Image Processing Reference
In-Depth Information
r
¼
y
0
A
0
u
0
(
7
:
4
)
and then form the sum of the squares of the residues,
Þ
T
S
¼
rr
T
¼
y
0
A
0
u
0
ð
Þ
y
0
A
0
u
0
ð
(
7
:
5
)
Now minimize S by differentiating Equation 7.5 with respect to
u
0
and set the resulting
equation to zero and solve for
u
0
. This process yields the standard least-squares solution,
1
A
0
y
0
u
0
¼
A
0
A
0
)
The term A
0
A
0
1
A
0
is called the pseudo-inverse of matrix A
0
. Equation 7.6 is used
to obtain an initial guess of the parameter matrix using the measured L*a*b* values
for test colors whose CMYK values are included in the A
0
matrix as u
1
¼
C; u
2
¼
M;
u
3
¼
Y; u
4
¼
K.
Using this technique, other parameter models can also be built easily. For
example, the partial quadratic model can be modeled as
(
7
:
6
u
1
u
2
u
3
u
4
A
0
¼
1
u
1
u
2
u
3
u
4
(
7
:
7
)
This is yet another form suitable for linear regression. Note, A
0
is a matrix of size
N
9 (as shown in Equation 7.7) when N number of test colors are used.
7.4.1.2 Recursive Least-Squares Estimation Algorithm
The RLS algorithm is an adaptive learning method that can be used to re
ne the
parameters of the model and perform system identi
cation with the data collected
with in situ sensors for a time-varying print engine. It can be used as a onetime
estimation algorithm to improve the numerical accuracy of a least-squares model.
Once the sensor data become available, it allows the system to update the parameters
to reduce the error between the outputs of the linear model and the actual printer to
an acceptable level. A schematic of the adaptation process is shown in Figure 7.5 for
a test color. Most adaptive algorithms are of the following form:
x
k
þ
1
¼
x
k
þ a
e
(
7
:
8
)
Color sensing
device
Printing
device
CMYK values
for target color
+
-
Parameterized
model
FIGURE 7.5
System block diagram showing adaptation process used to tune the parameters
of the linear model.
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