Image Processing Reference
In-Depth Information
(a)
(b)
(c)
FIGURE 1.1: Regular tessellations in 2-D: (a) triangular, (b) rectangular,
and (c) hexagonal.
this topic, we restrict out discussion to the geometry of digital space, which
is obtained through rectangular tessellation in the corresponding continuous
space.
1.2 Digital Grid
A digital grid is a finite subset of integral coordinate space, so that it
can be represented by a rectangular array of respective dimension. Let us
represent a digital grid in n-dimension as G
n
. For example, a set of points
G
2
= {(i,j) ∈ Z
2
|i = 0,1,2,...,M − 1;j = 0,1,2,...,N − 1} is a digital
grid in 2-D (Z
2
) of size M ×N. A point in a 2-D digital grid is called a pixel.
Similarly, a 3-D digital grid element is called a voxel. We define the border of
G
n
as the set of points whose coordinate values of one of the dimensions are
either 0 (the least value) or the maximum. The border of G
2
in the previous
example is B(G
2
) = {(i,j)|(i = 0) or (i = M −1) or (j = 0) or (j = N −1)}.
Two pixels are said to be 8 -adjacent, if they are distinct, and any one
of their coordinates (or both) differs (or differ) by 1. An 8-adjacent pixel of
p ∈G
2
is called its 8-neighbor. The coordinates of a 4-adjacent pixel of p differ
by 1 only in one of their dimensions and the corresponding pixel is named its
4-neighbor. In 3-D we have similar definitions of 6, 18, and 26 adjacency. A
