Image Processing Reference
In-Depth Information
7 Properties of the Fourier Transform
F
(
ω
)=(
Ω
n
δ
(
ω
−
nΩ
))
∗
F
=
Ω
n
δ
(
ω
−
nΩ
)
∗
F
=
Ω
n
F
(
ω
−
nΩ
)
(7.51)
As there is no overlap, the information relative to
F
is still intact after the periodiza-
tion of
F
. Since convolution in the
ω
-domain is equivalent to a multiplication in the
t
-domain Eq. (7.51) is equivalent to a multiplication in
t
-domain,
f
(
t
)=
m
δ
t
f
m
2
π
Ω
δ
t
m
2
π
Ω
m
2
π
Ω
−
−
(7.52)
m
where Eq. (7.50) has been used. However, a periodization of the
ω
-domain corre-
sponds to a sampling in the
t
-domain. Accordingly, we conclude that a multiplication
by a Comb distribution yields the mathematical representation of sampled function.
The following lemma, summarizes this and restates the Nyquist theorem:
Ω
2
,
Ω
2
Lemma 7.3.
Let
f
be a finite frequency function with
F
vanishing outside of
−
.
Then the sampled
f
is given by the distribution
Comb
T
(
t
)
f
(
t
)=
m
f
(
mT
)
δ
(
t
−
mT
)
(7.53)
with
T<
2
Ω
. A lossless reconstruction of
f
is achieved by convolving this distribu-
tion with the inverse FT of the characteristic function
χ
Ω
.
7.6 Hermitian Symmetry of the FT
The Hermitian symmetry of FCs were discussed in Sect. 5.4. There is an analogous
result even for FT. We suggest the reader to study it in the frame of the following
exercises.
Exercise 7.3.
Show that the
F
(
ω
)
that is the Fourier transform of
f
(
t
)
is Hermitian:
ω
)
∗
F
(
ω
)=
F
(
−
(7.54)
Do the same in 2D, i.e., when
F
(
ω
x
,ω
y
)
is the FT of
f
(
x, y
)
, show that
ω
y
)
∗
F
(
ω
x
,ω
y
)=
F
(
−
ω
x
,
−
(7.55)
Exercise 7.4.
Show that even DFT coefficients are Hermitian. Which coefficients rep-
resent
F
(
−
2
,
−
2)
in a 256
×
256 image (the first element is labelled as 0 in both
dimensions)?
