Classification Methods for Remotely Sensed Data, Second Edition

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with fuzzy means and fuzzy boundaries, and is less dependent on the initial state of clustering. The algorithm is described as follows.

Let , be a finite subset of R^d, the d dimensional real number vector space (e.g. where three features are used for clustering, d= 3). Let the integer c, n≥c>2, denote the number of fuzzy subsets. Thus, a fuzzy c partition of X can be represented by a (c×n) matrix U in which each entry of U, denoted by u_ik, satisfies the following two constraints:

In the case of image classification, n will be the number of pixels, and c is the number of clusters. The resulting matrix is shown in Figure 4.3. The value u_ik corresponding to the entry at the location (i, k) stores the kth pixel’s membership value for class i (see Figure 4.3). Note that all the entries in a given column must sum to 1 as specified in Equation (4.12).

The clustering criterion used in the fuzzy c-means algorithm is based on minimising the generalised within-groups sum of square error function J_m:

is the vector of cluster centres (i.e. the means of the clusters), with v_i R^d, and m is the membership weighting exponent, 1≤ m<∞. The term (x_k−_ki)² in Equation (4.13) can be replaced by the Mahalanobis distance (Chapter 2), the calculation of which requires the fuzzy covariance matrix, which is introduced in Equation (4.18). For m>1 and

Figure 4.3 The fuzzy c-means clustering membership matrix