Image Processing Reference
In-Depth Information
The second equation in Eq. 4.15 is also derived in the same way. Note here
Eq. 4.48 and Eq. 4.49 which will be derived later.
A similar relationship holds concerning the connectivity index (
R, H, Y
)
as is given below.
Property 4.4.
Let us adopt the same notation as in Property 4.3 above, and
let us express the connectivity index of the 1-voxel
x
20
by (
R
(
m
)
(
)
,H
(
m
)
X
,
x
)
,Y
(
m
)
(
)) instead of (
R
(
m
)
(
)
,H
(
m
)
(
)
,Y
(
m
)
(
(
X
,
x
X
,
x
x
x
x
)) so that depen-
dency on the set
X
may be explicitly shown. If we define a set
X
=
{
x
10
,
x
11
,...,
x
38
}
x
ij
−
x
ij
,where
=
1
as in Property 4.3, and if we de-
note by (
R
(
m
)
(
)
,H
(
m
)
(
)
,Y
(
m
)
(
X
,
x
X
,
x
X
,
x
)) the connectivity index of
x
with the local subpattern
X
in its 26-neighborhood, the following relations
hold.
R
(6)
(
1
=
H
(26)
(
Y
(26)
(
X
,
x
)
−
X
,
x
)
−
X
,
x
)
R
(26)
(
1
=
H
(6)
(
Y
(6)
(
X
,
x
)
−
X
,
x
)
−
X
,
x
)
1
=
H
(18
)
(
Y
(18
)
(
R
(18)
(
X
,
x
)
−
X
,
x
)
−
X
,
x
)
R
(18
)
(
1
=
H
(18)
(
Y
(18)
(
X
,
x
)
−
X
,
x
)
−
X
,
x
)
.
(4.17)
Furthermore,
Y
(
m
)
(
X
,
x
)=
1
if and only if
R
(
m
)
(
X
,
x
)=
H
(
m
)
(
X
,
x
)=
0
,
and otherwise,
Y
(
m
)
(
X
,
x
)=
0
[Toriwaki02a, Toriwaki02b].
Theorem 4.2.
Assuming that the density values of a 1-voxel
x
and its 26-
neighborhood are given, the 1-voxel
x
is
m
-deletable if and only if the con-
nectivity index (
R
(
m
)
(
)
,H
(
m
)
(
)
,Y
(
m
)
(
x
x
x
)) = (
1
,
0
,
0
).
(Proof) A 1-voxel
x
is not deletable if it is a 3D interior voxel, that is,
Y
(
m
)
(
)=
1
. Therefore,
Y
(
m
)
(
x
x
) should be equal to zero for
x
to be
m
-
deletable. Let us assume that by deleting a 1-voxel
with the connectivity
index (
l, n,
0),
α
connected components and
β
holes are created and
β
holes
and
γ
cavities vanish. Then,
l
=
α
+
β
+
1
,and
n
=
β
+
γ
.Avoxel
x
x
is
deletable if and only if
α
=
β
=
β
=
γ
=
0
.
(4.18)
Eq. 4.18 is equivalent to “
l
=
1
and
n
=
0
” [Toriwaki02a, Toriwaki02b].
Corollary 4.1.
1-voxel
x
is
m
-deletable if and only if
Nc
(
m
)
(
)=
1
and
R
(
m
)
=
1
.
x
(4.19)
For a 2D image, preservation of the values of the Euler characteristic is
a necessary and sucient condition of the topology preservation. That is, to
test the deletability of a 1-pixel in a 2D figure, calculation of the Euler char-
acteristic is enough. In a 3D image, on the contrary, preservation of the Euler
characteristic does not always mean the deletability of a 1-voxel. The rea-
son is the existence of a hole (handle). By deleting a 1-voxel, separation of a
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