Image Processing Reference
In-Depth Information
output colors were used to create the clusters) with each centroid, and then choosing the
cluster whose centroid has the shortest distance to the sample output color. After making
the right cluster assignment, run the RLS algorithm. The process for generating and
adapting piecewise linear models over time is illustrated in Examples 7.1 and 7.2.
Example 7.1
Gray [LBG]) algorithm [12] to divide the 10
4
input
Use K-means (Linde
output
printer modeling data into 10 clusters. Describe the critical steps in the algorithm.
-
Buzo
-
-
S
OLUTION
The method of classifying the color input
rst. Let the
training vectors, a mapping of the CMYK values to L*a*b* be denoted as a database
-
output data is described
2 R
nN
2 R
lN
U ¼ u
1
u
2
½
...
u
N
! Y ¼ y
1
½
y
2
...
y
N
(7
:
27)
where
u
1
, u
2
,..., u
N
are vectors containing input CMYK values for each color
containing four values denoted by n
y
1
, y
2
,...,y
N
are vectors containing output L*a*b* values for corresponding
colors containing three values denoted by l
10,000 for 10
4
characterization samples, which is a prede-
termined number based on the number of training samples that need to be classi
In this example, N¼
ed.
Generally, as the gamut becomes larger and more nonlinear, N will become larger.
N will be smaller for a calibrated printer. N could be 4
4
,8
4
,or16
4
and so on,
depending on the training set.
Using the K-means algorithm, described below, the training database is parti-
tioned into K clusters, U
k
for k¼
1, 2, 3, . . . , K as follows:
k
2 R
nN
k
U
k
¼ u
1
1
u
2
2
...
u
N
k
!Y
k
k
2 R
lN
k
¼ C
k
y
1
1
y
2
2
...
y
N
k
(7
:
28)
where u
1
u
2
½
k
are the vector elements containing the N
k
input CMYK
values for the kth cell. These clusters will be generated by the K-means algorithm.
After assigning the colors to the clusters, vectors u
1
, u
2
, . . . of Equation 7.27 do not
necessarily correspond to the u
1
, u
2
, . . . of Equation 7.28 because of the reordering
of the training sample data that occur during the clustering process. Hence, we used
the notation, u
1
1
, u
2
2
,...,u
N
k
to describe new input vectors, and [y
1
1
, y
2
2
,...,y
N
k
]
k
are the vector elements formed with L*a*b* values, each having three elements.
The number of elements could be higher when a re
...
u
N
ectance spectral database is
used for clustering (not considered in this example). Vector C
k
is the centroid of the
L*a*b* values in a kth cluster. The relationship between K and N is as follows:
N ¼
X
K
N
k
(7
:
29)
k¼
1
Figure 7.6 shows a
flowchart illustrating the method of determining the
centroids. These centroids will become the centers of the respective clusters of
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