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.
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