Image Processing Reference
In-Depth Information
1, into [(
D
x
+
iD
y
)
f
]
2
, to estimate Eq. (11.121) in a neighborhood:
I
20
=
n
e
in
arg(
x
+
iy
)
Γ
{
0
,σ
2
}
(
x, y
)[(
D
x
+
iD
y
)
f
]
2
dxdy
K
n
|
x
+
iy
|
(11.122)
By assuming 0
≤
n
and substituting Eq. (11.121) in Eq. (11.122), we obtain
I
20
(
x
,y
)
(2
πσ
1
)
K
n
=
(11.123)
n
e
in
arg(
x
+
iy
)
Γ
{
0
,σ
2
}
(
x, y
)
k
y
k
)
dxdy
h
k
Γ
{
0
,σ
1
}
(
x
|
x
+
iy
|
−
x
k
,y
−
Noting that
n
e
in
arg(
x
+
iy
)
Γ
{
0
,σ
2
}
(
x, y
)
=(
x
+
iy
)
n
Γ
{
0
,σ
2
}
(
x, y
)=(
|x
+
iy|
σ
2
)
n
Γ
{n,σ
2
}
(
x, y
)
−
(11.124)
where we used the definition of
Γ
{n,σ
2
}
in Eq. (11.97) and applied theorem 11.2, we
can estimate
I
20
on a Cartesian grid:
I
20
(
x
,y
)
(2
πσ
1
)
K
n
(
σ
2
)
n
−
h
k
Γ
{n,σ
2
}
(
x, y
)
Γ
{
0
,σ
1
}
(
x
−
=
k
x
k
,y
−
x
−
y
−
y
k
)
dxdy
=
k
h
(
x
k
,y
k
)(
Γ
{n,σ
2
}
∗
Γ
{
0
,σ
1
}
)(
x
−
x
k
,y
−
y
k
)
=
k
h
(
x
k
,y
k
)
Γ
{n,σ
1
+
σ
2
}
(
x
−
x
k
,y
−
y
k
)
(11.125)
Here Eq. (11.125) is obtained
13
by utilizing theorem 11.4. Equation (11.125) can be
computed on a Cartesian discrete grid by the substitution (
x
,y
)=(
x
l
,y
l
), yielding
an ordinary discrete convolution:
I
20
(
x
l
,y
l
)=
C
n
h
Γ
{n,σ
1
+
σ
2
}
(
x
l
,y
l
)
∗
(11.126)
with
C
n
=(2
πσ
1
)
K
n
(
σ
2
)
n
.
The result for
n<
0 is straightforward to deduce by following the steps after
Eq. (11.125) in an analogous manner. Likewise, the scheme of
I
11
−
is obtained by
following the same idea as for
I
20
.
13
We note that in the proof of theorem 11.4, theorem 11.3 is needed, so that all of the theorems
of this paper are actually utilized in the proof of this lemma.
