GEOMETRICAL PROPERTIES OF 3D DIGITIZED IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 92

There are two approaches to obtaining values of the Euler number, based
upon different viewpoints. Although the following procedures vary, the Euler
number is accurately obtained.

(1) Triangulation method : We regard a voxel as a cube in the continuous space
and digitization by voxels of cubes as a triangulation or a simplicial decom-
position of a 3D figure. Then the Euler number E is obtained by Eq. 4.5

in Definition 4.8.
(2) Simplex counting method : The number of digital simplex (as shown in
Fig. 4.5) included in an input image is counted.

We will present details of both methods in the following subsections.

4.7.1 Triangulation method

In the decomposition of a 3D object to cubes corresponding to 1-voxels, a
k -dimensional simplex ( k = 0 , 1 , 2 ) corresponds to a vertex ( k = 0 ), an edge
( k = 1 ), and a face ( k = 2 ) of a 1-voxel, respectively, and a three-dimensional
simplex reduces to a voxel itself. Thus, n k s in Eq. 4.5 are given as follows:

n 0 = number of vertexes of 1-voxels contained in a 3D object, (4.33)
n 1 = number of edges of 1-voxels contained in a 3D object, (4.34)

n 2 = number of faces of 1-voxels contained in a 3D object, (4.35)

and
n 3 = number of 1-voxels contained in a 3D object. (4.36)

The type of connectivity should be taken into consideration here. In the case
of Fig. 4.11, for example, the vertex V 1 should be counted twice for the 6-c,
18-c, and 18 -c, while only once for the 26-c, because it belongs to separating
two voxels X and Y in the first three cases and not a vertex for the 26-c case.
The edge e 1 should be counted twice for the 6-c, and 18-c cases, because it
belongs to both of two different voxels Y and Z. Thus the result is as shown
in Fig. 4.11.
To calculate the total sum of n k s over the whole of a given 3D image, it is
easiest to count them at each vertex of a 1-voxel. Let us consider, for example,
avertex V 1 and a set of 2 ( V ) sharing this vertex (Fig. 4.12),

and let ∆n k s( k = 0 , 1 , 2 , 3 ) denote the following quantities.
∆n 0 = number of vertexes of 1-voxels in

× 2 × 2 voxels S

S ( V )whichshare V. (4.37)

∆n 1 = (number of edges of 1-voxels in S ( V )

containing the vertex V ) × 1 / 2 . (4.38)

∆n 2 = (number of faces of 1-voxels in S ( V )

containing the vertex V ) × 1 / 4 . (4.39)

∆n 3 = (number of 1-vowels in S ( V )

containing the vertex V ) × 1 / 8 . (4.40)