Image Processing Reference
In-Depth Information
It follows from the above that the building blocks of
Z
are obtained as follows:
2
)=
(
D
ξ
f
(
ξ, η
)+
iD
η
f
(
ξ, η
))
2
dξdη
I
20
(
|
F
(
ω
ξ
,ω
η
)
|
(11.27)
and
|
2
)=
|
2
dξdη
I
11
(
|
F
(
ω
ξ
,ω
η
)
|
D
ξ
f
(
ξ, η
)+
i
L
η
f
(
ξ, η
)
(11.28)
Again, the real moments
μ
20
,
μ
02
, and
μ
11
, as well as the complex moments
I
20
and
I
11
are all taken w.r.t
harmonic coordinates
as in Eqs. (11.23), Eq. (11.27), and Eq.
(11.28). We must still find a way to obtain them directly by using Cartesian coor-
dinates. In different coordinate systems the representation of the Lie operators and
the linear symmetry operator may look quite different despite the identical physi-
cal effect when applied to a function. The representation of the Lie operators in the
Cartesian coordinates can be obtained by using the chain rule:
T
ξ
∇
∂
∂x
+
y
ξ
∂
∂y
=
ξ
x
ξ
x
+
ξ
y
∂
∂x
+
ξ
y
ξ
x
+
ξ
y
∂y
=
∇
∂
D
ξ
=
x
ξ
2
∇
(11.29)
ξ
T
⊥
∂
∂x
+
y
η
∂
∂y
=
ξ
y
ξ
x
+
ξ
y
∂
∂x
+
ξ
x
ξ
x
+
ξ
y
∂y
=
∇
∂
ξ
D
η
=
x
η
−
2
∇
(11.30)
∇
ξ
∂ξ
(
x,y
)
∂x
T
⊥
where the definitions
ξ
x
=
ξ
y
,ξ
x
) are used for simplicity.
Moreover, the partial derivatives of
ξ
and
η
are obtained by inverting the Jacobian
4
of
T
, and using the Cauchy-Riemann equations, Eq. (11.2):
and
∇
ξ
=(
−
=
x
ξ
x
η
y
ξ
y
η
=
∂
2
(
ξ, η
)
∂x∂y
−
1
=
ξ
x
ξ
y
η
x
η
y
−
1
=
ξ
x
−
∂
2
(
x, y
)
∂ξ∂η
1
ξ
x
+
ξ
y
ξ
y
ξ
y
ξ
x
We note that
D
η
does not explicitly depend on partial derivatives of
η
with respect
to
x
or
y
, which of course is the consequence of the HFP assumption that binds the
gradients of
ξ
and
η
together.
Example 11.6. For illustration we go back to Example 11.2, and see that the corre-
sponding infinitesimal operators are found by using Eqs. (11.29) and (11.30):
∂
∂x
+
y
∂
∂y
D
ξ
=
x
∂x
+
x
∂
∂
D
η
=
−
y
∂y
These are well-known scaling and rotation operators from differential geometry. By
applying
D
ξ
to the coordinate pair
x, y
we obtain an infinitesimal increment which
4
Represented by the symbol
∂
2
(
ξ,η
∂∂y
, a Jacobian of a 2D CT (
x, y
)
T
→
(
ξ, η
)
T
is defined as
the matrix
∂ξ
!
.
∂ξ
∂y
∂x
∂η
∂x
∂η
∂y
