Three-Dimensional Control of Color Management Systems - Control of Color Imaging Systems: Analysis and Design - page 365

2 3

@ L*
@ C

@ L*
@ M

@ L*
@ Y

@ L*
@ K

4 5

@ a*
@ C

@ a*
@ M

@ a*
@ Y

@ a*
@ K

B ¼ ( 7 : 89 )

@ b*
@ C

@ b*
@ M

@ b*
@ Y

@ b*
@ K

A nal

L*a*b*toCMYK LUT, which is the sum of the initial (or nominal) LUT containing
the GCR and the newly found correction

final multidimensional pro le LUT (i.e., ICC pro le) is generated based on the

D CMYK (vector V in Figure 7.41) through
iterations on the CMYK printer or the CMYK printer model.
The inversion approach uses a 3-to-4 control-based algorithm with a MIMO gain
matrix. The Jacobian matrix at the nominal CMYK values for each node color is used
to compute the gain matrix. The pole-placement algorithm, place() [105,106]
(see Section 5.2.3) or LQR methods (see Section 5.3.2) are candidate algorithms
used for computing the gain matrix. Furthermore, best actuator algorithm (similar to
the one described in Figure 7.32) would be required for node colors near the
boundary to select the best CMYK values during the iteration run. Gain scheduling
algorithms based on model-predictive control technology [100] are other methods
suitable for use in the control algorithm.

7.5.4.2 K-Restricted GCR

The function used to generate the black (K) component can be a parametric multi-
dimensional function known as the K-function or K-restricted GCR function. The
K-function produces the first (initial) estimated values for the black component based

on the L*a*b* values of the node colors. The initial estimated values for black (K)
are then used to produce values for other separations, C, M, and Y, in the initial
CMYK LUT of Figure 7.42.
The multidimensional K-function may be de

ned by the following equation:

1 1

2 U h

= L * 2

K ¼

1

( 7 : 90 )

a * 2 þ b * 2

U h e a þ

where

0
@

1
A

1

U ¼ 2 L *

L 0 L * 2

0 : 5 ( 7 : 91 )

e b

þ 1

and L 0 , are predetermined values, selected based on the color gamut.

Figure 7.42 shows two graphs of the function generated by Equations 7.90 and 7.91.
The graph shown on the left side is the L* response (vertical axis) with respect to K
values (horizontal axis). The graph on the right contains K values (vertical axis) as a

h , b , and a