Image Processing Reference
In-Depth Information
(Proof) The connectivity number
Nc
(
m
)
(
) was defined as the change in the
Euler number caused by the deletion of a 1-voxel
x
(Def. 4.10). Since the Euler
number is represented as Eq. 4.5, Eq. 4.6, and Eq. 4.7, the above relation is
derived immediately.
x
Remark 4.11.
The change in the Euler number caused by the deletion of a
1-voxel
(i.e., the connectivity number) is also written as follows using the
amount of change in Betti numbers.
x
)=
1
+
∆b
(
m
0
−
∆b
(
m
)
1
+
∆b
(
m
)
2
Nc
(
m
)
(
x
,
(4.13)
and
b
(
m
)
k
are Betti numbers of the order
k
before and after deletion of the 1-voxel
=
b
(
m
)
where
∆b
(
m
)
k
b
(
m
)
k
,k
=
0
,
1
,
2
,m
=
6
,
18
,
18
,
26
;
b
(
m
)
k
k
−
x
,
respectively, and
m
denotes the type of connectivity.
x
Theorem 4.1.
The following relation holds at an arbitrary 1-voxel
among
the connectivity number
Nc
(
m
)
(
x
) and the connectivity index (
R
(
m
)
(
x
)
,H
(
m
)
(
x
)
,
Y
(
m
)
(
x
)).
Nc
(
m
)
(
)=
R
(
m
)
(
H
(
m
)
(
)+
Y
(
m
)
(
x
x
−
x
x
)
)
(4.14)
where
m
(
m
=
6
,
18
,
18
,
26
) denotes the type of connectivity.
(Proof) See [Toriwaki02a, Toriwaki02b].
Property 4.3.
Let
x
20
is excluded) denote a set of
26 voxels in the 26-neighborhood of the voxel
X
=
{
x
10
,
x
11
,...,
x
38
}
(
x
20
in Fig. 4.2. Let us regard
in this notation an arbitrary 1-voxel
x
as
x
20
in Fig. 4.2
T
.Byusingthe
notation
Nc
(
m
)
(
) instead of
Nc
(
m
)
(
) so that we may show explicitly
that the connectivity number (CN) depends on a set
X
,
x
x
X
as well as
x
itself, the
following equations hold
Nc
(26)
(
Nc
(6)
(
X
,
x
−
X
,
x
)
,
)=
2
Nc
(18
)
(
Nc
(18)
(
X
,
x
)=
2
−
X
,
x
)
,
(4.15)
where
X
is a set of variables that are complements of elements of
X
,that
−
x
ij
.
Nc
(6)
(
is,
X
=
{
x
10
,
x
11
,...,
x
38
}
,where
x
ij
=
1
X
,
x
)representsthe
value of the CN for the configuration consisting of
x
and
X
, instead of
x
and
X
[Toriwaki02a, Toriwaki02b].
(Proof) Note that each term relating to a voxel
in the equations to calculate
the Euler number in Eq. 4.8 is also a function of the set
x
X
and the variable
x
. Denote this term by
∆
E
(
X
,
x
). Then, by the definition of the CN,
Nc
(26)
(
E
(26)
(
E
(26)
(
X
,
x
)=
∆
X
,
x
)
−
∆
X
,
x
)+
1
E
(6)
(
E
(6)
(
=
∆
X
,
x
)
−
∆
X
,
x
)+
1
(
Nc
(6)
(
=
−
X
,
x
)
−
1
)+
1
Nc
(6)
(
=
2
−
X
,
x
)
.
(4.16)
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