162.

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where α_k is single site clique potential parameter, which can be regarded as the first penalty term specially for label k. The larger the value of α_k, the less the probability that pixels in the image are assigned to label k. The term δ(a, b) is a step function defined as δ(a, b)=1 if ab, and −1 otherwise, and β (>0) is pair-wise clique potential parameter. β·δ(a, b) is the second penalty term. The larger the value of β, the stronger is the smoothing force. In other words, if w_r does not agree with its neighbours, higher prior energy (i.e. lower probability) will result. Note that, in most applications, the probability of each class is considered to be the same, and thus α_k is set to 0, unless otherwise specified.

The conditional distribution of the observed data d_r (e.g. pixel values at site r) given the true label w_r (w_r=class k) is often assumed to be Gaussian, and can be formulated as:

(6.15)

where ρ is the dimensionality of the feature space (e.g. the number of image bands), is the class-conditional covariance matrix for class k and

(6.16)

is the class-conditional or likelihood energy, where u_k is mean vector of class k.

By combining Equations (6.14) and (6.16), one obtains the posterior energy U(w|]d):

(6.17)

The MAP estimate, which maximises posterior probability P(w|d), and which is equivalent to minimising the posterior energy, is defined by:

(6.18)

The solution of Equation (6.18) requires a specially designed approach. An iterative algorithm is normally used because the labelling of each pixel has an effect on the labels to be assigned to its neighbours. Algorithms for the determination of energy minimisation are considered in Section 6.5.

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