Image Processing Reference
In-Depth Information
(1) separation of a connected component, hole, and cavity
(2) vanishment of a connected component, a hole, and a cavity
(3) creation of a new hole, cavity, and connected component
We have here a local feature which is effective in knowing shape features
relating to topology. Let us introduce several such features below.
Definition 4.9 (Deletability).
A1-voxel
is
deletable
(strictly,
m
-deletable
where
m
represents the type of connectivity;
m
=
6
,
18
,
18
,
26
), if its deletion
(= conversion into a 0-voxel) preserves topology of an input image.
x
Definition 4.10 (Connectivity number).
The
connectivity number
(CN)
Nc
(
m
)
(
x
)ata1-voxel
x
is defined as follows:
Nc
m
(
E
(
m
)
(
−E
(
m
)
(
x
)=
x
)
x
)+
1
,
(4.11)
E
(
m
)
(
E
(
m
)
(
where
) denote the values of the Euler number of a 3D
figure before and after deletion of a 1-voxel
x
)and
x
, respectively, and
m
denotes the
type of connectivity (
m
=
6
,
18
,
18
,and
26
).
x
Definition 4.11 (Connectivity index).
In a
3
×
3
×
3
neighborhood con-
sisting of a 1-voxel
and its 26-neighborhood, we define the following three
features, called
component index
,
hole index
,and
cavity index
, respectively.
Component index
R
(
m
)
(
x
x
)=thenumberof
m
-connected components that
and exist in the 18-neighborhood for
m
=
6
and
18
(in the 26-neighborhood for
m
=
18
and
26
). (The subpattern
Q
63
of
Table 4.4 is exceptionally regarded as a single connected component for
m
=
18
).
Hole index
H
(
m
)
(
are connected to
x
x
) = the number of holes that are newly created by deletion
of
, or equivalently the decrease in the number of
m
-connected compo-
nents of 0-voxels in this subarea caused by deletion of
x
x
.
Cavity index
Y
(
m
)
(
x
) = the number of cavities that are created by deleting
.
We call the triad (
R
(
m
)
(
x
)
,H
(
m
)
(
)
,Y
(
m
)
(
x
x
x
))
connectivity index
at a voxel
x
.
Let us discuss several properties of these features. Values of these features
depend on each 1-voxel
and the type of connectivity
m
.Theyrepresent
topological properties of a local shape at
x
x
and in its
3
×
3
×
3
neighborhood.
Property 4.2.
The connectivity number
Nc
(
m
)
(
x
x
is equal to
the total amount of change
∆
in the connectivity index at a voxel
)ata1-voxel
x
caused
by deletion of the 1-voxel
x
,thatis,
∆
= change in the number of connected components
−
change in the number of holes
+ change in the number of cavities +
1
(4.12)
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