Image Processing Reference
In-Depth Information
the same image as a landscape, illustrating that the gray variations are identical to
those in Fig. 10.1 across the isocurves.
The magnitudes of the Fourier transforms of
g
(
t
) and
g
(
k
T
x
) are also given in
Fig. 10.2 (bottom). The red color represents the value zero in the image. The bright
(yellowish)spotsrepresentthelargestvalues.Intheimage,the2DFouriertransform
magnitudes clearly equal zero outside of the two bright points.
Example 10.2. The 1D sinc function
g
(
t
)=
sin(
ωt
)
ωt
ω
=
2
π
,
with
12
,
(10.4)
is plotted in Fig. 10.3. The synthetic image represented by the function
g
(
k
T
r
) is
linearlysymmetric.ItsimageisillustratedbythegrayimageinFig.10.4(top)which
differs from the one in Fig. 10.2 only by the choice of
g
. The function values are
scaled and shifted to be rendered by the available 256 gray tones. The solid and
the dashed vectors represent
k
and
−
k
respectively. Both this image and the gray
image in Fig. 10.2 have the same vector
k
=(cos(
π
4
))
T
, which represents
the direction orthogonal to the isocurve direction. In the direction of
k
, any cross
sectionoftheimageisidenticaltothesincfunctionofFig.10.3,asillustratedbythe
3D graph in Fig. 10.4, which shows
g
(
k
T
r
) as a surface.
In the (2D) color image, we note that the magnitudes of the Fourier transformed
function equal zero (red color) outside of a line passing through the origin, indicated
by bright yellow in Fig. 10.4 (bottom, right). The line has the direction
k
. Along
the line, the Fourier transform magnitude has the same shape as the (1D) Fourier
transform magnitude (bottom, left).
)
,
sin(
π
4
We will bring further precision to the relationship between the 1D and 2D Fourier
transforms of the linearly symmetric functions below. For now we note the result of
this example, as illustrated by Fig. 10.5 bottom, right, is consistent with that of the
previous example, Fig. 10.2 bottom, right. Both Fourier transform magnitudes vanish
outside a central line having the direction
k
, whereas on the line itself both magni-
tudes have at least the same magnitude variations as their 1D counterparts shown on
the respective left. From the magnitudes of Fourier transformed functions, we can
in general not deduce the underlying complex values. However, there is one excep-
tion to that which is a result of the null property of norms, i.e., the magnitudes of
complex numbers are zero if and only if the complex numbers are zero. The Fourier
transforms of the two illustrated example images possessing linear symmetry must
consequently have not only magnitudes but also complex values that equal zero out-
side the referenced line.
The second example actually showed the same sinusoid as in the first one, with
the difference that in the second example, the sinusoid attenuates gradually as 1
/t
.
The Fourier transform magnitude of the Sinc example is therefore more spread as
compared to that of the pure sinusoid, which consists of a pair of Dirac pulses.
The sinusoid is neither a pure line nor a pure edge, but yet it has a direction. The
classical edge and line detection techniques in image processing model and detect
