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2
2
Φ
(
u
,
v
)
≡
|
G
x
(
u
,
v
)
|
+
|
G
y
(
u
,
v
)
|
|
F
(
u
,
v
)
|
2
√
3
2
√
3
2
2
(
u
2
+
v
2
4π
)
1
2
u
+
1
2
u
+
2
=
2
|
uH
a
(
u
,
v
)+(
v
)
H
b
(
u
,
v
)+(
−
v
)
H
c
(
u
,
v
)
|
2
(
u
2
+
v
2
9π
)
16
+
v
2
(
m
,
n
a
mn
φ
mn
)
2
=
(73)
u
2
Inconsistency
J
hex
thus is rewritten as
J
hex
G
x
(
2
G
y
(
2
=
D
(
|
u
,
v
)
|
+
|
u
,
v
)
|
)
dudv
=
D
Ψ
0
(
u
,
v
)
P
(
u
,
v
)
dudv
16
v
2
|
m
,
n
a
mn
ψ
mn
|
2
P
=
(
u
,
v
)
dudv
u
2
+
D
16
v
2
|
m
,
n
a
mn
ψ
mn
|
2
dudv
=
.
(74)
u
2
+
D
And gradient intensity
J
hex
1
is rewritten as
J
hex
2
2
=
D
(
|
(
,
)
|
+
|
(
,
)
|
)
G
x
u
v
G
y
u
v
dudv
1
=
D
Φ
(
u
,
v
)
P
(
u
,
v
)
dudv
16
v
2
{
m
,
n
a
mn
φ
mn
(
u
,
v
)
}
2
P
=
(
u
,
v
)
dudv
u
2
+
D
16
v
2
{
m
,
n
a
mn
φ
mn
(
u
,
v
)
}
2
dudv
=
.
(75)
u
2
+
D
The ratio of
J
1
to
J
corresponds to signal-to-noise ratio (SNR), which is defined as
follows:
1
2
log
2
J
1
J
.
SNR
≡
(76)
SNR
is used to compare the consistent gradient filters on square lattices[2] with the
gradient filters on hexagonal lattices derived here. The results for
J
J
1
and
SNR
are
listed in Table 1. First, the table shows that the values stated in [2] for the square
lattices could be reproduced. The derived filters on hexagonal lattices also achieved
higher
J
1
than similar size filters on square lattices. Since
J
1
is the integration of
signal intensity for each frequency that the filter lets pass, our filters on hexagonal
lattices are superior to similar size filters on square lattices with respect to frequency
permeability.
,
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