Image Processing Reference
In-Depth Information
w
(
z
)=
√
z
=
ξ
(
x, y
)+
iη
(
x, y
)=
√
r
exp
i
ϕ
2
=
√
r
cos
ϕ
2
+
i
√
r
sin
ϕ
2
generates the basis patterns that are illustrated in Fig. 11.2 (bottom). The pattern
familygeneratedbythispairconsistsofrotatedversionsofoneofthebasispatterns;
that is, the rotation angle corresponds to the direction of the vector (
a, b
).
11.2 Lie Operators and Coordinate Transformations
Here we summarize the essential parts of the coordinate transformation theory using
differential operators to detect symmetric pattern families. We will briefly present
the Lie operators that perform small (infinitesimal) CTs, which are all equivalent to
translations in HFP coordinates.
We start by performing translations of
ξ
and
η
, which are coordinates that an
arbitrary image
f
will later be represented by, beginning with
ξ
. The translated
1
coordinates are marked with
and yield:
T
1
(
ξ, η,
1
)=
ξ
=
ξ
+
1
,
(11.10)
η
=
η.
Two successive translations are equivalent to a single translation which can be ob-
tained by using the parameter combination rule:
φ
(
1
,δ
)=
1
+
δ
(11.11)
where
φ
is analytic with respect to both of its arguments and fulfills the group axioms
with
=0being the identity element of the group. These properties make
T
1
a
one-
parameter Lie group of transformations
[31]. With each Lie group of transformation
an
infinitesimal generator
is associated. In this case, this is
2
D
ξ
=
∂ξ
. When applied
to
f
the CT results in a translation of the isocurves of
f
along the basis vector,
ξ
, which is the
curvilinear basis
related to
ξ
.
D
ξ
applied to any function whose
isocurves consist of
ξ
(
x, y
)=
λ
delivers the tangent fields of
ξ
. An arbitrary amount
of translations in the
ξ
direction can be obtained by applying the exponential form of
D
ξ
:
f
(
ξ
,η
)=
1+
1
D
ξ
+
2!
D
ξ
+
f
(
ξ, η
)=exp(
1
D
ξ
)
f
(
ξ, η
)
···
(11.12)
which is a Taylor expansion of
f
(
ξ
+
1
,η
) around (
ξ, η
). The CT (
ξ
,η
) is com-
pletely determined by
D
ξ
's actions on the
ξ
,
η
coordinates. As a special case, if the
isocurves of
f
are given by
ξ
= constant, i.e.,
f
(
ξ, η
)=
g
(
ξ
) for some
g
, then we
1
The symbol
represents new coordinates after a CT is applied to original coordinates in
this chapter.
2
In studies of CTs, it is common to use the notation
L
1
for
D
ξ
, and
L
2
for
D
η
, where
L
is
a mnemonic for the “Lie operator”.
