In this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems. The target energy can be written either in terms of atomic measures and the Wasserstein distance or in terms of weighted points and power diagrams (generalized Voronoi diagrams). The latter formulation is more suitable for computation. We show that critica...

We propose a novel algorithm to compute Voronoi diagrams of order k in arbitrary 2D and 3D domains. The algorithm is based on a fast ordered propagation distance transformation called occlusion points propagation geodesic distance transformation (OPPGDT) which is robust and linear in the domain size, and has higher accuracy than other geodesic distance transformations published before. Our appr...

Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorit...

Given a set of compact sites on a sphere, we show that their spherical Voronoi diagram can be computed by computing two planar Voronoi diagrams of suitably transformed sites in the plane. We also show that a planar furthest-site Voronoi diagram can always be obtained as a portion of a nearest-site Voronoi diagram of a set of transformed sites. Two immediate applications are an O(n logn) algorit...

In this paper, we present a method to navigate a mobile robot using a webcam. This method determines the shortest path for the robot to transverse to its target location, while avoiding obstacles along the way. The environment is first captured as an image using a webcam. Image processing methods are then performed to identify the existence of obstacles within the environment. Using the Voronoi...

The Voronoi diagram is a famous structure of computational geometry. We show that there is a straightforward equivalent in graph theory which can be eeciently computed. In particular, we give two algorithms for the computation of graph Voronoi diagrams, prove a lower bound on the problem, and we identify cases where the algorithms presented are optimal. The space requirement of a graph Voronoi ...

Monte-Carlo Tree Search (MCTS) is a powerful tool in games with a finite branching factor. This paper describes an artificial player playing the Voronoi game, a game with an infinite branching factor. First, this paper shows how to use MCTS on a discretization of the Voronoi game, and the effects of enhancements such as RAVE and Gaussian processes (GP). A first set of experimental results shows...

The Voronoi diagram of a set of sites partitions space into regions one per site the region for a site s consists of all points closer to s than to any other site The dual of the Voronoi diagram the Delaunay triangulation is the unique triangulation so that the circumsphere of every triangle contains no sites in its interior Voronoi diagrams and Delaunay triangulations have been rediscovered or...

We modify the incremental algorithm for computing Voronoi diagrams in the Euclidean metric proposed by Ohya, Iri and Murota 6] in order to obtain an algorithm for computing Voronoi diagrams (resp. Delaunay triangulations) in Man-hattan and Maximum metric, that is rather simply to implement. We generalize the notions of \Voronoi diagram" and \Delaunay triangulation" in such a way that these stru...

Several methods based on the use of Voronoi diagrams, are proposed for the 3D and 2D reconstruction of natural objects. The main application domain is geology. A rst algorithm enables volumes to be constructed starting from the combined use of geological maps, cross-sections and drill-lines. Input data have associated colors (object characteristics), but may be sparse and heterogeneous, and do ...