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Ta b l e 5
Errors for
f
ex
3
, ω
3
=
π
/
(
8
·
90
)
,
A
3
=
255
Filter
Mean
Variance
Max
Min
Gradient intensity
Sqr 3
×
3 2.02389E1
3.60586E2
7.85115E1
2.91073E-4
Sqr 5
×
5 3.61619E1
1.05073E3
1.27089E2
1.04375E-4
Hex 1
1.45449E1
1.88399E2
5.56746E1
2.46549E-4
Hex 2
2.64606E1
5.85607E2
9.59810E1
7.97684E-3
Orientation by arctan (radian)
Sqr 3
×
3 1.17490E-2 9.75118E-5 8.30716E-2 0.00000E0
Sqr 5
×
5 1.20048E-3 3.78492E-6 3.03901E-2 0.00000E0
Hex 1
8.01422E-4 8.44267E-7 3.98812E-3 0.00000E0
Hex 2
1.11561E-4 2.05645E-8 1.69539E-3 0.00000E0
Orientation by Overington's method (radian)
Sqr 3
3 1.17166E-2 1.02811E-4 8.40553E-2 5.61363E-15
Sqr 5
×
5 2.14414E-3 5.88746E-6 3.01349E-2 5.61363E-15
Hex 1
×
1.79184E-3 3.83489E-6 1.01309E-2 0.00000E0
Hex 2
1.53078E-3 3.02628E-6 6.48015E-3 0.00000E0
lattices, with respect to accuracy, computational cost, and simplicity of circuit im-
plementation.
For applications where high accuracy in gradient intensity or orientation detec-
tion is needed, the filters derived in this chapter for hexagonal lattices are good
solutions.
10
Summary
Consistent gradient filters on hexagonal lattices were derived, the relationship be-
tween Staunton filters and the derived filters was investigated, and the derived filters
were compared with existing filters on square lattices theoretically and practically.
Staunton filters and the derived consistent gradient filters with radius of 1 have a
relationship that 30 degrees rotation make others, though the Staunton filters were
derived on image domain but the derived filters are derived on the frequency domain.
Thus both filters can calculate intensities and orientations with the same accuracy.
In a theoretical evaluation of the derived filters, these show better SNR in spite of
their smaller localization, in comparison to the consistent gradient filters on square
lattices. In the theoretical evaluation, an assumption that the frequency characteris-
tics of the input image is flat was adopted. To cope with this, the derived filters are
evaluated with artificial images that present various frequency characteristics. The
derived filters on hexagonal lattices achieve a higher accuracy in terms of gradient
intensity and orientation detection in comparison to the consistent gradient filters
on square lattices. Moreover, the derived filters on hexagonal lattices with Overing-
ton's method can detect the orientation with higher accuracy than gradient filters
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