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)=
∑
x
1
,
f
b
(
x
,
y
)=
h
b
(
x
,
y
)
∗
f
(
x
,
y
h
b
(
x
1
,
y
1
)
f
(
x
−
x
1
,
y
−
y
1
)
(3)
y
1
∈
RH
)=
∑
x
1
,
y
1
∈
RH
f
c
(
x
,
y
)=
h
c
(
x
,
y
)
∗
f
(
x
,
y
h
c
(
x
1
,
y
1
)
f
(
x
−
x
1
,
y
−
y
1
)
,
(4)
where
RH
is a set of pixels inside the filters. They are described in the frequency
domain as follows:
F
a
(
u
,
v
)=
H
a
(
u
,
v
)
F
(
u
,
v
)
(5)
F
b
(
u
,
v
)=
H
b
(
u
,
v
)
F
(
u
,
v
)
(6)
F
c
(
u
,
v
)=
H
c
(
u
,
v
)
F
(
u
,
v
)
.
(7)
(a) Hexagonal lattices
(b) Square lattices
Fig. 1
Hexagonal and square lattices. The distance between adjacent pixels is always 1 on
hexagonal lattices, while it is either 1 or
√
2 on square lattices.
The least-inconsistent image of discrete gradient images
f
a
(
x
,
y
)
,
f
b
(
x
,
y
)
,
and
f
c
(
x
,
y
)
is denoted as
g
(
x
,
y
)
. In a similar manner to Ando's definition of
g
for square
lattices,
g
(
x
,
y
)
is determined by minimizing the following criterion:
1
2
g
√
3
2
∞
∞
2
2
∂
∂
∂
∂
∂
∂
x
g
(
x
,
y
)
−
f
a
(
x
,
y
)
+
x
+
(
x
,
y
)
−
f
b
(
x
,
y
)
y
−
∞
−
∞
g
dxdy
√
3
2
2
1
2
∂
∂
∂
∂
+
−
x
+
(
x
,
y
)
−
f
c
(
x
,
y
)
.
(8)
y
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