Image Processing Reference
In-Depth Information
where
x
k
þ
1
¼
new estimate
x
k
¼
old estimate
a ¼
weight factor
measured
values
model
output
e
¼
The initial (estimated) parameters in the
u
0
matrix are obtained fromEquation 7.6. In
the adaptation context, we refer to
u
k
þ
1
as the new estimate.
When the new data arrive, let a
T
represent the regression matrix for the next set of data
while
u
k
as an old estimate, and
y is the corresponding output. This additional data can be incorporated using the
original equation (Equation 7.2) to obtain the overdetermined linear equation,
y
¼
A
u
(
7
:
9
)
This equation is similar in structure to Equation 7.2, with
;
A
0
a
T
y
0
y
A
¼
y
¼
(
7
:
10
)
u
k
þ
1
, are built by using all the available input and output
data points as in Equation 7.11.
The updated estimates,
"
#
1
T
A
0
a
T
y
0
y
A
T
A
1
A
0
a
T
A
0
a
T
A
T
y
¼
u
k
þ
1
¼
(
7
:
11
)
As expected, the equation for the new estimates again has similar structure as the
initial estimate Equation 7.6. The previously calculated inverse from Equation 7.6 is
in fact used in the new Equation 7.11. However, we will need to calculate the inverse
again as new data arrive. This means that unnecessary computation is required to
solve Equation 7.11. A more ef
cient way of calculating the new solution utilizes
Equation 7.6, which is shown next.
Few intermediate steps are necessary before the new estimates are determined.
From simple matrix manipulation of Equation 7.10, we can write
A
T
A
¼
A
0
A
0
þ
aa
T
; A
T
y
¼
A
0
y
0
þ
ay
(
7
:
12
)
A
T
A
1
1
P
0
¼
A
0
A
0
Also let
P
¼
and
(
7
:
13
)
Further manipulation is done as follows to reach a simple form for the new
u
matrix.
Rewriting Equation 7.11, we have
A
T
A
1
A
T
y
¼
A
0
A
0
þ
aa
T
1
A
0
y
0
þ
ay
u
k
þ
1
¼
(
7
:
14
)
Matrix inversion lemma (Equation 5.64) states that for any matrices A, B, and C
of appropriate size
1
CA
1
(
A
þ
BC
)
1
¼
A
1
A
1
BI
þ
CA
1
B
(
7
:
15
)
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