Image Processing Reference
In-Depth Information
[
Γ
{p,σ
2
}
]=
Γ
{p,σ
2
}
(
x, y
)
e
−iω
x
x−iω
y
y
dxdy
F
=2
πσ
2
(
−
i
σ
2
)
p
Γ
{p,
σ
2
}
(
ω
x
,ω
y
)
(11.101)
We note that, in the context of prolate spheroidal functions [204], and when con-
structing rotation-invariant 2D filters [52], it has been observed that the (integer)
symmetry order
n
of the function
h
(
ρ
) exp(
inθ
), where
h
is a one-dimensional func-
tion,
ρ
and
θ
are polar coordinates, is preserved under the Fourier transform. Accord-
ingly, the Fourier transforms of such functions are:
H
0
[
h
(
ρ
)](
ω
ρ
), where
H
0
is the
Hankel transform (of order 0) of
h
. However, a further precision is needed as to the
choice of the function family
h
, to render it invariant to Fourier transform.
Another analytic property that can be used to construct efficient filters by cascad-
ing smaller filters or simply to gain further insight into steerable filters or rotation-
invariant filters is the addition rule under the convolution. This is stated in the fol-
lowing theorem and is proved in the Appendix.
Theorem 11.4.
The symmetry derivatives of Gaussians are closed under the convo-
lution operator so that the order and the variance parameters add under convolution:
Γ
{p
1
,σ
1
}
∗
Γ
{p
2
,σ
2
}
=
Γ
{p
1
+
p
2
,σ
1
+
σ
2
}
(11.102)
11.10 Discrete GST for Harmonic Monomials
We discussed above how analytic functions
g
(
z
) generate curve families via a linear
combination of their real,
ξ
=
[
g
(
z
)] parts. The curve
families generated by the real part are locally orthogonal to those of the imaginary
part. Here we discuss a specific family. The next lemma, a proof of which is given in
the Appendix, makes use of the symmetry derivatives of Gaussians to represent and
to sample the generalized structure tensor. Sampled functions are denoted as
f
k
, i.e.,
f
k
=
f
(
x
k
,y
k
), as before.
[
g
(
z
)], and imaginary,
η
=
Lemma 11.6.
Consider the analytic function
g
(
z
)
with
d
dz
=
z
n
2
, and let
n
be an
. Then the discretized filter
Γ
{n,σ
2
}
k
integer,
0
,
±
1
,
±
2
···
is a detector for patterns
generated by the curves
a
[
g
(
z
)] =
constant, provided that a shifted
Gaussian is assumed as an interpolator and the magnitude of a symmetry derivative
of a Gaussian acts as a window function. The discrete scheme
[
g
(
z
)] +
b
2
)=
C
n
Γ
{n,σ
2
}
(
Γ
{
1
,σ
1
}
k
f
k
)
2
I
20
(
|
F
(
ω
ξ
,ω
η
)
|
∗
∗
(11.103)
k
Γ
{n,σ
2
}
k
Γ
{
1
,σ
1
}
k
|
2
)=
C
n
|
f
k
|
2
I
11
(
|
F
(
ω
ξ
,ω
η
)
|∗|
∗
(11.104)
