Two-dimensional line space Voronoi Diagram

Abstract

Given a set of points called sites, the Voronoi diagram is a partition of the plane into sets of points having the same closest site. Several generalizations of the Voronoi diagram have been studied, mainly Voronoi diagrams for different distances (other than the Euclidean one), and Voronoi diagrams for sites that are not necessarily points (line segments for example).In this paper we present a new generalization of the Voronoi diagram in the plane, in which we shift our interest from points to lines, that is, we compute the partition of the set of lines in the plane into sets of lines having the same closest site (where sites are points in the plane). We first define formally this diagram and give first properties. Then we use a duality relationship between points and lines to visualize this data structure and give more properties. We show that the size of this line space Voronoi diagram for n sites is in Θ(n 2 ) and give an optimal algorithm for its explicit computation.We then show a remarkable relationship between this diagram and the dual arrangement of the sites and give a new result on an arrangement of lines: we show that the size of the zone of a line augmented with its incident faces is still in O(n). We finally apply this result to show that the size of the zone of a line in the line space Voronoi diagram is in O(n).

Face Mole Detection, Classification and Application

“…To classify the detected moles on one's face, one of the better ways is by the nearest-neighbour classifier. Therefore, Voronoi diagram [10], [11] is used to partition the given sample mole-face into cells in which each defined sample mole is located at the center of the cell. Each detected mole dm is then classified as the defined mole m if dm falls within the region belonging to m. …”

On Voronoi Diagram in the Line Space and their Generalizations

“…For a set of point sites in the plane, its Voronoi diagram in the line space is defined as a partition of the latter into Voronoi regions, each corresponding to a distinct site ∈ and consisting of the lines being closer to than to any other site from . This kind of Voronoi diagrams was first introduced by Rivière and Schmitt [10]. In particular, they pointed out that such Voronoi diagrams can be easily computed and visualized in dual space (where lines map to points), and subsequently used for processing line localization queries.…”

“…One year later, Rivière [11] introduced and examined Voronoi diagrams of order in the line space, thereby also exploiting the concept of geometric duality.…”