Image Processing Reference
In-Depth Information
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
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