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Ta b l e 4 Errors for f ex 2 , ω 2 = π / 4, A 2 = 255
Filter
Mean
Variance
Max
Min
Gradient intensity
Sqr 3 × 3 4.78122E1
6.04649E2
1.03478E2
1.36418E-1
Sqr 5 × 5 7.96775E1
1.67478E3
1.61632E2
5.08181E-1
Hex 1
3.54782E1
3.00115E2
7.64858E1
1.51197E-1
Hex 2
6.13548E1
8.97091E2
1.27053E2
3.68400E-1
Orientation by arctan (radian)
Sqr 3 × 3 2.48541E-2 4.62197E-4 3.12839E-1 0.00000E0
Sqr 5
×
5 2.36221E-3 3.37087E-5 7.15883E-2 0.00000E0
Hex 1
3.35539E-3 5.45738E-5 1.18910E-1 0.00000E0
Hex 2
3.67423E-4 5.04084E-7 9.71053E-3 0.00000E0
Orientation by Overington's method (radian)
Sqr 3
3 2.46246E-2 4.76264E-4 3.12654E-1 5.61363E-15
Sqr 5 × 5 3.19489E-3 3.44523E-5 7.43153E-2 5.61363E-15
Hex 1
×
3.67976E-3 6.23404E-5 1.19508E-1 0.00000E0
Hex 2
1.59309E-3 3.31283E-6 1.31418E-2 0.00000E0
value of localization is, the smaller the mean error of gradient intensity is. This result
is expected, since the analytical gradient is defined as the first partial derivatives,
and the derivative is defined as the limit of the difference quotient. For applications
where a precise intensity is pursued, it is recommended to use the derived gradient
filter with radius of 1 on hexagonal lattices.
For orientation detection using arctangent, the larger the value of localization is,
the smaller the mean error is, regardless of the image. This result is consistent with
the result presented in section 7; namely, the larger the filter size is, the larger the
theoretical SNR becomes. Moreover, the derived gradient filters on hexagonal lat-
tices showed smaller mean errors in comparison with the consistent gradient filters
on square lattices. For better orientation detection, it is concluded that the derived
gradient filters on hexagonal lattices (with higher localization) are more appropriate.
The errors in orientation detection using Overington's method become smaller as
the value of localization of the filter gets larger. The same holds for orientation de-
tection using arctangent. Filters on hexagonal lattices perform better than the square
ones for f ex 2 and f ex 3 .For f ex 1 where luminance varies slowly, the results of errors
are similar for both types of lattices, however, the results for hexagonal lattices are
better than those for square ones.
Detecting the orientation of the gradient on hexagonal lattices with Overington's
method performs better than detecting the orientation on square lattices using arct-
angent for f ex 2 which mainly consists of high frequency components and f ex 3 which
consists of low to high frequency components. For f ex 1 which mainly consists of
low frequency components, this is not the case. The main advantage of using Over-
ington's method might be low computational cost, since it does not need to call
for arctangent. As a result, using Overington's method with the derived filters on
hexagonal lattices has advantages to using arctangent with gradient filters on square
 
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