Image Processing Reference
In-Depth Information
8.6.1 P
OLE
-P
LACEMENT
D
ESIGN
In pole-placement design,
the closed-loop poles are chosen to be at
locations
P
¼
[p
1
p
2
p
3
], where a pole is chosen for each of the three color patches that is
within the unit circle of the complex z-plane. We chose the poles to be positive real
numbers to avoid oscillations and possible overshoots. Typical values are between
0.3 and 0.6. We summarize below some key process steps used to generate gray-
balance TRCs using control approaches for a manual off-line sensor.
Step 1:
Select a set of N uniformly spaced gray-target patches whose gray-balance
targets are given. For equivalent neutral gray, the target values lie on the L* axis
(a*
N
i
¼
¼
b*
¼
0). Let the L* of these N gray patches be
{
d
i
}
1
, where d
1
¼
0 and
d
N
¼
100. For each target, a control loop is needed, resulting in N control loops
running in parallel.
Step 2:
Estimate an initial set of N nominal CMY values using the steps described
above.
Step 3:
Print these N patches with CMY values and measure their color (i.e., L*a*b*)
values.
Step 4:
Compute
D
E color difference between measured L*a*b* with the target
L*a*b*. If the color difference is small, stop the iteration process, save the printer
input CMY
i
¼
[C
i
M
i
Y
i
] corresponding to the ith patch, and go to Step 5; otherwise
process the error to generate V(k) and the new set of N printer input CMY values, and
continue with Step 3.
N
N
N
Step 5:
Form the three functions
{
D
i
, C
i
}
i
¼
1
,
{
D
i
, M
i
}
i
¼
1
, and
{
D
i
, Y
i
}
i
¼
1
, where
D
i
¼
d
i
). These functions are the three gray-balance TRCs corresponding
to cyan, magenta, and yellow channels, respectively. A smoothing algorithm can be
used to interpolate between points to generate full TRCs for all digital values from 0
to 255. If d
1
6¼
2.55 (100
find the TRC values below d
1
and above d
N
. Special highlight or shadow corrections described in Section 8.6.2
may be required prior to generating the
0ord
N
6¼
100, extrapolation is used to
final gray-balance TRCs.
Example 8.4
In this example, we select 7 gray targets with L*of25
90 in steps of 10 with
-
a*
0 as gray-balance targets. The Neugebauer model is used as the surro-
gate printer and the system is simulated in MATLAB. The pole-placement design is
used with three real poles, P¼
¼b*
¼
[0.2 0.3 0.2]. The nominal CMY values for the
ith patch is estimated to be
CMY
nominal
¼
½
255
2
:
55L
i
*
255
2
:
55L
i
*
255
. The B matrix is estimated at the nominal CMY using numerical
differentiation. The plot of e(k) for the
2
:
55L
i
*
first patch as a function of print number
is shown in Figure 8.13. The plot of
DE convergence for all the seven control
patches as a function of print (iteration) number k is shown in Figure 8.14. The
final gray-balanced TRCs for all three separations are shown in Figures 8.15
through 8.17.
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