Image Processing Reference
In-Depth Information
h
(
x
, λ
1
;
y
, λ
2
)
y
δ(
x
-λ
1
;
y
-λ
2
)
y
λ
2
λ
2
Imaging
system
x
x
λ
1
λ
1
FIGURE 2.1
PSF of an imaging system.
Next, the output for each weighted impulse is determined from the impulse response
f
(l
1
,
l
2
)d(
x
l
1
, y
l
2
) !
f
(l
1
,
l
2
)
h
(
x,
l
1
; y,
l
2
)
(
2
:
2
)
Finally, using the linearity principle, the output image is determined as the sum
(integral) of the outputs of all the impulses that make up the input.
1
1
g
(
x, y
) ¼
f
(l
1
,
l
2
)
h
(
x,
l
1
; y,
l
2
)dl
1
dl
2
(
2
:
3
)
1
1
This is a general input
output relationship for a linear imaging system. The integral
equation given by Equation 2.3 is known as Fredholm equation of
-
first kind. It
assumes that the shape of the PSF depends on its location in the input image plane.
Such a system is referred to as a linear space-variant system. Examples for such system
are the subject motion when a picture is being captured with a stationary camera, or
capturing a picture of a scene with objects at various planes. In many instances, the
shape of the PSF is independent of its location and is thus the same everywhere, that is,
h
(
x,
l
1
; y,
l
2
) ¼
h
(
x
l
1
, y
l
2
)
(
:
)
2
4
As a result of this, the PSF can be characterized as a function with two arguments x
and y, or simply h
(
x, y
)
. Such a system is known as a linear space-invariant (LSI)
system. The input
output relationship simpli
es for such a linear system and is
-
called convolution integral.
1
1
g
(
x, y
) ¼
f
(l
1
,
l
2
)
h
(
x
l
1
, y
l
2
)dl
1
dl
2
(
2
:
5
)
1
1
A major advantage of LSI systems is that they can be analyzed using Fourier
transform theory. For that purpose, many imaging systems are approximated by
LSI systems. Now, we give simple examples of LSI imaging systems.
2.2.1 P
OINT
S
PREAD
F
UNCTION OF A
D
EFOCUSED
L
ENS
A simple example of an LSI imaging system is a defocused lens. An exact method of
finding the PSF of such a system is based on physical optics and has been described
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