Image Processing Reference
In-Depth Information
Number of 1-simplex =
i
x
0
(
x
1
+
x
2
+
x
4
)
j
k
Number of 2-simplex =
i
x
0
(
x
1
x
2
x
3
+
x
2
x
4
x
6
+
x
1
x
4
x
5
)
j
k
[
7
l
=0
Number of 3-simplex =
i
x
l
]
,
(4.47)
j
k
where
i
means the summation over the whole of a given image. Sub-
j
k
stituting them into Eq. 4.5, we obtain Eq. 4.43.
[26-c case] Let
F
=
{
f
ijk
}
and
F
=
{
f
ijk
}
denote an input image
F
and its
inversion, that is,
F
=
{
1
−
f
ijk
}
. Denoting the Euler number of an image
F
for the
m
-c case (
m
=
6
,
18
,
18
,
26
)by
E
(
m
)
(
F
), following relations hold:
E
(6)
(
E
(26)
(
E
(26)
(
E
(6)
(
F
)=
F
)
,
F
)=
F
)
(4.48)
E
(18
)
(
E
(18
)
(
E
(18)
(
E
(18)
(
F
)=
F
)
,
F
)=
F
)
.
(4.49)
These are proved by calculating
∆
E
(
V
) of Eq. 4.41 for all
2
×
2
×
2
or
3
3
configurations given in Table 4.4. Eq. 4.46 is derived immediately
from Eq. 4.48.
[18
-c case] From the definition of the 18
-connectivity,
×
3
×
E
(18
)
(Eq. 4.45) is
E
(6)
(Eq. 4.43) the contribution of the configuration
shown in Fig. 4.4 which is a 2-simplex for the 18
-c case.
[18-c case] Eq. 4.44 is immediately derived from Eqs. 4.45 and 4.49.
obtained by adding to the
4.8 Algorithm of deletability test
A deletability test is performed by examining conditions in Theorem 4.2 or
Corollary 4.1 and is implemented as follows.
(a) Calculation of connectivity number and connectivity index by pseudo-
Boolean expression (Theorem 4.3).
(b) Local pattern matching on a
2
×
2
×
2
or a
3
×
3
×
3
local subarea.
(c) Test of shape features in the projection graph.
(d) Calculation of the adjacency matrix corresponding to the projection graph
[Yonekura80b, Yonekura80c, Yonekura82c].
Different combinations of these are used for different objectives. Rough
guidelines will be given as follows.
(i) Enumeration of
3
3
local patterns are used to derive a new procedure.
Reduction of a 3D arrangement of voxels to a 2D image by the use of the
projection graph can be used for convenience [Yonekura82c].
×
3
×
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