GEOMETRICAL PROPERTIES OF 3D DIGITIZED IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 86

4.5.3 Classification of the voxel state

The local shape of a 3D figure at a 1-voxel x 0 and its neighborhood is classified

by the arrangement of 0- and 1-voxels in the 3 × 3 × 3 neighborhood N 333 ( x 0 ).

Let us show several examples here.

(a) Interior and border point (voxel)

We can denote the type of the connectivity of a current figure using k ( k -
connectivity) and that of the background by k ( k -connectivity), respectively.
Then, a 1-voxel x 0 is called an interior point ( voxel ) if no 0-voxel exists in its

k -neighborhood (i.e., all voxels in the k -neighborhood are 1-voxels). Otherwise
the 1-voxel

x 0 is called a border point ( voxel ). As will be described later, a
cavity is created by the deletion of an interior point (voxel) (by inverting it
to a 0-voxel). Therefore, the connectivityindexofaninteriorpoint(voxel)is
( 1 , 0 , 1 ).

(b) Linear connecting point and a line segment

A1-voxel x 0 is called a linear connecting point if exactly two 1-voxels exist

in the 26-neighborhood and are not adjacent to each other. Such a voxel x 0

is considered as the middle of an ideal line figure in 3D space. If exactly one
1-voxel exists in the 26-neighborhood of

x 0 is considered as the location
at the end of a line figure and is called a linear edge point (voxel). A line figure
in general will have complicated shapes at a branching point, a crossing point,
and in the vicinity of these.
At a linear connecting point

x 0 ,

x 0 , let us denote by x 1 and x 2 two 1-voxels

in the 26-neighborhood of x 0 . Then we define a line element at x by a line

segment connecting a center point of voxels x 2 . Using the direction

of a line segment we can define the direction of a 3D line figure at a point
(voxel)

x 1 and

x 0 . All possible directions include 49 cases listed in Table 4.3.
A 3D line figure with some characteristic points and line elements are
illustrated in Fig. 4.10.

(c) Plane point

A figure with a unit thickness is ideally considered to be a plane or a surface.
This type of figure is rarely seen in practical applications. If such a situation
is realized at a 1-voxel x 0 is called a

plane point . No decisive method has been known for detection of a plane point.
Presently we only have a test to know whether any of 3D simplexes exists in
the neighborhood or not.

x 0 andinitsneighborhood N 333 ( x 0 ),