Image Processing Reference
In-Depth Information
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(
a
)
(
b
)
Fig. 5.17.
Surface neighborhood and starting voxel pair: (
a
) Surface neighborhood.
Assume the
m
-connectivity case. For
m
= 6, (i), (ii), (iii), and (iv); for
m
= 18,
(v)and (vi); for
m
=18
, (i) and (ii); and for
m
= 26, (v), (vi), and (vii) are used,
respectively; (
b
) possible pattern of a starting voxel pair in Algorithm 5.17.
Following are important properties of the surface neighborhood defined in
Def. 5.10.
Property 5.7.
Assume that a mark value
K
is given to each voxel of a con-
nected component
Q
of 0-voxels (assume that a voxel of
Q
(= set of voxels
that do not belong to
Q
) does not have the value
K
). Then if X
B
(
P, Q
)
for a 1-voxel X and an arbitrary border surface
B
(
P, Q
), then all voxels of
the value
K
in the surface neighborhood of X belong to the border surface
B
(
P, Q
).
∈
Property 5.8.
Assuming that each voxel in a connected component
Q
of 0-
voxels has the mark value
K
(
K
=
0
,
1
), let us consider a graph as follows
for a border surface
B
(
P, Q
).
(1) Put a node at the center of each voxel in
B
(
P, Q
).
(2) If two voxels in
B
(
P, Q
) is in the surface neighborhood mutually, connect
those two voxels with an edge.
Then this graph becomes a connected graph. Let us denote this graph by
G{
B
(
P, Q
)
,ε
}
,where
ε
is the set of all edges.
[Proofs of these properties were shown in [Matsumoto84]]
This property guarantees that we can extract all border voxels while keep-
ing adjacency relations among them by searching the graph
}
and extracting all nodes of the graph. Here each node corresponds to an in-
dividual border voxel. It should be noted that a mark
K
needs to be given to
G{
B
(
P, Q
)
,ε
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