Image Processing Reference
In-Depth Information
Fig. 4.9.
Examples of values of connectivity index for complicated local patterns.
Numbers in parenthesis show connectivity indexes (component index, hole index,
cavity index). From top to bottom, 6-, 18-, 18
-, and 26-connectivity cases are given.
Note that the 18-neighborhood is used in calculating the component index of the
6-connectivity case.
(b) Simplex: 3D simplexes and lower dimensional ones are found by searching
2
2
patterns.
(c) Marching cubes algorithm: In the field of computer graphics, a polygonal
surface approximating equidensity surfaces are derived by examining den-
sity values of
2
×
2
×
×
×
2
voxels. The well-known algorithm to execute this
is called a marching cubes algorithm [Lorensen87].
2
4.5.2
3
×
3
×
3
local patterns
(1)
The number of local patterns
: Let us consider a 1-voxel
x
0
and its
3
×
3
×
3
x
0
. It was shown by a simple ex-
haustive enumeration in [Toriwaki02b] that there are
2
,
852
,
288
patterns
excluding those derived by applying linear symmetrical transformation
from other patterns. Numbers of all those patterns classified according
to the number of 1-voxels and values of connectivity index are given in
[Toriwaki02b].
(2)
Number of
1
-voxels
: The simplest feature is the number of 1-voxels in the
neighborhood. It is obvious that the number of 1-voxels may take
0
neighborhood
N
333
(
x
0
), centered at
26
in the 26-neighborhood, and that the number of patterns including
k
1-
voxels is
26
C
k
. This includes patterns that are symmetrical to each other.
The number of patterns excluding those that are symmetrical to each
other is not reported yet.
(3)
Connectivity index
: All possible values of the connectivity index (
R
(
m
)
(
∼
x
)
,
H
(
m
)
(
)
,Y
(
m
)
(
x
x
)) are listed in Table 4.2.
The number of possible patterns for each set of these values are enumer-
ated and given in [Toriwaki02b]. Several examples of complicated patterns are
presented in Fig. 4.9.
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