This reduces the number parameters per mixture component celeux and govaert
374 Clustering
before a satisfactory clustering is produced. In any case, it is worthwhile trying several initialisations, since agreement between the resulting classifications lends more weight to the chosen solution. Celeux and Govaert (1992) describe approaches for developing the basic EM algorithm to overcome the problem of local optima.
Summary The approach detects a linear pattern in preprocessed images via modelbased clustering. It employs an approximate Bayes factor which provides a criterion for assessing the evidence for the presence of a defect.
The data Twodimensional point pattern data are generated by thresholding and cleaning, using mathematical morphology, images of fabric. Two fabric images are used, each about 500 ð 500 pixels in size.
Results were presented for some representative examples, and contrasted with a Hough transform.
10.5 


10.5.1 Clustering criteria
Let the n data samples be x1; : : : ; xn. The sample covariance matrix,O�, is given by
jD1
g n
Tr.SW/ D1 X X z jijxi � m jj2
jD1 iD1
g
D1 X Sj
jD1
where Sj DPn the minimisation of Tr.SW/ is equivalent to minimising the total withingroup sum of iD1z jijxi � m jj2, the withingroup sum of squares for group j. Thus,