Biomedical Engineering Reference
In-Depth Information
obtained from the Voronoï diagram of the nodal distribution, the influence-
domains are called 'influence-cells'. In order to fully understand the influence-cell
concept a brief presentation of the Voronoï diagram concept is required.
3.2.3 Natural Neighbours
The Voronoï diagram of a discrete nodal set is obtained using the natural neigh-
bour mathematical concept, which was firstly introduced by Sibson for data fitting
and field smoothing [
13
].
Consider the nodal set N ¼
f
n
1
;
n
2
;
...
;
n
N
g
discretizing the space domain X
d
with X ¼ x
1
;
x
2
;
...
;
x
f g 2
X. The Voronoï diagram of N is the partition of
the function space discretized by X in sub-regions V
i
, closed and convex. Each
sub-region V
i
is associated to the node n
i
in a way that any point in the interior of
V
i
is closer to n
i
than any other node n
j
2
N
^
j
6
¼ i. The set of Voronoï cells
V define the Voronoï diagram, V = {V
1
, V
2
, …, V
N
}. The Voronoï cell is defined
by,
R
;
d
V
i
:¼ x
I
2
X
R
:
k
x
I
x
i
k
\ x
I
x
j
8
i
6
¼ j
ð
3
:
3
Þ
being x
I
an interest point of the domain and
jj jj
the Euclidian metric norm. Thus,
the Voronoï cellV
i
is the geometric place where all points are closer to n
i
than to
any other node.
The Voronoï diagrams implications are extensive, with applications from the
natural sciences to engineering. In the literature it is possible to find detailed
descriptions of the properties and applications of such mathematical tool [
14
,
15
],
as well as efficient algorithms to construct Voronoï tessellations [
16
]. Since it is
easier to visualize, it is represented a two-dimensional space X
2
in order to
show how the Voronoï diagram can be generically obtained. Consider the nodal set
represented in Fig.
3.6
a. Since the objective is to determine the Voronoï cellV
0
of
node n
0
, the nodes on Fig.
3.6
a are chosen as potential neighbours of n
0
. Then one
of the nodes is selected as potential neighbour, for example node n
4
, Fig.
3.6
b, and
the vector u
40
is determined,
R
u
40
¼
ð
x
0
x
4
Þ
jj
x
0
x
4
jj
ð
3
:
4
Þ
being u
40
= {u
40
, v
40
, w
40
}. Using the normal vector u
40
it is possible to defined
plane p
40
,
u
40
x
þ
v
40
y
þ
w
40
z ¼
ð
u
40
x
4
þ
v
40
y
4
þ
w
40
z
4
Þ
ð
3
:
5
Þ
