GEOMETRICAL PROPERTIES OF 3D DIGITIZED IMAGES - Fundamentals of Three-Dimensional Digital Image Processing - page 70

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GEOMETRICAL PROPERTIES OF 3D
DIGITIZED IMAGES

A digitized binary image contains only two groups of voxels, 1-voxels and 0-
voxels. We assume for now that we have interest in the set of 1-voxels and
will call it a figure . The other set we call the background . Properties of a
digitized figure are often very different from those of ordinary figures in the
continuous space. Treating geometrical properties of digitized figures is called
digital geometry . In this chapter, we briefly introduce the basics of digital
geometry before discussing them further. The most important basic concepts
of digital geometry are shown in Fig. 4.1.
In Section 4.1, after introducing the basic concepts of neighborhood and
connectivity, we discuss three important topological features including genus,
connectivity indexes, and the relationship between them. Then we explain the
concept of deletability of 1-voxels and the condition under which an arbitrary
algorithm preserves topological features of a figure in a binary image. We also
show a simple proof of this condition using the above topological features.
In this chapter we will deal with binary images only. We assume that
images consist of cubic voxel arrays of I rows, J columns, and K planes. It
is also assumed that the first row, the I -th row, the first column, the J -th
column, the first plane, and the K -th plane are all filled with 0-voxels. These
rows, columns, and planes are called the frame of an image.

4.1 Neighborhood and connectivity

4.1.1 Neighborhood

As is shown in Fig. 4.1, we will start the discussion of geometrical properties of
a binary image with the neighborhood and the connectivity and will introduce
a connected component. As it was already stated in Section 2.4.2 we define
the neighborhood as follows:
Definition 4.1 (Neighborhood). The neighborhood of a voxel ( i, j, k ), de-
noted by N ijk (( i, j, k )), is defined by the equation