Direction in 2D - Vision with Direction - page 191

Fig. 10.29. Computing the second-order complex moments of the local power spectrum ob-
tained via a Cartesian separable Gabor decomposition according to Eqs. (10.84)-(10.85). The
sizes of the circles , representing the magnitudes of the filter responses become larger the closer
the corresponding filter sites get to the solid axis

but the direction estimations follow the true directions quite well, nevertheless. We
see a confirmation of these conclusions in the graphs of Fig. 10.32, which represent
arg( I 20 ) on the left, and , I 11 on the right, corresponding to sites along the

lines joining points 5 and 6 to the center, marked in the test image of Fig. 10.15.
The direction estimates are constant, as they should be and the quality of the fit as
represented by the magnitudes of

| I 20 |

and I 11 is high. These are close to each other
in the clean signal, whereas they are much smaller and not very close to each other
(relative the amplitude of

| I 20 |

).
Which structure tensor decomposition should one then use? The direct approach
is a cascade of 1D Gaussian filtering and directionally isotropic that is, free of
direction-dependent artifacts. Using this technique results in far fewer operations in
a sequential computer, such as a personal computer. Accordingly, if dense direction
tensor maps are needed, direct sampling of the structure tensor offers computational
advantages, while the accuracy of all elements of the structure tensor is virtually
unaffected by the minimum number of the filters needed (three filters are needed:
D x ,D y , and a Gaussian). However, a direct steering of the absolute frequency to
which the structure tensor is sensitive and the bandwidth of the frequency sensitiv-
ity range are more conveniently achievable by a log-polar mapping of the spectrum
and discretization via Gabor filtering. Such a steering is possible for direct sampling
of the structure tensor too, but indirectly. A Laplacian pyramidlike processing must
first be applied to the image to make sure that the frequency ranges of interest are
sufficiently well isolated. Also, dense structure tensors are not always needed. Some
applications can successfully be developed on sparse grids, e.g., the square grid sug-
gested in [144] or a log-polar sampling of the world [196, 216]. Tracking of humans
and their identification is, for example, achievable on a doubly log-polar sampling of
the image and its spectrum, i.e., on an image grid which samples the image plane in a
log-polar fashion the local spectrum is sampled in a log-polar fashion too [27, 205].

| I 20 |