Image Processing Reference
In-Depth Information
7
Properties of the Fourier Transform
In this chapter we study the FT to reveal some of its properties that are useful in
applications. As has been shown previously, the FT is a generalization of both FC and
DFT. There is an added value, to use FT indifferently both to mean FC and to mean
DFT. To do that, however, we need additional results. The first section contributes to
that by introducing the Dirac distribution [212]. This allows us to Fourier transform
a sinusoid, or a complex exponential, which is important to virtually all applications
of signal analysis in science and economics. Fourier transforming a sinusoid is not a
trivial matter because a sinusoid never converges to zero. In Sect. 7.2, we establish
the invariance of scalar products under the FT. This has many implications in the
practice of image analysis, especially in the direction estimation and quantification
of spectral properties. Finally, we study the concept of convolution in the light of
FT, and close the circle by suggesting the Comb tool to achieve formal sampling
of arbitrary functions. This will automatically yield a periodization in the frequency
domain via a convolution. With the rise of computers, convolution has become a
frequently utilized tool in signal analysis applications.
7.1 The Dirac Distribution
What is the area or the integral of the sinc signal that was defined in Eq. (6.21)? The
answer to this question will soon lead us to construct a sequence of functions that
will produce a powerful tool when working with the Fourier transform.
We answer the question by identifying the characteristic function of an interval
as the inverse FT of a sinc function via Eq. (6.23):
χ
T
(
t
)=
1
,
T
2
,
T
2
if
t
∈
[
−
];
0
,
otherwise
.
F
−
1
T
2
)
(
t
)=
T
2
π
sinc
ω
T
exp(
iωt
)
dω
2
π
sinc(
ω
T
=
(7.1)
2
Evaluating
χ
T
(0) yields the integral of a sinc function:
