Image Processing Reference
In-Depth Information
Let A
0
A
0
¼
A, B
¼
a, and C
¼
a
T
. Then substituting into Equation 7.15, we get
1
1
1
1
aI
þ
a
T
A
0
A
0
1
a
1
A
0
A
0
þ
aa
T
¼
A
0
A
0
A
0
A
0
a
T
A
0
A
0
(
7
:
16
)
Let us de
ne
P
0
a
K
¼
(
7
:
17
)
1
þ
a
T
P
0
a
Substituting P
0
, recognizing a
T
P
0
a is a scalar and then substituting K Equation 7.16
becomes
1
P
¼
I
Ka
T
P
0
A
0
A
0
þ
aa
T
(
7
:
18
)
Substituting Equation 7.18 into Equation 7.14, we get
P
0
A
0
y
0
þ
ay
u
k
þ
1
¼
I
Ka
T
¼
P
0
A
0
y
0
þ
P
0
ay
Ka
T
P
0
A
0
y
0
Ka
T
P
0
ay
(
7
:
19
)
þ
a
T
P
0
a].
Substituting these relationships in Equation 7.19, yields following expression for the
new
u
0
¼
P
0
A
0
y
0
and from Equation 7.17, P
0
a
¼
K[1
But, from Equation 7.6,
u
matrix. (Note: for convenience, we removed k
þ
1 from the notation.)
u ¼ u
0
þ
K y
a
T
u
0
(
7
:
20
)
where, again, the weight matrix K and the matrix P are given by
P
0
P
0
a
P
¼
I
Ka
T
K
¼
and
(
7
:
21
)
1
þ
a
T
P
0
a
Notice that the new equation for
(Equation 7.20) is very similar to the general
adaptive algorithm of Equation 7.8. The matrix P
0
is updated in Equation 7.21 for
each iteration. To show the iteration steps, let us replace the subscripts
u
0
with
''
K
''
''
''
to represent kth update and use
to denote the new updates. Equations 7.20
and 7.21 then yield the following form for the old estimate and the new estimate.
''
K
þ
1
''
u
k
þ
1
¼ u
k
þ
K
k
þ
1
y
k
þ
1
a
k
þ
1
u
k
(
7
:
22
)
P
k
P
k
a
k
þ
1
P
k
þ
1
¼
I
K
k
þ
1
a
k
þ
1
K
k
þ
1
¼
and
(
7
:
23
)
þ
a
k
þ
1
P
k
a
k
þ
1
1
Note:
y in Equation 7.20 is replaced with y in Equation 7.22 to represent the
measured values.
Simplifying Equation 7.23, it can be shown
K
k
þ
1
¼
P
k
þ
1
a
k
þ
1
(
7
:
24
)
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