65.

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grades to the interval [0, 255] and produce membership grade images for each class.

The choice of the number of clusters is not always easy, especially, when the user does not have any knowledge about the number of information classes. An alternative procedure is to allow the clustering algorithm to decide the number of clusters. In this case, an algorithm called ‘fuzzy partition, optimum number of clusters’ (Gath and Geva, 1989) can be adopted to solve the problem. This algorithm is a combination of the fuzzy c-means and fuzzy maximum likelihood algorithms. The latter is described in Section 4.3. The classification process is carried out in terms of unsupervised learning (Gath and Geva, 1989). The algorithm is similar to the fuzzy c-means algorithm described above, except that the distance measure used to define similarity between pixel x_k and a cluster centre v_i is defined as:

(4.16)

where P_i is the a priori probability of the ith cluster, defined as:

(4.17)

n is the number of pixels, F_i is the fuzzy covariance, expressed by:

(4.18)

and |F_i| denotes the determinant of the ith cluster covariance matrix.

Note that during the clustering process, the pixel membership grades, the cluster means, and the covariance matrix F_i have to be modified at each iteration. Once the algorithm has converged, the process is repeated using a different number of clusters. Finally, the optimal clustering scheme is chosen in terms of the minimisation of the overall determinants of the fuzzy covariance matrices, i.e. |F_i|, for all i.

The standard fuzzy c-means method assumes that every information class has been specified, so that each pixel can be completely described in terms of its membership of the given set of categories. This is not always the case, as Foody (2000b) points out. He suggests that the standard fuzzy c-means method produces membership grades that are analogous to relative proportions of the specified classes (with the possibility of error if some classes are omitted). The possibilistic c-means algorithm (Krishnapuram and

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