VORONOI ORDERED SPACE - Video Object Extraction and Representation: Theory and Applications - page 47

Chapter 3

VORONOI ORDERED SPACE

Starting from principles of Euclidean geometry, this chapter intro-
duces a central concept of Voronoi Ordered Space in our topic. The
Voronoi Ordered Space is a ℜ

2 2

in terms
of its projection onto a contour-based description of a shape. With the
amount of work and analysis done in the area of shape description, this
chapter is a subset of previous work (listed in Section 1.). We use the
concept of Voronoi Ordered Space as our interfaces for shape informa-
tion in our system designs: to integrate shape information into low-level
optimization algorithms in Chapter 4 and to derive shape representa-
tions from video object planes in Chapter 6. Our contribution is the
demonstration of how Voronoi Ordered Space can express concepts of
shape through concrete system functionalities.

space that describes a point in ℜ

1. PREVIOUS WORK
The relationship between Euclidean distance and skeletonization has
been studied with the Medial Axis Transform (MAT) (also known as
the grassfire algorithm) [Blum and Nagel, 1978] [Blum, 1973] [Malladi
et al., 1995] [Lee, 1982]. The hierarchical representation of MAT-derived
shape was studied [Shapiro, 1980]; this work culminated in an excellent
treatment that clearly shows its multiresolution properties [Ogniewicz
and Kubler, 1995]. Related concepts such as Voronoi Cells and thin-
ning processes are well-known in image processing [Aurenhammer, 1991]
[Rosenfeld, 1986].
Our contribution is the demonstration of how Voronoi Ordered Space
can express concepts of shape through these concrete system functionali-
ties. This chapter simplifies and condenses a subset of the previous anal-
ysis, packaging it into a single concept called Voronoi Ordered Spaces.