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where
√
3
2
√
3
2
1
2
u
1
2
u
G
1
(
u
,
v
)
≡
(
uH
a
(
u
,
v
)+(
+
v
)
H
b
(
u
,
v
)+(
−
+
v
)
H
c
(
u
,
v
))
F
(
u
,
v
)
.
(17)
(16) gives the condition on
g
for (9) to be minimal.
Errors of gradient filters are attributed to the inconsistency and to the smoothing
effect. Applying a gradient filter on discrete image means that the resultant gradient
value is actually smoothed, as the gradient filter for discrete images is not equal to
the gradient defined on the continuous domain. As Ando described[2], the smooth-
ing effect of gradient filters is not important in comparison to the inconsistency.
Similarly, in the present derivation, the smoothing effect is therefore, also discarded.
To reduce the inconsistency, the minimization of (9) is thus targeted.
4
Point Spread Function
The aim of this chapter is to derive gradient filters for the three orientations on the
hexagonal lattice: 0
◦
,
60
◦
and 120
◦
. It is supposed that the gradient filters for 60
◦
and
120
◦
are obtained by rotating the gradient filter derived for 0
◦
. It is also supposed
that the gradient filter for 0
◦
is symmetric with respect to the x-axis and antisym-
metric with respect to the y-axis (See Figure 4). And
a
mn
is taken as a discrete
coefficient of a gradient filter. Using a
-function as the common impulse function,
makes it possible to write a set of elements of a gradient filter as follows.
The point spread functions
h
mn
a
δ
h
mn
b
and
h
mn
c
,
, which define elements of gradient
filters in the respective directions of 0
◦
,
60
◦
and 120
◦
are described as
√
3
n
4
)+
δ
(
√
3
n
4
m
2
,
m
2
,
h
mn
a
(
x
,
y
)=
a
mn
{−
δ
(
x
−
y
−
x
+
y
−
)
√
3
n
4
)+
δ
(
√
3
n
4
m
2
,
m
2
,
−
δ
(
x
−
y
+
x
+
y
+
)
},
(18)
√
3
2
√
3
n
4
√
3
2
√
3
n
4
h
mn
b
1
2
(
−
m
2
)
−
m
2
)+
1
2
(
−
(
x
,
y
)=
a
mn
{−
δ
(
x
+
(
−
)
,
y
+
(
−
))
√
3
2
√
3
n
4
√
3
2
√
3
n
4
1
2
(
m
2
)
−
m
2
)+
1
2
(
−
+
δ
(
x
+
(
−
)
,
y
+
(
))
√
3
2
√
3
n
4
√
3
2
√
3
n
4
1
m
m
1
−
δ
(
x
+
2
(
−
2
)
−
(
)
,
y
+
(
−
2
)+
2
(
))
√
3
2
√
3
n
4
√
3
2
√
3
n
4
1
m
m
1
+
δ
(
x
+
2
(
2
)
−
(
)
},
y
+
(
2
)+
2
(
))
}
(19)
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