Image Processing Reference
In-Depth Information
in Sect. 11.10, though not by gray image correlations as these filters were originally
intended for, but via GST field correlations. We can conclude that the angular band-
limitedness condition is a sufficient but not a necessary condition for steering the
rotations of 2D patterns or for their detection.
11.9 Symmetry Derivatives and Gaussians
In this section we describe a set of tools that will be useful when recognizing intri-
cate patterns that certain harmonic curve families and the Lie operators are capable
representing.
Definition 11.5 (Symmetry derivative).
We define the first symmetry derivative as
the complex partial derivative operator:
∂x
+
i
∂
∂
D
x
+
iD
y
=
(11.94)
∂y
The symmetry derivative resembles the ordinary gradient in 2D. When it is applied
to a scalar function
f
(
x, y
), the result is a complex field instead of a vector field.
Consequently, the first important difference is that it is possible to take the (positive
integer or zero) powers of the symmetry derivative, e.g.,,
(
D
x
+
iD
y
)
2
=(
D
x
−
D
y
)+
i
(2
D
x
D
y
)
(11.95)
(
D
x
+
iD
y
)
3
=(
D
x
−
3
D
x
D
y
)+
i
(3
D
x
D
y
−
D
y
)
(11.96)
···
Second, being a complex scalar, it is even possible to exponentiate the result of the
symmetry derivative, i.e., (
D
x
+
iD
y
)
n
f
, to yield nonlinear functionals:
[(
D
x
+
iD
y
)
n
f
]
m
Had it not been for the two mentioned exponentiation properties, there would not be
any real reason to introduce the symmetry derivatives concept, because the ordinary
gradient operator,
, would be sufficient in practice. As will be seen, however, the
symmetry derivatives posses properties that make them elegant tools when modeling
and detecting intricate patterns.
The operator (
D
x
+
iD
y
)
n
will be defined as the
n
th order symmetry derivative
since its invariant patterns (those that vanish under the linear operator) are highly
symmetric. In an analogous manner, we define, for completeness,
the first conjugate
symmetry derivative
as
D
x
− iD
y
=
∇
∂x
− i
∂y
∂
and the
nth conjugate symmetry
iD
y
)
n
. We will, however, only dwell on the properties of the
symmetry derivatives. The extension of the results to conjugate symmetry derivatives
are straightforward.
derivative
as (
D
x
−
