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.
Search WWH ::




Custom Search