Image Processing Reference
In-Depth Information
FIGURE 7.3: Voxelation of
pickup-van
shown in Fig. 7.1. (See color insert.)
3-cell c
p
centered at p for g = 1, each face of c
p
is a grid face lying on a face
plane, which is parallel to one of the three coordinate planes. It may be noted
that a UGC consists of g×g×g voxels and each UGC-face consists of g×g
voxels.
Two 3-cells c
1
and c
2
having centers at (x
1
,y
1
,z
1
) and (x
2
,y
2
,z
2
), respec-
tively, are α-adjacent if and only if c
1
∩ c
2
contains an α-cell
(α ∈ {0,1,2}) [115]. The grid points (x
1
,y
1
,z
1
) and (x
2
,y
2
,z
2
) are in a k-
neighborhood where k ∈{6,18,26}; k = 6 denotes 2-adjacency, k = 18 denotes
1-adjacency and k = 26 denotes 0-adjacency with respect to the cell model. As
introduced in Chapter 1 and Chapter 2, the set of 6-adjacent points of a point
p(x
1
,y
1
,z
1
) is called the 6-neighborhood of p, which is given by N
6
(p). The
set of 18-adjacent points is called 18-neighborhood given by N
18
(p). The set
of 26-adjacent points is called 26-neighborhood given by N
26
(p). Each point
in N
k
(p) is said to be a k-neighbor of p where k ∈ {6,18,26}. Two points p
and q are k-connected in a digital set A ⊂ Z
3
if and only if there exists a
sequence p := p
0
,p
1
,...,p
n
:= q⊆ A such that p
i
= c
2
and c
1
∈ N
k
(p
i−1
) for 1 6 i 6 n.
For any point p ∈A, the set of points that are k-connected to p ∈ A is called
a k-connected component of A. In other words, a k-connected component of
a nonempty set A⊆ Z
3
is a maximal k-connected set of A. If A has only one
connected component, it is called a k-connected set.
7.1 Voxelation and Approximation of 3-D Surface
Voxelation (also called voxelization or 3-D scan conversion) has been ad-
dressed in the literature since the 1980s [47, 48, 106, 107, 108, 145]. Recently,
several interesting works have been done on discrete volume polyhedrization,
which deals with construction of a polyhedron P enclosing a set of voxels S