Image Processing Reference
In-Depth Information
∞
+∞
∫
∫
FhxFfx gx
{( )}
=
()(
′
− ′
xx
)
d
′
=
hxe
()
ik x
x
(
)
d
x
−∞
−∞
(A.12)
+∞
∞
∫∫
′
=
−
′
fx gx
()(
xxe
)d
′
�
�
ik x
x
(
)
d
x
−∞
−∞
Let
x
-
x
′ be written as
y
then d
y
= d
x
and we can write
+∞
∞
∫∫
Fhx
{
()
}
=
f xgyye
()()
′
d
�
�
ik yx
x
(
(
+
′
))
d
x
′
−∞
−∞
+∞
+∞
∫
∫
=
=
f
(()
xe
′
ik x
(
′
)
d
xgye
′
()
i ky
(
)
d
yFkGk
()()
x
x
x
x
−∞
−∞
Graphical illustration of convolution
:
If
g
=
f
, then a correlation function is termed as an autocorrelation function,
and its Fourier transform is an energy spectrum (Wiener-Khinchin Theorem)
Examples of convolutions
:
triangle()
x
=
rect
()*(
x
rect
x
)
(A.13)
F
{
triangle
(
x
)}
=
sin( )sin ()
c
u
⋅
c
u
=
sinf
cu
2
()
(A.14)
fx
()*(
δ− =
xx
)
fx
(
−
x
)
(A.15)
0
0
In the Fourier domain, these convolutions are simple products of the func-
tions' Fourier transforms, thus
1
�
u
b
-
‚
if
fx
()
=
rect()
bx
,()
Fu
=
sinf
c
b
If
Then
h
(
t
) real
H
(−
u
) = [
H
(
u
)]′
h
(
t
) imaginary
H
(−
u
) = −
[
H
(
u
)]′
h
(
t
) even
H
(−
u
) =
H
(
u
) (even)
h
(
t
) odd
H
(−
u
) = −
H
(
u
) (odd)
h
(
t
) real and even
H
(
u
) real and even
h
(
t
) real and odd
H
(
u
) imaginary and odd
h
(
t
) imaginary and even
H
(
u
) imaginary and even
h
(
t
) imaginary and odd
H
(
u
) real and odd
Figure A.3
Fourier transform symmetries.
Search WWH ::
Custom Search