Image Processing Reference
In-Depth Information
Fig. 6.2. The image on the
right
is obtained by applying the forward FT to the image on the
left
two times. The
red lines
are added for comparison convenience
Example 6.1. First, we illustrate the symmetry of FT by showing how one would
need a simple rearrangement to make the ordinary FT a replacer of the inverse FT.
Subsequently,weusethesamelemmatoestablishathirdversionoftheFCtheorem.
•
Figure 6.2 represents a forward FT applied to an image twice. The second FT
actsasaninverseFT,exceptforareflection.TheFThasbeenappliedtoallthree
color components, in RGB.
Because of the symmetry of FT, the conclusions on FT remain valid even if the roles
of
f
(
t
) and
F
(
ω
) are interchanged. The FC theorem can therefore be formulated for
limited frequency functions.
Theorem 6.2 (FC II).
There exists a set of scalars
f
(
t
m
)
that can synthesize a func-
tion
F
(
ω
)
having the finite extension of
Ω
,
Ω
m
F
(
ω
)=
1
f
(
t
m
) exp(
−it
m
ω
)
,
(Synthesis)
(6.12)
where
f
(
t
m
)=
Ω
2
F
(
ω
) exp(
it
m
ω
)
dω,
(Analysis)
(6.13)
Ω
2
−
using
t
m
=
m
2
Ω
.
Exercise 6.2.
Prove theorem 6.2.
HINT: Expand
ΩF
(
ω
)
in
exp
−
it
m
ω
.
6.2 Sampled Functions and the Fourier Transform
Via theorem 5.2, we established that there exists a unique set of values capable of
reconstructing any
f
(
t
) by means of complex exponentials, provided that
f
has an
