Image Processing Reference
In-Depth Information
isocurves fulfill the Laplacian equation, as this is difficult to verify for an arbitrary
digital prototype, and it is not very helpful when
ξ
is not strictly harmonic. The tilde
in
ξ
j
is used in order to emphasize that we do not know whether
ξ
is harmonic or not.
We can exploit the fact that the computation of
I
20
is essentially a matching between
the ILST image and the sampled version of the normalized ILST of the prototype.
As an algorithm implementing the discrete ILST operator according to Eq. (11.54)
is assumed to exist, we can apply such an algorithm to
ξ
j
to obtain:
(
ξ
)(
x
j
,y
j
)=[(
ξ
x
j
+
iξ
y
j
)]
2
,
(11.60)
We can then proceed as if
ξ
j
is a sampled harmonic function and find the kernel
w
20
j
as
=
exp(
i
2tan
−
1
(
ξ
x
j
, ξ
y
j
))
if
∇
ξ
(
x
j
,y
j
)
=0;
w
20
j
(11.61)
0
,
if
∇
ξ
(
x
j
,y
j
)=0.
Accordingly, in analogy with Eq. (11.56),
I
20
can be computed as
I
20
=
j
(
f
)(
x
j
,y
j
)(
w
2
j
)
∗
(11.62)
The classical alternative to the case in this subsection is to directly match (cor-
relate) the two digital images,
f
j
and
ξ
j
, without filtering them through the ILST
operator. However, matching the ILST image with an appropriate kernel has certain
advantages. In the ILST approach it is the edges of
f
j
and not the gray values which
are aligned in case of match. As a consequence we can expect a high localization.
However, this is only a byproduct; the main advantage is the complex voting process
and its rich interpretability, as will be discussed in Sect. 11.5.
Case 3: Estimation of
I
11
According to Eq. (11.47),
I
11
is obtained as
I
11
=
1
dxdy
=
w
20
(
x, y
)
|
(
f
(
x, y
))
|·
|
(
f
)(
x, y
)
||
|
dxdy
(11.63)
Consequently, the continuous kernel of
I
11
is
w
11
=
w
20
|
|
(11.64)
By using the reconstructed
|
, and in analogy with Eq. (11.56),
I
11
=
j
(
f
)
|
w
11
j
|
(
f
)(
x
j
,y
j
)
|
(11.65)
where the discrete kernel,
=
w
11
j
w
20
(
x, y
)
ψ
(
x
−
x
j
,y
−
y
j
)
|
|
dxdy
(11.66)
