Image Processing Reference
In-Depth Information
Using the trigonometric identity
v
x
cos uþv
y
sin u ¼ v
r
cos (uw)
(
2
:
19
)
where
q
v
v
r
¼
2
x
þv
2
y
and
w ¼ tan
1
v
y
v
x
we have
ð
1
1
ð
2
p
2
p
h
(
r
)
e
jr
v
r
cos(uw)
r
d
r
du ¼
e
jr
v
r
cos(uf)
dud
r
H
(v
x
,
v
y
) ¼
rh
(
r
)
(
:
)
2
20
0
0
0
0
Since function
cos (u)
is periodic with period of 2
p
and integration is over one
period, the inner integral is independent of
f
. Therefore, we set
f ¼
0.
1
ð
2
p
e
jr
v
r
cosu
dud
r
H
(v
x
,
v
y
) ¼
rh
(
r
)
(
2
:
21
)
0
0
The inner integral is given by
ð
2
p
e
jr
v
r
cosu
du ¼
2
p
J
0
(
r
v
r
)
(
2
:
22
)
0
where J
0
is the Bessel function of
first kind and order 0. Hence,
1
H
(v
r
) ¼
2
p
rh
(
r
)
J
0
(
r
v
r
)d
r
(
2
:
23
)
0
This is known as Hankel transform. Thus, the OTF of a rotationally symmetric PSF
is also rotationally symmetric and is related to its 1-D PSF through 1-D Hankel
transform. The inverse Hankel transform can be used to express the PSF in terms of
OTF. This is given by
1
0
v
r
H
(v
r
)
J
0
(
r
v
r
)dv
r
1
2
h
(
r
) ¼
(
2
:
24
)
p
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