Information Technology Reference
In-Depth Information
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)
Search WWH ::
Custom Search