Image Processing Reference
In-Depth Information
Remark 4.4 (Equivalence relationship).
Given a set, assuming two arbi-
trary elements of the set, we can decide whether a predetermined relationship
“
∼
” holds or not. If the relationship “
∼
” satisfies all of the following laws, we
call it an
equivalence relationship
.
(i)
x
∼
x
(reflective law)
(ii) If
x
∼
y
,then
y
∼
x
(symmetric law)
(iii) If
x
∼
y
and
y
∼
z
,then
x
∼
z
(transitive law)
If
x
∼
y
,itissaidthat
x
is equivalent to
y
. The set of all voxels equivalent
to a voxel
x
is called an
equivalence class
of
x
.
Definition 4.3 (Connected component).
All voxels in an image can be
classified into different classes by making voxels connected to each other be-
long to the same class. Each class derived from this procedure is called a
con-
nected component
. More precisely, each equivalence class of voxels defined by
6-connectedness is called a 6-connected component. An 18-connected compo-
nent and a 26-connected component are defined in a similar way. A connected
component of 0-voxels is called a 0
-component
, and that of 1-voxels is called
a1
-component
.
Definition 4.4 (Cavity).
Any connected component of 0-voxels that is not
connected to the frame of an image is called a
cavity
.
Definition 4.5 (Hole, handle).
Let us consider a connected component of
1-voxels
C
. Then consider a figure
F
C
in the continuous space obtained by
combining all 1-voxels in
C
. Each 1-voxel is treated as a cube here. The
figure
F
C
is called a
continuous figure corresponding to
C
. The surface of this
continuous figure
F
C
is a closed curved surface in the continuous 3D space.
Then if this closed surface has a hole (handle), we call this hole (handle) a
hole
(
handle
) of the connected component
C
(Fig. 4.3).
Definition 4.6 (Simply connected, multiply connected).
A connected
component of 1-voxels with none of hole and cavity is said to be
simply con-
nected
, and otherwise,
multiply connected
.
Remark 4.5 (
18
-Connectivity).
Although any of 6-, 18-, or 26-connectivity
may be adopted for the analysis of any particular problem, care must be taken
to avoid a contradiction concerning the connectivity of 1-voxels and that of 0-
voxels. It is required that only the pairs of connectivity listed in Table 4.1 are
used. Here the 18
-connectivity is introduced for preserving theoretical con-
sistency. Definitions of the 18
-connectivity and the 18
neighborhood are the
same as those of the 6-connectivity except the configuration given in Fig. 4.4.
This local configuration of six voxels is regarded as a plane of six voxels in the
18
-connectivity case and as a loop of six voxels in the 6-connectivity case.
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