Image Processing Reference
In-Depth Information
Fig. 9.5. FMTEST image, which consists of frequency-modulated planar waves in all direc-
tions
f
(
x
)=
f
(
Q
T
y
)=
f
(
y
)
(9.26)
and then convolving the rotated image separably. In this case, the 1D Gaussians
having the arguments
y
i
, the right-hand side of Eq. (9.25), as discussed in Section
7.3.
Directional Isotropy
It is easy to equip the Gaussian filter with another desirable property, isotropy. From
Eq. (9.25) one can conclude that if
σ
1
=
σ
2
=
···
=
σ
N
=
σ
then we can write:
exp
N
(
x
T
v
i
)
2
2
σ
g
(
x
)=
a
−
i
=1
exp
=
a
exp
N
(
y
i
)
2
2
σ
i
−
y
2
2
σ
2
=
a
−
(9.27)
i
=1
Consequently, Gaussians can not only be made separable in
N
D, but they can also
be made fully invariant to rotations, i.e., the function values depend only on the
distance to the origin. Gaussians are the only functions that are both separable and
rotationally invariant, in 2D and higher dimensions. Used as filters, this property
allows us to identically weight all frequencies having the same distance to the origin
in the
-domain. This is significant to image analysis because there is no reason to
systematically disfavor a direction compared to the others.
ω
Example 9.3. We illustrate isotropy in image analysis operators by using an omni-
directional test image consisting of frequency-modulated planar waves in Fig. 9.5.
We compare Gaussians with sinc functions when they are used as lowpass filters.
