Image Processing Reference
In-Depth Information
4.6.3 Calculation of the connectivity number
The connectivity number (CN)
Nc
(
m
)
(
x
) is obtained by counting simplexes
and using the following equation.
Property 4.5.
Let
n
(
m
)
k
and
n
(
m
)
k
denote the number of
k
-simplexes (
k
=
0
,
1
,
2
,
3
) contained in a 3D object before and after the deletion of a 1-voxel
x
by
Nc
(
m
)
(
. Denoting the CN at a 1-voxel
x
x
),
Nc
(
m
)
(
)=
∆n
(
m
1
−
∆n
(
m
)
2
+
∆n
(
m
)
3
x
,
(4.20)
where
∆n
(
m
)
k
=
n
(
m
k
−
n
(
m
)
k
, and m represents the type of connectivity.
(Proof) Denoting the Euler number of an object before and after the deletion
of a 1-voxel
by
E
(
m
)
(
)and
E
(
m
)
(
x
x
x
), respectively,
)=
n
(
m
0
−
n
(
m
)
1
+
n
(
m
2
−
n
(
m
)
3
E
(
m
)
(
x
(4.21)
)=
n
(
m
0
−
n
(
m
)
1
+
n
(
m
2
−
n
(
m
)
3
E
(
m
)
(
x
,
(4.22)
and
n
(
m
0
−
n
(
m
)
0
1
. This and the definition of the CN imply Eq. 4.20.
Note here that changes in the number of simplexes due to the deletion of
avoxel
=
−
x
occur in the neighborhood of
x
only.
Remark 4.15.
For the 6-connectivity case,
∆n
[6]
k
is given as follows.
∆n
[6]
1
x
.
= number of 1-voxels 6-adjacent to
(4.23)
∆n
[6]
2
×
x
.
= number of the set of
2
2
1-voxels including
(4.24)
∆n
[6]
3
×
×
x
.
= number of the set of
2
2
2
1-voxels including
(4.25)
Property 4.6.
(1) The hole index
H
(
m
)
(
x
x
is equal to the
amount of increase in the 1D Betti number in the 26-neighborhood of
)ata1-voxel
x
.Inotherwords,
H
(
m
)
(
caused by deleting
x
x
) is equal to the number of
holes in the
3
×
3
×
3
local area consisting of
x
and its 26-neighborhood
created by the deletion of
. It equals the number of separate connected
components of 0-voxels that are connected by the deletion of
x
x
.
H
(
m
)
(
(2)
0
≤
x
)
≤
7
.
(4
.
26)
R
(
m
)
(
(3)
0
≤
x
)
≤
8
.
(4
.
27)
Nc
(
m
)
(
(4)
−
6
≤
x
)
≤
8
.
(4
.
28)
[Toriwaki02a]
Theorem 4.3.
The connectivity number
Nc
(
m
)
(
x
)ata1-voxel
x
is calcu-
lated by the following equations [Toriwaki02a, Toriwaki02b].
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