Image Processing Reference
In-Depth Information
Γ
{p
0
+1
,σ
2
}
(
x, y
)=
x
2
+
y
2
2
σ
2
1
2
πσ
2
(
D
x
+
iD
y
)(
D
x
+
iD
y
)
p
0
exp(
=
−
)
(
D
x
+
iD
y
)
−
p
0
exp
(11.110)
x
2
+
y
2
2
σ
2
1
2
πσ
2
x
σ
2
+
i
−
y
σ
2
=
−
where we indicated by brackets []the terms on which the differential operators
act, is obtained by applying the induction assumption (point 2). Consequently,
and by using the chain rule as well as the linearity of the partial differential
operators, we obtain:
Γ
{p
0
+1
,σ
2
}
(
x, y
)
(
D
x
+
iD
y
)
−
p
0
exp
x
2
+
y
2
2
σ
2
1
2
πσ
2
x
σ
2
+
i
−
y
σ
2
=
−
···
−
p
0
(
D
x
+
iD
y
)exp
(11.111)
x
2
+
y
2
2
σ
2
1
2
πσ
2
x
σ
2
+
i
−
y
σ
2
+
−
By repeated applications of the algebraic rules that govern the differential oper-
ators, we obtain:
Γ
{p
0
+1
,σ
2
}
(
x, y
)
D
x
−
p
0
+
iD
y
−
p
0
exp
1
2
πσ
2
x
σ
2
+
i
−
y
σ
2
x
σ
2
+
i
−
y
σ
2
x
2
+
y
2
2
σ
2
=
−
···
+
σ
2
−
p
0
−
exp
x
2
+
y
2
2
σ
2
x
σ
2
+
i
−
y
σ
2
x
σ
2
+
i
−
y
σ
2
−
p
0
−
−
p
0
−
1
+
i
2
p
0
−
−
p
0
−
1
1
2
πσ
2
1
σ
2
x
σ
2
+
i
−
y
σ
2
1
σ
2
x
σ
2
+
i
−
y
σ
2
=
···
exp
+
−
p
0
+1
exp
x
2
+
y
2
2
σ
2
x
2
+
y
2
2
σ
2
1
2
πσ
2
x
σ
2
+
i
−
y
σ
2
×
−
−
−
x
σ
2
p
0
+1
exp
x
2
+
y
2
2
σ
2
1
2
πσ
2
+
i
−
y
σ
2
=
−
(11.112)
Consequently, when it holds for
p
=
p
0
, Eq. (11.98) will also hold for
p
=
p
0
+1.
Now we turn to the general case, lemma 11.5.
x
2
+
y
2
2
σ
2
1
2
πσ
2
Q
(
D
x
+
iD
y
)
exp(
−
)=
N−
1
1
2
πσ
2
x
2
+
y
2
2
σ
2
a
n
(
D
x
+
iD
y
)
n
] exp(
=
[
−
)
n
=0
and obtain
