Image Processing Reference
In-Depth Information
¼
Ee
(
n
)
e
T
(
n
)
Ee
2
(
n
)
"
#
"
#
X
X
P
P
y
T
(
n
) þ
y
T
(
n
i
)
A
i
Ey
(
n
) þ
A
i
y
(
n
i
)
i
¼
1
i
¼
1
X
X
X
X
P
P
P
P
A
i
R
yy
(
i
) þ
R
yy
(
i
)
A
i
þ
A
i
R
yy
(
j
i
)
A
j
¼
R
yy
(
0
) þ
(
7
:
81
)
i
¼
1
i
¼
1
i
¼
1
i
¼
1
where R
yy
(i) is the 3
3 correlation matrix of the output of the system at lag i and is
given by
2
4
3
5
r
LL
(
i
)
r
La
(
i
)
r
Lb
(
i
)
R
yy
(
i
) ¼
r
aL
(
i
)
r
aa
(
i
)
r
ab
(
i
)
(
:
)
7
82
r
bL
(
i
)
r
ba
(
i
)
r
bb
(
i
)
Since y(n) is real, R
yy
(
i
) ¼
R
yy
(
i
)
.
The diagonal elements of the positive de
nite matrix R
yy
(i) are the autocorrela-
tion of the three components of the color vector L*a*b* and the off diagonal
elements are measure of correlation between the three coordinates of the L*a*b*
color vector. We now optimize the cost function given by Equation 7.81 with respect
to matrix A
i
. The result is similar to the Yule
-
Walker equations for the scalar case
and is given by
2
4
3
5
2
4
3
5
¼
2
4
3
5
R
yy
(
0
)
R
yy
(
1
)
R
yy
(
P
)
A
0
A
1
.
A
P
0
R
yy
(
1
)
R
yy
(
0
)
R
yy
(
P
1
)
(
7
:
83
)
.
.
.
0
R
yy
(
P
)
R
yy
(
P
)
R
yy
(
)
1
0
where
A
0
is a 3
3 matrix with A
0
(i, j)
¼
1
0isa3
3 matrix of zero elements
S
is the 3
3 covariance matrix of the prediction error signal e(n)
S
is the covariance matrix of the prediction error function and is given by
)
A
1
þ
R
yy
(
)
A
2
þþ
R
yy
(
P
)
A
P
S ¼
R
yy
(
0
) þ
R
yy
(
1
2
(
7
:
84
)
P
i
¼
1
, we can predict the new values
of L*(n), â*(n), and b*(n) from P previous values of the output using the following
equation:
After estimating the VAR matrix coef
cients A
fg
X
P
y
(
n
) ¼
c
A
i
y
(
n
i
)
(
7
:
85
)
i
¼
1
[L*(0) a*(0) b*(0)]
T
.
Comments about the experimental validation of AR drift models are shown next.
where c
¼
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