Digital Signal Processing Reference
In-Depth Information
Figure 3.4.
Theorem 3.1 (Distinctness of Minimal Distance Projections on C) Given
→
→
→
→
two points
a
,
b
and their respective minimal distance points on C,
y
and
z
, we prove
that the line segments AY and BZ cannot intersect.
|
AY
|=|
AX
| + |
XY
| ≥ |
AX
|+|
XZ
| ≥ |
AZ
|
(3.4)
Let us consider the two cases of |
AY
| |
AZ
|. Case #1: if |
AZ
|=
|
AY
|, then ||
AX
||+ ||
XZ
|| = ||
AZ
||, ||
XZ
|| = ||
XY
|| and ∠
AXZ
= 180
°
,
forcing
≥
→
→
→
y
and
→
z
to coincide and contradicting that
y
and
z
are distinct.
→
Case #2:
if |
AY
|>
AZ
|, then
y
not the minimally distant point on
→
C to
a
.
C
OROLLARY
3.1
(Distinctness of Voronoi Projections) Given a con-
tour C
C
and two diferent points with diferent Voronoi Orders,
the
Voronoi Projections of the two points do not intersect.
Proof.
This result shows that the Voronoi Ordered Space acts as an alterna-
tive coordinate system: Voronoi Projections of different Voronoi Centers
do not intersect and provide a consistent axis of the Voronoi Order for
an alternative coordinate system.
This corollary is a special case of Theorem 3.1.
T
HEOREM
3.2 (C
ONNECTIVITY OF
V
ORONOI
O
RDER
R
EGIONS
)
Level
sets of the Voronoi Order form connected regions.
→
Proof.
Consider any two points,
a
and
→
b,
with the same Voronoi Order.
Since they have the same Voronoi Order, they have the same Voronoi
Center,
→
x.
Consider the path of AX and XB. All points of AX and XB
Search WWH ::
Custom Search