Image Processing Reference
In-Depth Information
Dunn (Dunn, 1973) and Bezdek (Bezdek J.C., 1992) is known as the fuzzy c-means (FCM)
algorithm. The FCM algorithm is only capable of generating spherical clouds of points that
results from employing the Euclidean metric in the calculations. This can be a significant
limitation because the spherical shape does not necessarily represent the data in the optimal
way. An alternative algorithm, proposed by Gustafson and Kessel (GK) (Gustafson and
Kessel, 1979), has a different metric that permits the shape of clusters to be ellipsoidal which
can be better fitted to the data. Additionally, for ellipsoidal clusters the loss of information
represented by a box formed by overlapping regions is lower, i.e. the box is rectangular and
not square as for FCM (see. Fig. 6.3) . Each cluster is induced by a symmetric and positive
definite matrix defined as a norm of its own:
||y|| B = p y T By
FIGURE 6.3: Illustration of the loss of information for ellipsoidal clusters (Hoeppner et al.,
Using this approach the distance value d(x, v) is defined as the Mahalanobis metric
d(x, v) =||x−v|| B = q ||x−v|| T B||x−v||
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