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[
,
]
and the target image
g
[cf. Eq. (
6.3
)]. In the current work, however, ECAs are
extracted from the two images before the CCF values are calculated. Both images
are spatially normalized so that the pixel values outside the ECAs are taken to be
zero. This normalization process results in the output images,
f
[
i
k
and
g
[
,
]
,
]
,
which are then used to obtain the CCF value for similarity measurement. This is
explained in Sects.
6.3.1
and
6.3.2
.
i
k
i
k
6.3.1
Local Normalization with Cross-Covariance Function
The CCC is referred to as a bias-independent measure for two-dimensional discrete
data [
175
]. For extraction of ECAs, the CCC is first used for characterization of
the similarity of the individual parts of the image in order to eliminate the local
irrelevant region, thus defining the locally-normalized image data. Specifically, each
coefficient
CCC
s
for
s
=
1
,...,
S
is associated with a subset
R
s
ↂ X
, where
X
denotes the set of points in a
N
y
×
N
x
image lattice:
X
=
(
N
x
i
,
k
)
:1
≤
i
≤
N
y
,
1
≤
k
≤
(6.33)
and where the regions
R
s
forms a partition of
X
R
s
1
∩R
s
2
= ∅
,
s
1
=
s
2
(6.34)
S
1
R
s
=
X
(6.35)
s
=
Here, image lattice
X
is partitioned into non-overlapping square blocks of size
n
×
n
.
The selection strategy is to first classify each region
as
relevant or non-relevant, and then derive the normalized image data. The preliminary
classification was performed by adopting the following local
CCC
s
measure [
176
]
for each region:
R
s
for
s
∈{
1
,...,
S
}
∑
n
2
k
2
(
q
1
[
k
]
−
q
2
[
k
])
=
1
CCC
s
(
f
s
,
g
s
)=
1
−
(6.36)
n
2
k
2
(
q
1
[
k
])
∑
=
1
where
{
f
s
[
i
,
k
]
}
denotes the grayscale value of the reference image within the region
R
s
, that is,
{
f
s
[
i
,
k
]
}
=
{
f
[
i
,
k
]
:
(
i
,
k
)
∈ R
s
}
, and
{
g
s
[
i
,
k
]
}
denotes the grayscale
value of the target image within the region
R
s
, that is,
{
g
s
[
i
,
k
]
}
=
{
g
[
i
,
k
]
:
(
i
,
k
)
∈
R
s
}
sequence in Eq. (
6.36
) is a feature sequence whose elements in
descending order are obtained from the auto-covariance matrix of the reference
image
.The
{
q
1
[
k
]
}
{
f
s
[
i
,
k
]
}
; that is,
C
ff
[
i
,
k
]
defined as [
171
]:
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