Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework

KIPPI: KInetic Polygonal Partitioning of Images

International audienceRecent works showed that floating polygons can be an interesting alternative to traditional superpixels, especially for analyzing scenes with strong geometric signatures , as man-made environments. Existing algorithms produce homogeneously-sized polygons that fail to capture thin geometric structures and over-partition large uniform areas. We propose a kinetic approach that brings more flexibility on polygon shape and size. The key idea consists in progressively extending pre-detected line-segments until they meet each other. Our experiments demonstrate that output partitions both contain less polygons and better capture geometric structures than those delivered by existing methods. We also show the applicative potential of the method when used as preprocessing in object contouring

On Kinetic Delaunay Triangulations: A Near Quadratic Bound for Unit Speed Motions

Let $P$ be a collection of $n$ points in the plane, each moving along some straight line at unit speed. We obtain an almost tight upper bound of $O(n^{2+\epsilon})$, for any $\epsilon>0$, on the maximum number of discrete changes that the Delaunay triangulation $\mathbb{DT}(P)$ of $P$ experiences during this motion. Our analysis is cast in a purely topological setting, where we only assume that (i) any four points can be co-circular at most three times, and (ii) no triple of points can be collinear more than twice; these assumptions hold for unit speed motions.Comment: 138 pages+ Appendix of 7 pages. A preliminary version has appeared in Proceedings of the 54th Annual Symposium on Foundations of Computer Science (FOCS 2013). The paper extends the result of http://arxiv.org/abs/1304.3671 to more general motions. The presentation is self-contained with main ideas delivered in Sections 1--