GEOMETRICAL PROPERTIES OF 3D DIGITIZED IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 78

For the whole of a binary image that includes more than one figure in it,

the following is obtained by adding the above equation over all the figures in
F

F

. The Euler number of an image F ,

E ( F ) = the number of connected components in F

− the number of handles in F

+ the number of cavities in F (4.8)

Remark 4.9. The Euler number of a 3D figure C is alternatively given by
the Euler number of a continuous figure defined in Definition 4.5. If we need
to differentiate the Euler number by Def. 4.8 and the one using the above
method, the one by Def. 4.8 is called the volume Euler number ,andtheabove
one the face Euler number . A 3D figure of the minimum size occupies at least
one voxel. Its boundary surface is a closed surface and has a cavity inside it.
Therefore, the face Euler number is

1 − 0 (= number of handles) + 1 (= number of cavities) = 2 . (4.9)

If a figure does not have a handle, the face Euler number is twice the volume
Euler number.

Remark 4.10. For a 2D image the Euler number E 2D was defined as

E 2D = the number of connected component − the number of holes . (4.10)

The natural extension of this to a 3D image is the volume Euler number.

4.4 Local feature of a connected component and
topology of a figure

A topological property of a 3D binary image is summarized in numbers of
connected components, handles, and cavities. The number of connected com-
ponents is apparent in the labeling. Algorithms will be presented in the next
chapter. The number of cavities also can be found in the same way after in-
verting 1 and 0 of a given image. The number of handles is presently di cult
to count directly.
Counting numbers of simplexes ( n 0 , n 1 , n 2 , n 3 , in Eq. 4.5) is an exhaustive
procedure and requires much computation time. Then, by using Eq. 4.5 and
by solving Eq. 4.6 or 4.7 with respect to the number of holes, the number of
holes is indirectly known.
An image processing algorithm changes shapes of 3D figures in an input
image by replacing parts of 1-voxels by 0-voxels and 0-voxels by 1-voxels.
Then, if none of the following occur by the execution of the algorithm, this
algorithm (or processing) is said to be topology-preserving (or to preserve
topology ):