VORONOI ORDERED SPACE - Video Object Extraction and Representation: Theory and Applications - page 50

Figure 3.1, Level sets of the Voronoi Ordered Space: a) for a given contour C, b)

→

φ v ( C,y ) = c, level sets of the Voronoi Order, the first component of the Voronoi

Ordered Space, c) δ V ( C, y ) =

→

c, level sets of Voronoi Distance, the second component

of the Voronoi Ordered Space.

2

We define a mapping based upon a contour C C from the ℜ image

2

space to another ℜ space.

D EFINITION 3.2 (V ORONOI O RDERED S PACE ) Given a contour C
C , we define a mapping ℜ 2 → ℜ 2 onto a Voronoi Ordered Space
(see Figure 3.1), as follows:

(3.1)

(3.2)

(3.3)

i.e., for a given point, the mapping of Voronoi Ordered Space can be
separated into two mappings ℜ 2 → ℜ: the first (φ V ) is called the Voronoi
Order and is determined by minimum value of the arc length parameter
of the closest point on C; the second (δ ) is called the Voronoi Distance

and is the shortest distance from the point to the contour C. With this
mapping, we define a new coordinate system where we replace the x-
axis of our Cartesian coordinate system with a contour C. This pairing
of Voronoi Order and Voronoi Distance creates a new description of the
image space, warped w.r.t. the contour C. As shown in Figure 3.1, the
level sets of the Voronoi Order, i.e., a partitioning of the space into
sets of points of the same value, are the orthonormal projections of the
contour C and the Voronoi Distances are the equidistant rings of C. If
we overlay the level sets of Voronoi Order with level sets of the Voronoi

V