Image Processing Reference
In-Depth Information
The argument “
ω
”in
δ
(
ω
) is there to mark the “hot point” of the delta distri-
bution, not to tempt us to conclude that
δ
is a function delivering a useful value at
the argument. For example,
δ
(
ω
ω
) becomes infinity when
ω
=
ω
(whereas it
vanishes elsewhere). In fact, taking a close look at the rule that defines the Dirac dis-
tribution, we note that it is not based on real numbers as arguments but on functions,
represented by
f
. Whereas functions are rules that deliver numbers for arguments
that are numbers, the Dirac distribution is a rule that delivers numbers for “argu-
ments” that are functions appearing in scalar products, Eq. (7.1). In plain English,
this is what the Dirac distribution does. No matter what the integration domain is,
as long as the domain contains the “hot-point” of the
δ
-distribution, the distribution
kicks out the value of its “fellow integrand” at the hot point. Example behaviors
include
−
f
(
x
)
δ
(
x
−
y
)
dx
=
f
(
y
)
(7.9)
and
f
(
x
)
δ
(
x
y
)
dxdy
=
−
f
(
y
)
dy.
(7.10)
Theorem 7.1.
The Fourier transform of the constant function “1” is a Dirac distri-
bution:
1
2
π
F
(1)(
ω
)=
exp(
−
iωt
)
dt
=
δ
(
ω
)
(7.11)
Exercise 7.1.
The proof of the theorem is obtained by Egs. (7.4) and Eq. (7.6). In
particular, provide the following:
i) Prove that the Fourier transform of the complex exponential
exp(
iωt
)
is a shifted
delta distribution.
ii) Prove that the Fourier Transform of a Dirac-
δ
is a constant.
iii) Create a
δ
in a 2D image. Compute the real part of its Fourier transform (use
DFT on a large image). Compute the imaginary part of its Fourier transform.
What are the parameters that determine the direction and the frequencies of
planar waves?
7.2 Conservation of the Scalar Product
In the case of Fourier series, we were able to show that the scalar products were
conserved between the time domain and the Fourier coefficients. Now we show that
scalar product conservation
is valid even for the Fourier transform. The scalar prod-
ucts are defined as
=
∞
−∞
f
(
t
)
∗
g
(
t
)
dt
f, g
(7.12)
and
