Image Processing Reference
In-Depth Information
As an example of a circularly symmetric OTF, consider a defocused lens. The OTF is
given by Hankel transform as
1
ð
R
v
r
ð
R
1
p
R
2
J
0
(
r
v
r
)d
r
¼
2
H
(v
r
) ¼
2
p
rh
(
r
)
J
0
(
r
v
r
)d
r
¼
2
p
r
xJ
0
(
x
)d
x
(
2
:
25
)
2
r
R
2
v
0
0
0
Using the known relationship xJ
0
(
x
) ¼
d[
xJ
1
(
x
)]
d
x
,wehave
R
v
r
ð
2
d
xJ
1
(
x
)
d
x
2
2
J
1
(
R
v
r
)
R
v
r
R
v
r
0
H
(v
r
) ¼
d
x
¼
R
2
xJ
1
(
x
)j
¼
(
2
:
26
)
2
r
R
2
2
r
v
v
0
The MTF is given by
¼
H
(v
r
)
H
(
2
J
1
(
R
v
r
)
R
v
r
MTF(v
r
) ¼
(
2
:
27
)
0
)
where J
1
(
x
)
is a Bessel function of the
first kind of order 1 and its Taylor series
expansion is
1
i
¼
0
(
x
22iþ1i!(iþ
þ
1
2
22iþ1i!(iþ
þ
1
i
!(
i
þ
i
J
1
(
x
) ¼
1
)
(
2
:
28
)
1
)!
Figure 2.3 shows the MTF of a defocused lens.
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
R
ω
ρ
FIGURE 2.3
MTF of defocused lens.
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