Digital Signal Processing Reference
In-Depth Information
Figure 3.3.
Terminology for our proofs: Voronoi Center, the minimally distant point
on C of minimal arc length value to the given point Voronoi Order, the arc length
parameter of the Voronoi Center; Voronoi Distance, the distance from
→
→
x
to the Voronoi
→
Center; Voronoi Projection, the line segment from
to the Voronoi Center
by shape to the given contour. Systems may integrate shape information
by only considering this restricted set of contours.
To aid our analysis, we define two structures related to the Voronoi
Order:
Voronoi Center
and
Voronoi Projection
(see Figure 3.3). Given
a point on the image plane, the
Voronoi Center
is the point on the
contour C that is closest to
→
→
of minimum arc length value; the
Voronoi
Projection
is the line segment from
to its Voronoi Center and its length
→
is the Voronoi Distance.
T
HEOREM
3.1 (D
ISTINCTNESS OF
M
INIMAL
D
ISTANCE
P
ROJECTIONS
)
→
Given a contour C
C and two different points in
ℜ
2
, a and b, if the
→
two points do not have the same minimally distant points on C, y and
z, respectively, then AY and BZ do not intersect.
→
→
Proof by Contradiction.
As shown by Figure 3.4, we assume the
line segments AY,BZ intersect at a point
x.
Without loss of generality,
we assume that |
XY
| ≥ |
XZ
|. By Schwartz Inequality,
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