Image Processing Reference
In-Depth Information
6.4.3.1 Selection of Step Size Parameter
m
The updating equation for y(k) is given by
) ¼
y
(
k
) m
J
k
{
P
[
y
(
k
)]
x
}
y
(
k
þ
1
(
6
:
40
)
De
ne the error e(k)as
e
(
k
) ¼
P
[
y
(
k
)]
x
(
6
:
41
)
Use the
first-order linear approximation
e
(
k
þ
1
)
e
(
k
) ¼
P
[
y
(
k
þ
1
)]
P
[
y
(
k
)]
J
k
[
y
(
k
þ
1
)
y
(
k
)]
(
6
:
42
)
Using the updating law given by Equation 6.38, we have
)
e
(
k
) m
J
k
J
k
e
(
k
)
e
(
k
þ
1
(
6
:
43
)
or
e
(
k
)
) ¼
I
m
J
k
J
k
e
(
k
þ
1
(
6
:
44
)
The Jacobian matrix J
k
is not changing rapidly from one iteration to another;
therefore, we assume that J
k
¼
J
0
. With this assumption,
e
(
k
)
) ¼
I
m
J
0
J
0
e
(
k
þ
(
:
)
1
6
45
Therefore, for the iterations to converge, the error e(k) must approach zero as k
!1
.
This requires that all eigenvalues of matrix I
m
J
0
J
0
must lie inside the unit circle
in the complex plane, that is,
<
l
i
I
m
J
0
J
0
1
i
¼
1, 2, 3
(
6
:
46
)
Therefore, the parameter
m
should be selected to satisfy the following condition:
2
0
< m <
(
6
:
47
)
l
max
J
0
J
0
, a more conservative, but easier to compute, method
for obtaining an upper bound for parameter
Tr
J
0
J
0
J
0
J
0
Since
l
max
m
is
2
0
< m <
(
6
:
48
)
Tr
J
0
J
0
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