302.

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(2.9)

where Xi is the observed vector of the ith pixel and μj is the current mean vector of the jth cluster. The dimension of vector Xi is equal to the number of bands being used as input. For instance, if three band images are used, then Xi will be of dimension 3×1. Another choice for the calculation of distance between pixel and cluster centre is the Mahalanobis distance DM, given by:

(2.10)

where T denotes the matrix transpose, and Cj1 is the inverse of the variance-covariance matrix for cluster j, respectively. Matrix Cj is obtained by Equation (2.2) (note that only pixels belonging to cluster j are used in the calculation of Cj).

The Mahalanobis distance takes into account the shape of the frequency distribution (assumed to be Gaussian) for a given cluster in feature space, resulting in ellipsoidal clusters, whereas the use of the Euclidean distance assumes equal variances and a correlation of 0.0 between the features, giving circular clusters (Figure 2.6). Here, although the distances between the pixel and two cluster centres are the same, the pixel will be assigned to cluster a when the Mahalanobis distance measure is used (Figure 2.6a) because cluster a is of higher value than cluster b in variance-covariance measure (thus the shorter the distance). If the Euclidean distance measure is used, then the decision to place the pixel in cluster a or b will be ambiguous.

2.3.1.2 Cluster refinement

Additional criteria may be used to refine the ISODATA procedure. For instance, one may set a tolerance for maximal standard deviation, σmax, of a cluster, and also for minimal distance, dmin, between clusters. If a cluster has a standard deviation larger than σmax in any dimension, the cluster is split into two along that dimension. A merge occurs where the distance dij between cluster centres i and j is less than dmin. This split and merge operation allows the procedure to refine the number of clusters. The choice of the split and merge parameters σmax and dmin is difficult. A trial and error process can be used, with the values of the two parameters being altered in a systematic way until a satisfactory result is achieved. The definition of ‘satisfactory’ implies that the result meets the user’s prior expectations, in the sense that it is ‘interpretable’. It is apparent that the procedure is not entirely automatic and independent of the user.

An alternative approach for merging the clusters can be performed iteratively using the following equations (Mather, 1976):

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Classification Methods for Remotely Sensed Data
Classification Methods for Remotely Sensed Data, Second Edition
ISBN: 1420090720
EAN: 2147483647
Year: 2001
Pages: 354

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