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a
tmp
a
(
i
+
1
)
mn
=
(58)
mn
∑
m
,
n
ma
tmp
2
mn
is a constant and
a
tmp
where
mn
is a temporary variable. The minimization stops when
all
a
mn
that compose the gradient filter satisfy the following condition.
α
a
(
i
+
1
)
a
(
i
mn
|
−
| <
ε
,
(59)
mn
where
ε
is a constant. Under the assumption that
(
,
)=
,
P
u
v
1
the actual values of the gradient filter are obtained, as Ando described on square
lattices. This assumption means that the input image's spectrum is equivalent to
white noise, so the derived filters are not specialized for a specific frequency.
The optimization was performed with Octave[1] with
10
−
12
.
α
=
0
.
01 and
ε
=
a
=
.
333011
b
=
0
.
166989
(a) Hex 1: Filter
of radius 1
0
a
=
0
.
272665
b
=
0
.
136340
c
=
0
.
030332
(b) Hex
√
3: Filter of
radius
√
3
a
=
0
.
182049
b
=
0
.
091025
c
050753
d
=
0
.
024889
e
=
0
.
012444
=
0
.
(c) Hex 2: Filter of radius 2
Fig. 4
Forms of derived gradient filters on hexagonal lattices
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