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

(6.24)

FIGURE 6.3: Illustration of the loss of information for ellipsoidal clusters (Hoeppner et al.,

1999)

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||

(6.25)

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