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SKELETON REPRESENTATION BASED ON COMPOUND
BEZIER CURVES
Leonid Mestetskiy
Department of Mathematical Methods of Forecasting, Lomonosov Moscow State University, Moscow, Russia
Keywords: Polygonal Figure, Continuous Skeleton, Radial Function, Parabolic Edges, Bezier Curves, Control Graph.
Abstract: A new method to describe the skeleton of a polygonal figure is presented. The skeleton is represented as a
planar graph, whose edges are linear and quadratic Bezier curves. The description of a radial function in
Bezier splines form is given. An algorithm to calculate control polygons of Bezier curves is proposed. Also,
we introduce a new representation of skeleton as a straight planar control graph of a compound Bezier
curve. We show that such skeleton representation allows simple visualization and easy-to-use skeleton
processing techniques for image processing.
1 INTRODUCTION
A closed domain on Euclidean plane
2
such that
its boundary consists of one or more simple
nonintersecting polygons is called a polygonal
figure. The set of polygonal figure points that have
two or more closest boundary points of figure is
called the skeleton or medial axis. Polygonal figures
and their skeletons are widely used in image shape
analysis and recognition (Pfaltz, Rosenfeld, 1967).
To construct the skeleton of a polygonal figure
the concept of a Voronoi diagram of line segments is
commonly used (Drysdale, Lee, 1978, Kirkpatrick,
1979). The polygonal figure boundary is a union of
linear segments and vertices, which are considered
as the Voronoi sites. The Voronoi diagram of these
sites is generated and the skeleton is extracted as a
subset of the diagram. The skeleton of a polygonal
figure with n sides can be obtained from the
Voronoi diagram taking
)(nO time. By-turn, there
are known effective
)log( nnO algorithms to
construct the Voronoi diagram for the general set of
linear segments (Fortune, 1987, Yap, 1987) as well
as for the sides of a simple polygon (Lee, 1982) or
multiply-connected polygonal figures (Mestetskiy,
Semenov, 2008).
Geometric construction of a polygonal figure
skeleton is simple enough: it is a planar graph with
straight-line and parabolic edges (figure 1).
However, such analytical description of skele-
tons presents some difficulties. Presence of parabolic
edges gives rise to certain problems in constructing,
storing, processing, and utilizing skeletons in image
analysis. The general form for a parabola is
described by an implicit equation. This is not handy
for calculation of parabolas intersections, for
drawing and analysis.
Figure 1: A polygonal figure and its skeleton.
This shortcoming generates the tendency to
handle skeletons having no parabolic edges. This
idea is implemented in the concept of straight
skeleton (Aichholzer, Aurenhammer, 1996). But the
straight skeleton suffers from certain shortcomings,
videlicet: complexity of mathematical definition,
low algorithmic efficiency, regularization
complexity if noise effects are available.
In this paper, we propose a different method of
describing a skeleton in the form of a planar graph
with straight edges. To construct such a graph,
computing parabolic edges is not necessary either at
the step of the Voronoi diagram computing, or at the
steps of skeleton storing, drawing and processing,
respectively. This can be achieved as follows.
1. The skeleton of a polygonal figure is the union
of a set of the first and second order elementary
44
Mestetskiy L. (2010).
SKELETON REPRESENTATION BASED ON COMPOUND BEZIER CURVES.
In Proceedings of the International Conference on Computer Vision Theory and Applications, pages 44-51
DOI: 10.5220/0002831600440051
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SciTePress