Image Processing Reference
In-Depth Information
Fig. 4.2.
Notations of voxels in the neighborhood and three basic neighborhoods:
(
a
) 6-neighborhood =
S
1
;(
b
) 18-neighborhood =
S
1
∪ S
2
;(
c
) 26-neighborhood =
S
1
∪ S
2
∪ S
3
.
x
∈N
[6]
(
If two voxels
x
and
y
satisfy the relation “
y
),” it is said that
x
and
y
are 6-adjacent to, each other, or that
x
is 6-adjacent to
y
, etc. The 18-
and 26-adjacency are also defined in the same way.
Remark 4.2.
If the neighborhood of a voxel (
i, j, k
) is symmetrical in respect
to (
i, j, k
),
x
∈N
[
k
]
(
y
∈N
[
k
]
(
), and vice versa. This is not always
true otherwise or if the neighborhood is asymmetrical.
y
) implies
x
x
Remark 4.3.
As was stated in Section 2.4.2, a neighborhood of a voxel
in
x
general may or may not contain the voxel
itself. For example, the 6- and the
26- neighborhood of
x
does not contain the central voxel
x
, but the
K
×
L
×
M
neighborhood of
x
includes the voxel
x
.
4.1.2 Connectivity and connected component
Let us define the concept of connectivity between two voxels based on the
neighborhood.
Definition 4.2 (Connectivity).
Two voxels
x
2
with a common value
are said to be 6
-connected
(18
-connected
,26
-connected
), if a sequence of voxels
y
0
(=
x
1
and
x
1
)
,
y
1
,...,
y
n
(=
x
2
) exists, such that each
y
i
is in the 6-neighborhood
( 18-neighborhood, 26-neighborhood ) of
y
i−
1
(
∀
i
(
1
≤
i
≤
n
)andall
y
i
s
have the same value as
x
2
. It is clear that 6-connectedness, 18-
connectedness, and 26-connectedness thus defined gives a kind of
equivalence
relationship
among voxels with the value
1
(or value
0
).
x
1
and
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