Image Processing Reference
In-Depth Information
Definition 11.6.
We apply the
p
th symmetry derivative to the Gaussian and define
the function
Γ
{p,σ
2
}
as
1
2
πσ
2
e
−
x
2
+
y
2
2
σ
2
Γ
{p,σ
2
}
(
x, y
)=(
D
x
+
iD
y
)
p
(11.97)
with
Γ
{
0
,σ
2
}
being the ordinary Gaussian.
Theorem 11.2.
The differential operator
D
x
+
iD
y
and the scalar
−
1
σ
2
(
x
+
iy
)
op-
erate on a Gaussian in an identical manner:
(
D
x
+
iD
y
)
p
Γ
{
0
,σ
2
}
=(
−
1
σ
2
)
p
(
x
+
iy
)
p
Γ
{
0
,σ
2
}
(11.98)
The theorem, proved in the Appendix (Sect. 11.13), reveals an invariance property of
the Gaussians w.r.t. symmetry derivatives. We compare the second-order symmetry
derivative with the classical Laplacian, also a second-order derivative operator, to
illustrate the analytical consequences of the theorem. The Laplacian of a Gaussian,
+
x
2
+
y
2
σ
4
2
σ
2
(
D
x
+
D
y
)
Γ
{
0
,σ
2
}
=(
)
Γ
{
0
,σ
2
}
−
(11.99)
can obviously not be obtained by a mnemonic replacement of the derivative symbols
D
x
with
x
and
D
y
with
y
in the Laplacian operator. As the Laplacian already hints,
with an increased order of derivatives, the resulting polynomial factor, e.g.,, the one
on the right-hand side of Eq. (11.99), will resemble less and less the polynomial form
of the derivation operator. Yet, it is such a form invariance that the theorem predicts
when symmetry derivatives are utilized. By using the linearity of the derivation op-
erator, the theorem can be generalized, see Sect. 11.13, to any polynomial as follows:
Lemma 11.5. Let the polynomial
Q
be defined as
Q
(
q
)=
N−
1
n
=0
a
n
q
n
. Then
Q
(
D
x
+
iD
y
)
Γ
{
0
,σ
2
}
(
x, y
)=
Q
(
−
1
σ
2
(
x
+
iy
))
Γ
{
0
,σ
2
}
(
x, y
)
(11.100)
That the Fourier transformation of a Gaussian is also a Gaussian has been known
and exploited in information sciences. It turns out that a similar invariance is valid
for symmetry derivatives of Gaussians, too.
A proof of the following theorem is omitted because it follows by observing that
derivation w.r.t.
x
corresponds to multiplication with
iω
x
in the Fourier domain and
applying Eq. (11.98). Alternatively, theorem 3.4 of [208] can be used to establish it.
Theorem 11.3.
The symmetry derivatives of Gaussians are Fourier transformed on
themselves, i.e.,
