Image Processing Reference
In-Depth Information
Construction of the gain matrix: We use the LQR for designing the gain matrix of the
feedback controller that is used to compute the CMYK values at the pro
le nodes for
the updated printer model. The linear quadratic controller (Section 5.3.2) minimizes
a selected quadratic objective function for a node color over the iteration length,
N, which is shown below
2
P
N
1
J
¼
1
0
x
T
(
k
)
Qx
(
k
) þ
u
T
(
k
)
Ru
(
k
)
(
:
)
7
92
k
¼
where x(k) is the state vector containing L*a*b* values and u(k) is the actuator vector
for four-color system. The state space formulation of Figure 7.41 is used for
obtaining the gain matrix with LQR design.
Since our problem is focused on suppressing black, we set the values in the
Q and R matrices as follows:
Q
¼ diag
q
1
½
q
2
q
3
(
7
:
93
a)
R
¼ diag
r
1
½
r
2
r
3
a
(
7
:
93
b)
fixed values for the elements (e.g., q
1
¼
q
2
¼
The Q matrix is 3
3 with very small
10
3
). Varying the scale value, r, can allow the K to be suppressed (or not).
Generally, when the user
q
3
¼
1
finds excessive black in neutrals, they can change the
values of the parameter r. The R matrix contains
a
as the weight which is used to
suppress black.
The gain matrix equation is obtained by using the procedure described in
Chapter 5. We state the
final equations below.
Gain matrix:
1
K
(
k
) ¼
R
1
B
T
P
(
k
þ
I
þ
BR
1
B
T
P
(
k
þ
1
)
1
)
A
(
7
:
94
a)
Recursive equation to compute P(k):
P
(
k
) ¼
A
T
P
(
k
þ
)
A
A
T
P
(
k
þ
)
BR
1
I
þ
B
T
P
(
k
þ
)
BR
1
1
1
1
B
T
P
(
k
þ
)
A
þ
Q
(
:
b)
1
7
94
Boundary condition:
P
(
N
) ¼
0
(
7
:
94
c)
It turns out that the state space model for each node color has an A matrix which is
equal to an identity matrix (i.e., A
¼
diag[1 1 1]). Note that k refers to iteration
number in the above equation. Figure 7.54 illustrates the neutral response without the
scum dots. Thus, by using the LQR design, we are able to automatically provide low
gain for black dots in the regions where black is not desirable. The new gain matrix
uses the weight pro
le to emphasize the removal of black dots via the appropriate
values in R matrix.
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