Moreover, for those small regions with boundary points in the edge map (Canny edge detector is applied in this study), this edge constraint can also protect them from being merged with its large neighboring regions. In summary, the a prior probability for MRF can be written as:

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