Image Processing Reference
In-Depth Information
Notice that the basis change resulted in a decoupling of the terms according to
their dependency on coordinates. The first term depends only on
ξ
, computable with-
out a knowledge of
f
, whereas the second term is dependent only on the image
f
,
computable without a knowledge of
ξ
. For convenience, in what follows we drop the
mnemonic subscripts of
x,y
when it is clear from the context that the derivations
are w.r.t.
x, y
. Consequently,
I
20
is given by
I
20
=
ξ,η
(
f
)(
ξ, η
)
dξdη
(11.39)
=
∗
(
ξ
)(
x, y
)
|
∗
(
ξ
)(
x, y
)
(
f
)(
x, y
)det(
∂ξ
(
x, y
)
∂x∂y
|
2
)
dxdy
(11.40)
=
∗
(
ξ
)(
x, y
)
|
∗
(
ξ
)(
x, y
)
|
(
f
)(
x, y
)
dxdy
(11.41)
=
w
20
∗
(
x, y
)
(
f
)(
x, y
)
dxdy
(11.42)
with
w
20
(
x, y
)=
(
ξ
)(
x, y
)
|
(11.43)
(
ξ
)(
x, y
)
|
Here the Cauchy-Riemann equations applied to the functional determinant:
det
∂ξ
(
x, y
)
∂x∂y
=
ξ
x
ξ
y
−
=
ξ
x
+
ξ
y
=
|
∗
(
ξ
)(
x, y
)
|
(11.44)
ξ
y
ξ
x
have been used. Accordingly, we have the following result:
Lemma 11.2.
The complex moment
I
20
of the power spectrum in harmonic coordi-
nates can be estimated in Cartesian coordinates as follows
I
20
=
w
20
∗
(
x, y
)
(
f
)(
x, y
)
dxdy
(11.45)
where
w
20
(
x, y
)=
(
ξ
)(
x, y
)
|
(11.46)
(
ξ
)(
x, y
)
|
In Eq. (11.45), we note that
I
20
is obtained by projecting the ILST of the image
in Cartesian coordinates on the kernel
w
20
, which consists of the ILST of the target
curve pattern,
ξ
, except for a magnitude normalization. This is a scalar product be-
tween a function, which only depends on the image and is independent of the target
pattern family
ξ
, and a kernel that encodes the directional information of the pattern
family. Consequently, once the ILST of the image is available it can be tested for
matching a multitude of pattern families by changing the corresponding kernel
w
20
without recalculating the ILST of the image. This is not surprising as the ILST of
the image represents “universal” information for pattern matching, namely the local
edges and the double of their directions [90, 137].
Similarly,
I
11
which is the average ILST magnitude, can be obtained. We state
this result as a lemma.
