Image Processing Reference
In-Depth Information
Fig. 4.1.
Basic concept of digital geometry.
∈S
ijk
}S
⊂S
N
ijk
≡{
(
i
+
p, j
+
q, k
+
r
); (
p, q, r
)
(
≡
I
×
I
×
I)
,
(4.1)
S
ijk
where I is the set of all integers and
is a suitably given set of integer
triads.
Here we consider only the neighborhood that does not depend on the
position (
i, j, k
). Examples of the neighborhood were given in Table 2.4 and
Fig. 4.2. Three of them - the 6-neighborhood, the 18-neighborhood, and the
26-neighborhood - are most frequently used in practical applications and often
denoted in the following way (Fig. 4.2)
N
[6]
(
x
0
)=
{
x
p
,p
∈
S
1
}
,
(4.2)
N
[18]
(
x
0
)=
{
x
p
,p
∈
S
1
∪
S
2
}
,
(4.3)
N
[26]
(
x
0
)=
{
x
p
,p
∈
S
1
∪
S
2
∪
S
3
}
,
(4.4)
where
S
1
,
S
2
,and
S
3
are sets of integers and show the numbers given to voxels
in
3
3
neighborhood as in Fig. 4.2. We also use two other numbering
systems as shown in the system
T
and
U
in Fig. 4.2.
×
3
×
Remark 4.1.
Considering a voxel is a cube in a digitized image, then
(i) any of voxels in the 6-neighborhood of a voxel
x
shares at least one face
,
(ii) any of voxels in the 18-neighborhood of a voxel
of the voxel
x
x
shares at least one edge
,
(iii) any of voxels in the 26-neighborhood of a voxel
of the voxel
x
x
shares at least one
vertex of the voxel
x
.
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