VORONOI ORDERED SPACE - Video Object Extraction and Representation: Theory and Applications - page 58

→→ →

Figure 3.8. Voronoi Areas of a point y, Φ( y ): If a) y is equidistant from four differ-

ent distinct points on the contour C, then b) there are six different Voronoi Areas
associated with y.

→

know that this skeleton formed by the non-zero values of χ V can be

mapped to a graph that is a tree as shown in Figure 3.7. The analysis in

this section shows the value of χ V is related the depth of a point within

multiresolution version of that tree:

V value of a parent in the tree
is always greater than or equal to value of its children. From χ V values,
we can derive this multiresolution tree and use it as a robust shape rep-
resentation. This section condenses proofs presented by Ogniewice and
Kubler [Ogniewicz and Kubler, 1995].
To show the connection between χ V and the multiresolution tree,
we must define the underlying structures of the contour called Voronoi
Areas:

the χ

D EFINITION 3.4 (V ORONOI A REAS P IVOTS ) Given a contour C

AND

→ C , a point on the interior of C, let S be the set of points on C that

are minimally distant from

→

→

. As shown in Fig. 9.8, we define Φ( C,y ) ,

the Voronoi Areas of , are the set of regions of finite area described by
all pair-wise combinations of S through the following construction (for a
pair of points a, b ):

→

→

→

1. the line segment AY

2. the line segment BY

→

3. a segment of the contour C from a to b whose length is less than or

→

equal to half the perimeter of the contour C.