203.

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the prior probability. However, as demonstrated in Chapter 6, the smooth contextual concept can be used to model prior probability using smooth-prior modelling in terms of Markov random fields (MRF) so as to achieve a MRF-MAP classification. The concept of MRF can also be applied to multisource classification.

The method for deriving an MRF-MAP estimate, as described in Section 6.1, requires the specification of a posterior energy function. The posterior energy shown in Equation (6.17) is valid only for classifying a single data source. In the case of multisource classification, one has to construct a multisource posterior energy function. Equation (7.13) can be used to achieve this aim. According to Bayesian probability theory, is can be shown that [P(w_j|x_i)/P(w_j)] α P(x_i|w_j), so that the right-hand side of Equation (7.13) can be expressed as:

 

or, equivalently,

(7.14)

U(w_j|x₁, x₂,…, x_n) denotes the multisource posterior energy, and U(w_j) and U(x_i|w_j) are the prior energy and the class-conditional energy. Each of these terms is defined in Chapter 6 (Equations (6.14) and (6.16), respectively). Using this definition of multisource posterior energy, one can use the classification algorithms described in Chapter 6 using one of the methods outlined in Figures 6.18, 6.19 or 6.21 to perform multisource MRF-MAP estimation (note that one has to substitute the multisource posterior energy in place of the original single source posterior energy in the algorithm being applied). Schistad et al. (1996) use a similar equation in multisource classification.

From the foregoing description, it can be seen that not only will the posterior probability of each source influence the result of multisource consensus, but the weighting parameter a, will also contribute to the final decision. The problem is: how to specify a reliability measure in order to obtain a more reliable multisource fusion result. Note also that in the single source MRF-MAP estimate one has to choose a suitable potential parameter (Chapter 6) for the energy functions in order to obtain a good classification result. Here, in the case of multisource MRF-MAP estimation, both the potential parameter and the source-associated weighting factors have to be determined. The parameter estimation issue in multisource classification is considered in Section 7.5.

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