Image Processing Reference
In-Depth Information
p
x
Fp
0
x
(a) Original sampled signal
t
(b) First coefficient Fp
0
2
⋅
10
2
⋅
10
⋅
e
j
⋅
t
⋅
e
j
⋅
t
⋅
Re
Fp
1
Re
Fp
0
+ Fp
1
⋅
t
t
(c) Second coefficient
Fp
1
(d) Adding
Fp
1
and
Fp
0
2
⋅π
10
⋅
u
3
2
⋅ π
10
⋅
5
Fp
u
⋅
e
j
⋅
t
⋅
Σ
u=0
e
j
⋅
t
⋅
u
Re
Σ
u=0
Re
Fp
u
⋅
t
t
(e) Adding
Fp
0
,
Fp
1
,
Fp
2
and
Fp
3
(f) Adding all six frequency components
Figure 2.13
Signal reconstruction from its transform components
which are the horizontal and vertical spatial frequencies, respectively. Given an image of
a set of vertical lines, the Fourier transform will show only horizontal spatial frequency.
The vertical spatial frequencies are zero since there is no vertical variation along the
y
axis.
The two-dimensional Fourier transform evaluates the frequency data,
FP
u
,
v
, from the
N
×
N
pixels
P
x
,
y
as:
2
N
-1
N
-1
v
-
j
( +
ux
y
)
=
1
N
FP
P
e
(2.22)
v
u
,
N
xy
,
x
=0
y
=0
The Fourier transform of an image can actually be obtained
optically
by transmitting a
laser through a photographic slide and forming an image using a lens. The Fourier transform
of the image of the slide is formed in the front focal plane of the lens. This is still restricted
to transmissive systems whereas reflective formation would widen its application potential
considerably (since optical computation is just slightly faster than its digital counterpart).
The magnitude of the 2D DFT to an image of vertical bars (Figure
2.14
(a)) is shown in
