Image Processing Reference
In-Depth Information
Figure
2.14
(b). This shows that there are only horizontal spatial frequencies; the image is
constant in the vertical axis and there are no vertical spatial frequencies.
(a) Image of vertical bars
(b) Fourier transform of bars
Figure 2.14
Applying the 2D discrete fourier transform
The
two-dimensional (2D) inverse DFT
transforms from the frequency domain back to
the image domain. The 2D inverse DFT is given by:
2
NN
-1
-1
j
( +
ux
v
y
)
Σ Σ
v
N
P
=
FP
e
(2.23)
v
xy
,
u
,
u
=0
=0
One of the important properties of the FT is
replication
which implies that the transform
repeats
in frequency up to
infinity
, as indicated in Figure
2.9
for 1D signals. To show this
for 2D signals, we need to investigate the Fourier transform, originally given by
FP
u
,
v
,
at
integer multiples of the number of sampled points
FP
u
+
mM
,
v
+
nN
(where
m
and
n
are integers).
The Fourier transform
FP
u
+
mM
,
v
+
nN
is, by substitution in Equation 2.22:
2
N
-1
N
-1
v
=
1
-
j
((
u
+
mN
)
x
+(
+
nN
)
y
)
N
FP
P
e
(2.24)
umN
+, +
v
nN
xy
,
N
x
=0
y
=0
so,
2
N
-1
N
-1
-
j
(+)
ux
v
y
=
1
N
-2(
j
x
+ )
y
FP
P
e
e
(2.25)
umN
+, +
v
nN
xy
,
N
x
=0
y
=0
and since
e
-
j
2π (
mx
+
ny
)
= 1 (since the term in brackets is always an integer and then the
exponent is always an integer multiple of 2π ) then
FP
u
+
mN
,
v
+
nN
=
FP
u,
v
(2.26)
which shows that the replication property does hold for the Fourier transform. However,
Equation 2.22 and Equation 2.23 are very slow for large image sizes. They are usually