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8.2
Detection of Gradient Intensity and Orientation
Ando's gradient filters (3
5) for square lattices and the derived gradient
filters (radius 1 and 2) for hexagonal lattices are evaluated as follows. The gradi-
ent intensity and orientation are determined from a filtered image as follows. The
differential values in the x and y directions are taken as
f
x
and
f
y
respectively, on
square lattices,
f
r
×
3and5
×
60
◦
,and
120
◦
directions, respectively, on hexagonal lattices. The functions for calculating
the gradient intensity on the square and hexagonal lattices, respectively, are taken as
Int
sqr
f
s
and
f
t
are taken as the differential values in the 0
◦
,
,
and
Int
hex
as follows:
Int
sqr
(
f
)
≡
(
f
x
)
2
+(
f
y
)
2
(92)
2
3
2
1
/
2
f
r
+
2
1
√
3
(
1
2
(
Int
hex
(
f
)
≡
f
s
−
f
t
)
+
f
s
+
f
t
)
(93)
Similarly, the orientation is given as
Ori
sqr
(
f
)
≡
arctan
(
f
x
,
f
y
)
(94)
arctan
2
3
f
r
+
1
2
(
1
√
3
(
Ori
hex
(
f
)
≡
f
s
−
f
t
)
,
f
s
+
f
t
)
.
(95)
Both functions rely on arctan, a function known for its high computational cost.
In the following section, the accuracy of the orientation calculated by Overington's
method[17], which has a smaller calculation cost, is evaluated.
8.3
Overington's Method of Orientation Detection
Overington[17] proposed an orientation-detection method specially designed for
hexagonal lattices. The method performs better in terms of calculation time than
methods using arctangent, such as (94) and (95). While Overington focused on
hexagonal lattices (six axes), his approach is extended here to any number of axes.
First, it is supposed that
n
gradient filters are available in
n
directions, uniformly
sampled. Each filter's orientation is at angle
π
/
n
from its neighbors. The differen-
f
tial value in the direction
θ
is taken as
z
θ
f
. Accordingly, the following equations
are defined:
f
1
≡
θ
argmax
(
|
z
|
)
(96)
f
f
θ
θ
f
0
≡
θ
f
1
−
π
/
θ
n
(97)
f
2
≡
θ
f
1
+
π
/
θ
n
(98)
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