Image Processing Reference
In-Depth Information
we can add to
x, y
to obtain the new coordinates (
x
,y
)
T
. This will yield an in-
finitesimal scaling:
D
ξ
x
=
x
D
ξ
y
=
y
x
=
x
+
xdx
y
=
y
+
ydy
⇒
(11.31)
Similarly, a small rotation is obtained as
D
η
x
=
x
=
x
−
y
ydx
y
=
y
+
xdy
−
⇒
(11.32)
D
η
y
=
x
x
=
x
y
=
y
+
xdy.
D
η
x
=
−
y
⇒
−
ydx,
and
D
η
y
=
x
⇒
(11.33)
We extend the definition of the ILST introduced in Chap. 10 to harmonic coordi-
nates
Definition 11.4.
The infinitesimal linear symmetry tensor w.r.t. the harmonic coordi-
nates
ξ
,
η
is defined as
ξ,η
(
f
)(
ξ, η
)=[
D
ξ
f
(
ξ, η
)+
iD
η
f
(
ξ, η
)]
2
(11.34)
The subscripts in
ξ,η
represent the derivation variables, i.e., they remind that the
two derivations of the operator are taken with respect to
ξ
and
η
variables, respec-
tively.
Lie operators can be translated to Cartesian coordinates even if they were initially
formulated in canonical coordinates. Accordingly, we can also translate
ξ,η
, which
consists of the square of a Lie operator applied to an image, as follows:
ξ,η
(
f
)(
ξ, η
)=(
D
ξ
f
+
iD
η
f
)
2
=[(
∇
⊥
ξ
)
T
f
]
2
/
4
.
∇
ξ
+
i
∇
∇
ξ
(11.35)
However, since
∇
⊥
ξ
=
ξ
x
−
=(
ξ
x
−
iξ
y
)
1
iξ
y
∇
ξ
+
i
(11.36)
i
(
ξ
x
−
iξ
y
)
i
where
ξ
x
−
iξ
y
is complex-valued (scalar function), we have the result
(1
,i
)
f
x
f
y
2
=
(
ξ
x
−
ξ,η
(
f
)(
ξ, η
)=
(
ξ
x
−
iξ
y
)
2
iξ
y
)
2
(
f
x
+
if
y
)
2
|
ξ
x
−
iξ
y
|
4
|
ξ
x
−
iξ
y
|
4
ξ,η
(
ξ
)(
x, y
)
|
x,y
(
ξ
)(
x, y
)
=
|
2
x,y
(
f
)(
x, y
)
.
(11.37)
which we restate in the following lemma.
Lemma 11.1.
Under a harmonic conjugate basis change given by
ξ, η
, the ILST
changes basis according to:
ξ,η
(
f
)(
ξ, η
)=
x,y
(
ξ
)(
x, y
)
|
x,y
(
ξ
)(
x, y
)
|
2
·
x,y
(
f
)(
x, y
)
(11.38)
