Image Processing Reference
In-Depth Information
3
1
2
0.8
1
0.6
0
0.4
−1
−2
0.2
−3
3 π /2
5 π /2
π /2
0
2
3
4
5
6
7
2
3
4
5
6
7
Fig. 10.26. The graphs in the left image represent the estimated (arg( I 20 r ) , solid ) as well as
the ideal direction angle ( dashed ) on the circle passing through 1 to 4 in Fig. 10.15 when using
three tune-on directions. The graphs on the right show
|I 20 |
( solid ) and I 11 ( dashed )onthe
same ring
dient in the 45 and 30 directions respectively. We show the frequency coordinates
z kl (marked by
), representing the tune-on frequencies of the filters. Note that the
indices k and l now represent frequencies in the horizontal and vertical directions of
the image instead of explicitly encoding the direction and absolute frequencies of the
filters.
Marked explicitly by arrows in Fig. 10.29, we show the frequency coordinates
z kl . The sizes of the circles are modulated by the magnitudes of the filter responses,
|
×
F k,l
. The magnitudes of the responses are larger when the corresponding filter tune-
on frequencies are close to the solid line because we assumed that we have a linearly
symmetric image, which has a spectral energy concentrated to the shown solid line.
To obtain a reasonably good estimate of the structure tensor, the filters should
cover all directions, not necessarily all frequencies. This is because most images
with directions consist of a wide range of frequencies. Some studies have used other
tessellations which are a mixture of Cartesian and log-polar tessellation [124, 144].
Along the positive horizontal axis, they divided the frequency coordinate in expo-
nentially increasing cell sizes. In each such cell they then placed Cartesian-separable
Gaussians,
|
ω y
2 σ ω y
G k ( ω x y )=exp ( ω x
exp
ω k ) 2
2 σ ω x
(10.90)
symmetric around the cell centers, ( ω k , 0) T representing the horizontal tune-on di-
rections. This set of filters is then rotated with an angular increment to obtain the
Gabor filters for other directions, allowing an angular tessellation of the frequency
domain. However, we emphasize that the filter functions themselves are Cartesian
because they are fully symmetrical Gaussians in the frequency domain, although
they are placed at frequency sites that tessellate the spectrum in a log-polar fash-
ion. One advantage is implementation efficiency because all horizontal filters are
Cartesian-separable enabling a computation by cascades of 1D filtering. Other direc-
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