Image Processing Reference
In-Depth Information
Now we will discuss how this model can be applied to predict the printer drift. Say
we have the outputs of the system in L*a*b* and density, d,measuredusingacolor
sensor at time index n corresponding to time t. The initial time t
0
is assumed to be zero,
with corresponding time index n
¼
0. We predict the output in each channel (say cyan),
independent of the other channels (i.e., magenta, yellow, and black), using the above
model (Equation 7.74) with y(n)
¼
L*(n)
L*(0) for predicting L*, y(n)
¼
a*(n)
a*(0)
for predicting a*, y(n)
d(0) for
predicting print density represented by the symbol d. After estimating the AR
coef
¼
b*(n)
b*(0)
for predicting b*, y(n)
¼
d(n)
cients for the three channels using Equation 7.74 or Equation 7.76, we can
predict the new values L*(n), â*(n), b*(n), d(n) from the P previous values of the
outputs using Equation 7.62. That is
X
P
y
(
n
) ¼
c
a
i
y
(
n
i
)
(
7
:
77
)
i
¼
1
where c is the initial value of L* for predicting L* and the corresponding initial values
for predicting a*, or b*, and density d. This approach ignores the interaction between
L*a*b* and d treating them independent of each other. The model order P can be
determined using statistical techniques including the minimization of an order selec-
tion criterion [70] (not discussed in this topic).
7.4.4.2 Vector Autoregressive Model Applied to Printer Drift Prediction
In the vector AR model, we consider the dependence of one channel on the other
channels while predicting the output of the printer. So in the vector AR model, the
measured output process is a 3
1 vector random process de
ned by
2
4
3
5
(
n
)
L*
(
)
L*
0
y
(
n
) ¼
(
n
)
a*
(
)
(
:
)
a*
0
7
78
b*
(
n
)
b*
(
0
)
The predicted output in terms of the P (previously) measured output values is given
by the following equation:
X
P
y
(
n
) ¼
A
i
y
(
n
i
)
(
7
:
79
)
i
¼
1
The error signal between measured and predicted outputs is given by
X
P
e
(
n
) ¼
y
(
n
)
y
(
n
) ¼
y
(
n
) þ
A
i
y
(
n
i
)
(
7
:
80
)
i
¼
1
where A
i
for i
¼
cients.
The prediction error is assumed to be zero-mean white noise process with unknown
covariance matrix
1,...,P are 3
3 matrices which de
ne the VAR matrix coef
. The error is minimized in the least squares sense (similar to the
scalar case). The MSE is given by
S
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