Image Processing Reference
In-Depth Information
Fig. 10.23. The 2D spectrum and the Gabor filter responses when an input image is linearly
symmetric with a direction between filter tune-on directions. The
crosses
represent the tune-on
frequencies. The
arrows
represent the coordinate vectors,
z
k
, of some of these, whereas the
circles
illustrate, via their radii, the magnitudes of the respective filter responses,
|F
{k,l}
|
Structure Tensor by Log-Polar Gabor Decomposition
The sums in Eqs. (10.84)-(10.85) are taken over the entire range of Gabor filters.
Assuming log-polar separable filter tune-on frequencies discussed in Sect. 9.6, one
could, however, split the sum over the direction and the frequency components, as
shown for
I
20
:
I
20
=
l
2
=
l
z
kl
|
F
{k,l}
|
I
l
20
(10.86)
k
with
=
k
I
l
20
z
kl
|
F
{k,l}
|
2
(10.87)
being the complex moment contribution from filters on a “ring” of Gabor tune-on
frequency sites. Analogously, we can obtain
I
11
=
l
|z
kl
|
2
|F
{k,l}
|
2
=
l
I
l
11
(10.88)
k
with
