Image Processing Reference
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h ( t - τ)
(c)
(a)
(b)
h (- τ )
x ( τ )
1
1/2
1/2
τ
t
τ
1
-1
τ
x ( τ )
x ( τ )
x ( τ )
h ( t 1 - τ )
h (2 t 1 - τ )
h (3 t 1 - τ )
τ
τ
3 t 1
τ
t 1
(d)
2 t 1
(c)
(e)
x ( τ )
x ( τ )
h (4 t 1 - τ )
h ( - τ )
y ( t )
4 t 1
τ
τ
(f )
(b)
1/2
x ( τ )
x ( τ )
h (- t 1 - τ )
h (5 t 1 - τ )
(a)
(g)
τ
- t 1
5 t 1
τ
- t 1
0
t 1
2 t 1
3 t 1
4 t 1
5 t 1
t
Figure A.4
Example of convolution of rect function.
f ( x )
f ( x )
f ( x - x 0 )
=
*
x
x
x
x 0
x 0
Figure A.5
Example of rect function convolved with impulse function.
1 sin

u
b
-
fx
(
−→ =
x
)
F ue
()
ix u
e
ixu
c
0
0
0
b
Multiplication with a linear phase function in x or u leads to a shift of
the function in the respective Fourier spaces and vice versa. We have
δ(
F.T
.
xx
(physically this is a “tilted” plane wave).
There are a number of useful symmetries associated with Fourier transforms.
±¨→
)
¨¨ ±
e
ix u
0
0                                                                                    Search WWH ::

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