VORONOI ORDERED SPACE - Video Object Extraction and Representation: Theory and Applications - page 48

While ignoring some of the deeper mathematics, we limit our analysis
to the design of our two systems for video object extraction and rep-
resentation. Using analysis from this chapter, Chapters 4 and 6 show
how the Voronoi Ordered Spaces can express shape information within
system design.

2. NOTATION

2

First, we will be working in a ℜ space that represents an image.

Notation for this chapter is as follows:

1. Lowercase English letters ( a, b , . . .) are scalars.

2. Lowercase Greek letters (α, β, . . .) are functions that return scalars.

→

→

b , . . .) are points in ℜ 2 .

3. Lowercase English vectors ( a ,

→

→

4. Lowercase Greek vectors (α, β, . . .) are functions that return points
or vectors, depending on the context.

5. A pair of uppercase letters are a line segment between the points of
corresponding lowercase letters, e.g., AB is the line segment between

→

points

→

a and b .

6. A single uppercase letter ( A , B , . . .) is a set of points such as contour

or an area.

7. A boldface single uppercase letter ( A , B , . . .) is a set of sets of points

such as a set of contours.

8. Uppercase Greek letters (Λ, Φ, . . .) are functions that return a set of

sets of points such as sets of areas.

3. DEFINITION OF VORONOI ORDERED

SPACE

We define a restricted set of contours to represent our shape.

D EFINITION 3.1 (T HE C ONTOURS C ) We define C as the set of OF

contours that are defined below:

2

1. a simple closed contour in ℜ

2. piecewise continuous.

→

3. parametrized by arc length, such that C ={γ( s )|0 ≤ s < || C || } , where

|| C || is the length of the contour C.