Classification Methods for Remotely Sensed Data, Second Edition
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Figure 6.3 The cliques C1, C2, C3 and C4 derived from a 3×3 window centred on the pixel r of interest.
that a unique GRF exists for every MRF as long as the GRF is defined in terms of cliques on a neighbourhood system. Several authors present the proof of this theorem (Besag, 1974; Moussouris, 1974; Kindermann and Snell, 1980). A proof that the GRF is a MRF is given below.
Let P(w) be a Gibbs distribution on the set of sites (image pixels) S with respect to the neighbourhood system N. Consider the conditional probability defined by:
(6.5) |
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