Image Processing Reference
In-Depth Information
FIGURE 7.1: Some typical examples of 3-D models (triangular faces shown
randomly colored). From top-left to bottom-right:
icosahedron
(20 identical
equilateral triangular faces),
teapot
,
turbine
,
cow
,
pickup-van
. (See color
insert.)
3-cell is a unit cube consisting of six unit squares called 2-cells, twelve unit
edges called 1-cells, and eight vertices called 0-cells [115].
In this chapter, we define the isothetic distance between two points
p(x
1
,y
1
,z
1
) and q(x
2
,y
2
,z
2
) as the Minkowski norm L
∞
given by d
⊤
(p,q) =
max{|x
1
|}. This metric, along with other distance
metrics, are discussed in detail in Chapter 2. The (isothetic) distance
of a point p from an object A is therefore defined as d
⊤
(p,A) =
min{d
⊤
(p,q) : q ∈A}, and the distance between two connected components
A
1
and A
2
is d
⊤
(A
1
,A
2
) = min{d
⊤
(p,q) : p ∈A
1
,q ∈A
2
−x
2
|,|y
1
−y
2
|,|z
1
−z
2
}.
In Chapter 1 (refer to Section 1.2), we discussed the digital grid and pre-
sented it as a point set. In this chapter, we provide a fuller exposition of this
grid in 3-D by considering it as a cellular complex of cubic cells or 3-cells.
Some of the basic definitions related to this complex are introduced here. In
this representation, since the isothetic cover of an object is obtained with re-
spect to an underlying grid in 3-D digital space, we define it as follows. A
digital grid in 3-D consists of three orthogonal sets of equispaced grid lines,
