Image Processing Reference
In-Depth Information
Fig. 7.2. FT of a real function is Hermitian, i.e., the points mirrored through the origin have
the same absolute value, shown as the
sizes of circles
, but negative arguments, shown as the
red
hue and its negative hue,
cyan
Exercise 7.5.
Take a digital image and apply the DFT to it. Put all strictly negative
horizontal frequencies to zero, i.e., half the frequency plane. Inverse Fourier trans-
form this, and display the real part and the imaginary part. The complex image you
obtained is called the
analytic signal
in signal processing literature. Comment on
your result.
Exercise 7.6.
Now fill the points in the half-plane with zero values with values from
the other half-plane according to Eq. (7.55). Comment on your result.
7.7 Correspondences Between FC, DFT, and FT
We started with the Fourier series and derived from it the FC transform, theorem 5.2.
By theorem 6.1, we showed that a generalization of the FCs leads to FT if the finite
extension restriction imposed on functions is removed. In that we also established
that not only the
t
-domain but also the
ω
-domain can be continuous. This led to
the idea of reformulating the FC theorem for the finite frequency functions, yielding
theorem 6.2. From both of the theorems FC I and FC II, we could derive a fully
discrete transform pair, DFT, where the number of FCs synthesizing the current finite
extension function is finite, theorem 6.4.
