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The resulting consistent gradient filters for the 0 ,
60 ,
120 directions are given in
Figure 4.
7
Theoretical Evaluation
7.1
Signal-to-Noise Ratio
(
,
)
(
,
)
In the evaluation of signal-to-noise ratio, first, f i
are taken as
discrete gradient images in the x and y directions on square lattices, respectively,
and F i
x
y
and f j
x
y
(
u
,
v
)
and F j
(
u
,
v
)
are taken as their Fourier transforms, respectively. Next,
g sqr
,and G sqr
(
x
,
y
)
is taken as the least inconsistent image of f i (
x
,
y
)
and f j (
x
,
y
)
(
u
,
v
)
are taken as its Fourier transform. Ando[2] transformed the error
2 dxdy
2
x g sqr
y g sqr
(
x
,
y
)
f i (
x
,
y
)
+
(
x
,
y
)
f j (
x
,
y
)
(60)
by applying Parseval's theorem and defined inconsistency as
2 |
1
/
2
1
/
2
2
J sqr
uiG sqr
=
2
π
(
u
,
v
)
F i
(
u
,
v
) |
1
/
2
1
/
2
dudv
2
viG sqr
+
π
(
u
,
v
)
F j (
u
,
v
)
(61)
where the domain of integration is a unit of repeated spectra due to the sampling
theory. The gradient intensity on square lattices is defined as
2 dudv
+ G sqr
y
)
1 / 2
1 / 2
J sqr
1
G sqr
x
2
=
|
(
u
,
v
) |
(
u
,
v
,
(62)
1
/
2
1
/
2
where G sqr
and G sqr
are the Fourier transforms of g sqr
and g sqr
,
which are the partial differentials of the least inconsistent image in the x and y
directions, respectively, on square lattices.
Similarly,
(
u
,
v
)
(
u
,
v
)
(
x
,
y
)
(
x
,
y
)
x
y
x
y
2
2
3 (
1
2 (
x g
(
x
,
y
)
f a (
x
,
y
)+
f b (
x
,
y
)
f c (
x
,
y
)))
2 dxdy
1
3 (
+
y g
(
x
,
y
)
f b (
x
,
y
)+
f c (
x
,
y
))
(63)
 
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