Image Processing Reference
In-Depth Information
Table 4.1.
Acceptable pairs of connectivity for 1-voxels and 0-voxels.
1-voxel
m
0-voxel
m
6-connectivity 26-connectivity
18-connectivity 18
-connectivity
18
-connectivity 18-connectivity
26-connectivity
6-connectivity
Remark 4.6.
Labels usually will be positive integers because the value
0
is
used as the label of the background.
There may exist more than one holes and cavities in one component. A
continuous figure corresponding to a 1-component in Definition 4.5 has the
same number of closed surfaces of cavities as that of cavities, in addition to
the outside surface of the component. A hole may exist on either of an outside
surface and a surface of a cavity. Examples of a handle are a center hole of
a doughnut, a floater, a coffee cup with a handle (one hole), a pot with two
handles, an eyeglass frame, pants (two holes), and the sponge (many holes).
Conceptually, a hole in a 2D figure corresponds to a cavity of a 3D figure. A
hole (handle) is the concept specific to a 3D figure.
Remark 4.7.
It seems that an explicit definition of a handle (or a hole) in a
3D figure has not been given in literature. In [Lee93], for example, the number
of handles was explained from the viewpoint of the “nonseparating cut.” Also
linkage
and the
knot
of a 3D figure were discussed also, although algorithms
to treat them were not been shown.
For a continuous 3D figure, on the other hand, the following is known as
one possible definition. Consider a division of a 1-component
C
into a set
of simplexes. Then the number of handles (holes) of the 1-component
C
is
defined as the number of independent 1D homology classes.
4.2 Simplex and simplicial decomposition
Simplex is one of most basic concepts in the study of topological properties for
a continuous figure. This is a typical example of a figure representing all figures
of each dimension as a concrete form realizing the concept of dimensionality.
For example, a point (zero dimension), a closed segment (one-dimension), a
triangle (two-dimension) and a tetrahedron (three-dimension) are simplexes
of lower dimensions.
We need to extend these simplexes to a digitized image. One way to per-
form this is given in Fig. 4.5.
Definitions of them are given as follows.
Definition 4.7 (Simplex).
Simplexes
used for decomposition of a 3D digi-
tized image should be defined as follows:
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