Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 315

the partitioned database. Various initialization values are entered, including , K,

e

m, D 0 , i, and E.

is a distortion threshold, which indicates the maximum allowable

e

distortion, as de ned by the criterion associated with desired system performance.

K is the number of clusters into which the database is to be partitioned. The larger
the parameter K is, the better the results obtained from the model (see Figures
7.9 7.11), but it will increase the processing time. m, D 0 , i, and E are values used

in the algorithm. Speci

-

cally, m and i are simply iteration counters, and may be
initially set at 0 and 1, respectively. D 0 is an initial distortion setting, and is set at
an arbitrary large positive number, such as 1000. E is a distortion value, which is
initially set at 0.
Empty sets A 1 , A 2 ,..., A K are established apriori. These are the clusters of the
database, which will be filled with initial values and then updated until certain

algorithm criteria are met, as described below. Initial cluster centroids C 0 are set
equal to C k ,wherek¼ 1,2,...,K. Thus, one centroid C k is assigned to each empty

set. The centroids C k may be arbitrary, or may be set using a

based
on previous experience. For each training sample yi, i , expressed as a color vector, the
Euclidean distance d to each cluster centroid C k is determined, and from
the Euclidean distances d. The cluster A J having the minimum Euclidean distance is
identi

best guess

'' ''

ed as follows:

J ¼ argmin

k

d ¼ argmi k dy i , C k ð Þ (7 : 30)

Then, y i is accumulated into A J . Next, the distortion E is determined by

E ¼ E þ d min (7 : 31)

where d min is the minimum distortion value d obtained in earlier steps. Accumu-
lation of the other training sample yi i into the appropriate cluster A J continues until
all training samples have been collected into the appropriate clusters. When i i¼N,
that is, when all training samples have been accumulated, an updated cluster
centroid C k is determined for each cluster A 1 , A 2 ,...,A K . This determination may
be performed according to the following equation:

P L k
i¼ 1 A k (i)

L k

C k ¼ (7 : 32)

where L k is the number of vectors in cluster A k . The average distortion D m is
obtained by

E
N

D m

¼ (7 : 33)

Then, it is determined whether distortion is within the distortion threshold . This

determination may be made by determining whether the following relation is
satis

e

ed:

D m - 1 D m

D m

e (7 : 34)

If Equation 7.34 is not satis ed, the process sets E¼ 0 and m¼mþ 1, and repeats

the previous steps as shown in the flowchart, beginning this time with the updated