Image Processing Reference
In-Depth Information
To minimize the functional H, we need to differentiate H with respect to the
three variables x
(
k
)
, u
(
k
)
,
l(
k
)
, and x
(
N
)
and set the results equal to zero.
q
H
q
x
(
k
)
¼
Qx
(
k
) þ
A
T
l(
k
þ
1
) l(
k
) ¼
0
(
5
:
53
)
q
H
q
u
(
k
)
¼
Ru
(
k
) þ
B
T
l(
k
þ
1
) ¼
0
(
5
:
54
)
q
H
q
l(
k
)
¼
Ax
(
k
1
) þ
Bu
(
k
1
)
x
(
k
) ¼
0
(
5
:
55
)
q
H
q
x
(
N
)
¼
Sx
(
N
) l(
N
) ¼
0
(
5
:
56
)
The above equations can be solved by assuming that
l(
k
) ¼
P
(
k
)
x
(
k
)
(
5
:
57
)
Substituting Equation 5.57 into Equation 5.53 results in
Qx
(
k
) þ
A
T
P
(
k
þ
1
)
x
(
k
þ
1
)
P
(
k
)
x
(
k
) ¼
0
(
5
:
58
)
From Equation 5.54, we have
u
(
k
) ¼
R
1
B
T
l(
k
þ
) ¼
R
1
B
T
P
(
k
þ
1
1
)
x
(
k
þ
1
)
(
5
:
59
)
By substituting Equation 5.59 into Equation 5.51, we have
) ¼
Ax
(
k
)
BR
1
B
T
P
(
k
þ
x
(
k
þ
1
1
)
x
(
k
þ
1
)
(
5
:
60
)
Equation 5.60 is now used to solve for x
(
k
þ
1
)
. That is
) ¼ [
I
þ
BR
1
B
T
P
(
k
þ
)]
1
Ax
(
k
)
x
(
k
þ
1
1
(
5
:
61
)
We now substitute Equation 5.61 into Equation 5.58 to obtain
Qx
(
k
) þ
A
T
P
(
k
þ
)[
I
þ
BR
1
B
T
P
(
k
þ
)]
1
Ax
(
k
)
P
(
k
)
x
(
k
) ¼
1
1
0
(
5
:
62
)
Equation 5.62 must be satis
ed for all x
(
k
)
. Therefore, we must have
P
(
k
) ¼
A
T
P
(
k
þ
)[
I
þ
BR
1
B
T
P
(
k
þ
)]
1
A
þ
Q
1
1
(
5
:
63
)
The above equation can be simpli
ed using the matrix inversion lemma which states
that for any A, C,andD matrices of appropriate dimensions, we have
(
A
þ
CD
)
1
¼
A
1
A
1
C
(
I
þ
DA
1
C
)
1
DA
1
(
5
:
64
)
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