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2
u
2
GG
∗
−
iuGF
a
+
iuG
∗
F
a
+
F
a
F
a
4
π
2
π
2
π
2
1
2
u
v
2
i
1
v
GF
b
√
3
2
√
3
2
GG
∗
−
+
4
π
+
2
π
2
u
+
i
1
v
G
∗
F
b
+
√
3
2
F
b
F
b
+
2
π
2
u
+
2
v
2
i
v
GF
c
√
3
2
√
3
2
1
2
u
1
2
u
GG
∗
−
+
4
π
−
+
2
π
−
+
i
v
G
∗
F
c
√
3
2
1
2
u
F
c
F
c
+
−
+
+
2
π
(11)
In a similar manner to Ando's method, differentiating with respect to
G
and
G
∗
gives
the following conditions for
g
(
x
,
y
)
to be the least inconsistent gradient image.
√
3
2
√
3
2
1
2
u
1
2
u
G
∗
−
iuF
a
−
F
b
−
F
c
=
2
u
2
v
2
6
π
(
+
)
2
π
2
π
i
(
+
v
)
2
π
i
(
−
+
v
)
0
(12)
√
3
2
√
3
2
1
2
u
1
2
u
2
u
2
v
2
6
π
(
+
)
G
+
2
π
iuF
a
+
2
π
i
(
+
v
)
F
b
+
2
π
i
(
−
+
v
)
F
c
=
0
(13)
The sum and difference of these two expressions are given respectively as follows:
2
u
2
v
2
G
∗
)+
F
a
)
6
π
(
+
)(
G
+
2
π
iu
(
F
a
−
√
3
2
√
3
2
1
2
u
1
2
u
F
b
)+
F
c
)=
+
2
π
i
(
+
v
)(
F
b
−
2
π
i
(
−
+
v
)(
F
c
−
0
(14)
2
u
2
v
2
G
∗
)+
F
a
)
6
π
(
+
)(
G
−
2
π
iu
(
F
a
+
√
3
2
√
3
2
1
2
u
1
2
u
F
b
)+
F
c
)=
+
2
π
i
(
+
v
)(
F
b
+
2
π
i
(
−
+
v
)(
F
c
+
0
(15)
Equation (14) consists only of real parts, while Equation (15) consists of imaginary
terms only. For
g
to be least inconsistent, both equations must equal zero. Summing
these two expressions, gives the following condition:
−
i
G
(
u
,
v
)=
G
1
(
u
,
v
)
,
(16)
π
(
u
2
+
v
2
)
3
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