Image Processing Reference
In-Depth Information
complex. Therefore a specific notion of group direction θ is needed to differentiate
the image direction from individual line normal directions.
The third column shows a more elaborate line configuration. It consists of three
distinct directions, none of which is more dominant than the other two. The images
are rotated w.r.t. each other with the angle
π
8
+ l 2 π
6
, with l being an integer. In analogy
with the above, we can obtain the group direction, uniquely and continuously deter-
mining the image and the line constellation directions, by hexa-angle representation ,
θ =6 ϕ (14.3)
The group directions of the images at the top and the bottom then become 0 and 6 π
8
,
respectively.
14.2 Test Images by Logarithmic Spirals
In case of single direction occuring in a local image, we previously discussed the
accuracy of direction estimation methods by using a frequency-modulated test chart,
see FMTEST in Fig. 10.15. The idea was to observe the output of a method to local
image inputs having all possible directions at all possible frequencies. Similarly, we
present here two test charts to evaluate methods estimating group directions: one
for group directions of local images containing two maximally distinct directions,
e.g., the second column of Fig. 14.1, and one for the same, but with third maximally
distinct directions, e.g., the 3rd column of Fig. 14.1.
To construct test charts with two distinct directions, we add wave patterns that
are orthogonal to each other, namely the first two images of Fig. 14.2 in the reading
direction. Each constituent image is locally a sinusiod with isocurves that are ap-
proximately parallel. All directions and a wide range of frequencies are represented.
When added, the local result contains two wave patterns that are mutually orthogonal
with a unique group direction and granularity (frequency), see FMTEST2 shown as
the fifth image of Fig. 14.2. The group direction and the granularity change indepen-
dently in the angular and in the radial directions, respectively.
Similarly, to construct test charts with three distinct directions we add wave pat-
terns that intersect with the angle
π
3
with each other, namely the first, the third, and
the fourth images of Fig. 14.2 in reading direction. Each of these spiral 1 images
are sinusiod patterns in log-polar coordinates with isocurves that are approximately
parallel, locally. The result contains three sinusoid patterns with maximally distinct
wave front directions, i.e. the spirals intersect one another at
π
3
. There is a contin-
uous change in the group direction (angularly) and the granularity (radially), see
FMTEST3 shown as the sixth image in Fig. 14.2.
In the test charts of FMTEST2 and FMTEST3, multiple wave patterns are added.
By contrast, in the test chart of FMTEST there is only one sinusoidal wave pattern,
locally. Although there are two gradient vectors with opposing directions present
in all neighborhoods of FMTEST, there is a unique group direction for each local
1 A circle is a special case of a spiral.
 
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