Image Processing Reference
In-Depth Information
Image Domain
Transform Domain
1
0.8
0.6
0.4
0.2
0
10
20
30
2
1
0
0
20
20
30
30
0
10
20
30
square
ft_square
(b) 2D sinc function
(a) Square
1
0.8
0.6
0.4
0.2
0
1.5
1
0.5
0
0
20
20
30
30
0
10
20
30
10
20
30
Gauss
ft_Gauss
(c) Gaussian
(d) Gaussian
Figure 2.17
2D Fourier transform pairs
2.6
Other properties of the Fourier transform
2.6.1
Shift invariance
The decomposition into spatial frequency does not depend on the position of features
within the image. If we shift all the features by a fixed amount, or acquire the image from
a different position, the magnitude of its Fourier transform does not change. This property
is known as
shift invariance
. By denoting the delayed version of
p
(
t
) as
p
(
t
- τ ), where τ
is the delay, and the Fourier transform of the shifted version is
F
[
p
(
t
- τ )], we obtain the
relationship between a time domain shift in the time and frequency domains as:
F
[
p
(
t
- τ
)] =
e
-
j
ωτ
P
(ω )
(2.29)
Accordingly, the magnitude of the Fourier transform is: