Image Processing Reference
In-Depth Information
y
x
z
Printer inverse
map
Forward
printer map
L
*
a
*
b
in
CMY
L
*
a
*
b
out
FIGURE 6.14
Forward and inverse printer maps.
where input L*a*b* points are on a 3-D grid of size h
h
h having a dynamic range
0
128, which is created to compensate for
the nonlinearity in P, as shown schematically in Figure 6.14.
Another interpretation for the inverse is that, for a given target color x
¼
L*a*b
in
L
100,
127
a
128, and
127
b
*
,
we seek an inverse printer map P
1
such that the printed color z and the target x are
as close as possible (or same). When this happens, we say there is a colorimetric
match between the target color and the printed color. This inverse printer map P
1
is
a key component of many color control algorithms used in digital printers. The
problemofinterestistocomputeastructuredinverseLUT,P
1
,suchthatthe
composition P
1
[P(
)] is as close as possible to the identity matrix in the minimum
mean-square error (MMSE) sense. A simple cost function to be minimized can be
formulated using the
D
E
a
*
color difference formula between the requested input
*
and the printed output color z
¼
P
(
y
) ¼
L*a*b
out
*
:
color x
¼
L*a*b
in
2
1
1
2
[
P
(
y
)
x
]
T
[
P
(
y
)
x
]
E
(
y
) ¼ D
E
[
P
(
y
)
, x
] ¼
2
k
P
(
y
)
x
k
¼
(
6
:
35
)
Notice that we do not have an exact model for P; only an approximation given by
the forward LUT is available. The interpolation methodology used to solve the three-
to-three inverse problem is summarized as follows:
(1) Obtain the forward LUT P for the given color printer. To achieve this, we
grid the CMY color space at the input of the GCR algorithm. Process the
CMY grids through the GCR algorithm to create print ready CMYK values
at each CMY grid node. For this discussion, we can also restrict the printer
forward map without the GCR algorithms. That is, simply, a CMY to
L*a*b* printer whose structured inverse, L*a*b*toCMY, is required so
that the structured inverse conforms to the performance index de
ned by
Equation 6.35. The printed color z
j
for each grid node y
j
is obtained from
experiments on the actual printer.
(2) Select
the
collection
of
target
colors
x
i
,
i
¼
1, 2, . . . , h, where
½
T
. For this, we can uniformly sample the CIELab color
space. We assume that every grid point in the sampled CIELab color space is
either inside the printer gamut or it has been mapped to a point inside the
gamut by an appropriate gamut mapping algorithm (Section 7.6).
(3) Obtain the corresponding value of yi¼
i
¼
x
i
¼
L
i
*
a
i
*
b
i
*
[C
i
M
i
Y
i
] by minimizing the cost
function given by Equation 6.35.
An approach to solve the three-to-three optimization problem is described next.
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