Parallel Shortest Path Algorithm for Voronoi Diagrams with Generalized Distance Functions

International audienceVoronoi diagrams are fundamental data structures in computational geometry with applications on different areas. Recent soft object simulation algorithms for real time physics engines require the computation of Voronoi diagrams over 3D images with non-Euclidean distances. In this case, the computation must be performed over a graph, where the edges encode the required distance information. But excessive computation time of Voronoi diagrams prevent more sophisticated deformations that require interactive topological changes, such as cutting or stitching used in virtual surgery simulations. The major bottleneck in the Voronoi computation in this case is a shortest-path algorithm that must be computed multiple times during the deformation. In this paper, we tackle this problem by proposing a GPU algorithm of the shortest-path algorithm from multiple sources using generalized distance functions. Our algorithm was designed to leverage the grid-based nature of the underlying graph used in the simulation. Experimental results report speed-ups up to 65x over a current reference sequential method.Les Diagrammes de Voronoï sont des structures de données fondamentales de la géométrie algorithmique, avec des applications dans différents domaines. Des nouveaux algorithmes de simulation d'objets déformables, en temps réels, nécessitent le calcul des diagrammes de Voronoï sur des images 3D avec des distances non euclidiennes. Dans ce cas, le calcul doit être effectué sur un graphe, où les arêtes codent l'information de distance requise. Cependant, le temps de calcul des diagrammes de Voronoï est trop coûteux et empêche des déformations plus complexes qui nécessitent des modifications topologiques interactives, telles que la coupe ou la couture utilisée dans les simulations de chirurgie virtuelle. Le goulot d'étranglement majeur dans le calcul de Voronoï dans ce cas est un algorithme du plus court chemin qui doit être calculé plusieurs fois au cours de la déformation. Dans cet article, nous nous attaquons à ce problème en proposant un algorithme de GPU pour le probléme du plus court chemin à partir de plusieurs sources utilisant une fonctions de distance généralisées. Notre algorithme a été conçu pour tirer parti de la nature basé sur une grille du graphe sous-jacent utilisé dans la simulation. Les résultats expérimentaux indiquent des accélérations jusqu'à 65x sur une méthode séquentielle de référence

Parallel Shortest Path Algorithm for Voronoi Diagrams with Generalized Distance Functions

International audienceVoronoi diagrams are fundamental data structures in computational geometry with applications on different areas. Recent soft object simulation algorithms for real time physics engines require the computation of Voronoi diagrams over 3D images with non-Euclidean distances. In this case, the computation must be performed over a graph, where the edges encode the required distance information. But excessive computation time of Voronoi diagrams prevent more sophisticated deformations that require interactive topological changes, such as cutting or stitching used in virtual surgery simulations. The major bottleneck in the Voronoi computation in this case is a shortest-path algorithm that must be computed multiple times during the deformation. In this paper, we tackle this problem by proposing a GPU algorithm of the shortest-path algorithm from multiple sources using generalized distance functions. Our algorithm was designed to leverage the grid-based nature of the underlying graph used in the simulation. Experimental results report speed-ups up to 65x over a current reference sequential method.Les Diagrammes de Voronoï sont des structures de données fondamentales de la géométrie algorithmique, avec des applications dans différents domaines. Des nouveaux algorithmes de simulation d'objets déformables, en temps réels, nécessitent le calcul des diagrammes de Voronoï sur des images 3D avec des distances non euclidiennes. Dans ce cas, le calcul doit être effectué sur un graphe, où les arêtes codent l'information de distance requise. Cependant, le temps de calcul des diagrammes de Voronoï est trop coûteux et empêche des déformations plus complexes qui nécessitent des modifications topologiques interactives, telles que la coupe ou la couture utilisée dans les simulations de chirurgie virtuelle. Le goulot d'étranglement majeur dans le calcul de Voronoï dans ce cas est un algorithme du plus court chemin qui doit être calculé plusieurs fois au cours de la déformation. Dans cet article, nous nous attaquons à ce problème en proposant un algorithme de GPU pour le probléme du plus court chemin à partir de plusieurs sources utilisant une fonctions de distance généralisées. Notre algorithme a été conçu pour tirer parti de la nature basé sur une grille du graphe sous-jacent utilisé dans la simulation. Les résultats expérimentaux indiquent des accélérations jusqu'à 65x sur une méthode séquentielle de référence

Parallel Voronoi Computation for Physics-Based Simulations

International audienceVoronoi diagrams are fundamental data structures in computational geometry, with applications in such areas as physics-based simulations. For non-Euclidean distances, the Voronoi diagram must be performed over a grid-graph, where the edges encode the required distance information. Th e major bottleneck in this case is a shortest path algorithm that must be computed multiple times during the simulation. We present a GPU algorithm for solving the shortest path problem from multiple sources using a generalized distance function. Our algorithm was designed to leverage the grid-based nature of the underlying graph that represents the deformable objects. Experimental results report speed-ups up to 65× over a current reference sequential method

A Pseudopolynomial Algorithm for Alexandrov's Theorem

Alexandrov's Theorem states that every metric with the global topology and local geometry required of a convex polyhedron is in fact the intrinsic metric of a unique convex polyhedron. Recent work by Bobenko and Izmestiev describes a differential equation whose solution leads to the polyhedron corresponding to a given metric. We describe an algorithm based on this differential equation to compute the polyhedron to arbitrary precision given the metric, and prove a pseudopolynomial bound on its running time. Along the way, we develop pseudopolynomial algorithms for computing shortest paths and weighted Delaunay triangulations on a polyhedral surface, even when the surface edges are not shortest paths.Comment: 25 pages; new Delaunay triangulation algorithm, minor other changes; an abbreviated v2 was at WADS 200

Metric combinatorics of convex polyhedra: cut loci and nonoverlapping unfoldings

This paper is a study of the interaction between the combinatorics of boundaries of convex polytopes in arbitrary dimension and their metric geometry. Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a `convex polyhedral pseudomanifold'. We prove that S has a polyhedral nonoverlapping unfolding into R^d, so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in R^d by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point v in S, which is the exponential map to S from the tangent space at v. We characterize the `cut locus' (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of 3-polytopes into R^2. We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic non-polynomial complexity of nonconvex polyhedral manifolds.Comment: 47 pages; 21 PostScript (.eps) figures, most in colo

Searching edges in the overlap of two plane graphs

Consider a pair of plane straight-line graphs, whose edges are colored red and blue, respectively, and let n be the total complexity of both graphs. We present a O(n log n)-time O(n)-space technique to preprocess such pair of graphs, that enables efficient searches among the red-blue intersections along edges of one of the graphs. Our technique has a number of applications to geometric problems. This includes: (1) a solution to the batched red-blue search problem [Dehne et al. 2006] in O(n log n) queries to the oracle; (2) an algorithm to compute the maximum vertical distance between a pair of 3D polyhedral terrains one of which is convex in O(n log n) time, where n is the total complexity of both terrains; (3) an algorithm to construct the Hausdorff Voronoi diagram of a family of point clusters in the plane in O((n+m) log^3 n) time and O(n+m) space, where n is the total number of points in all clusters and m is the number of crossings between all clusters; (4) an algorithm to construct the farthest-color Voronoi diagram of the corners of n axis-aligned rectangles in O(n log^2 n) time; (5) an algorithm to solve the stabbing circle problem for n parallel line segments in the plane in optimal O(n log n) time. All these results are new or improve on the best known algorithms.Comment: 22 pages, 6 figure

Path Planning for Mobile Robot Navigation using Voronoi Diagram and Fast Marching

For navigation in complex environments, a robot need s to reach a compromise between the need for having efficient and optimized trajectories and t he need for reacting to unexpected events. This paper presents a new sensor-based Path Planner w hich results in a fast local or global motion planning able to incorporate the new obstacle information. In the first step the safest areas in the environment are extracted by means of a Vorono i Diagram. In the second step the Fast Marching Method is applied to the Voronoi extracted a reas in order to obtain the path. The method combines map-based and sensor-based planning o perations to provide a reliable motion plan, while it operates at the sensor frequency. The m ain characteristics are speed and reliability, since the map dimensions are reduced to an almost uni dimensional map and this map represents the safest areas in the environment for moving the robot. In addition, the Voronoi Diagram can be calculated in open areas, and with all kind of shaped obstacles, which allows to apply the proposed planning method in complex environments wher e other methods of planning based on Voronoi do not work.This work has been supported by the CAM Project S2009/DPI-1559/ROBOCITY2030 I

Collision-free path planning

Motion planning is an important challenge in robotics research. Efficient generation of collision-free motion is a fundamental capability necessary for autonomous robots;In this dissertation, a fast and practical algorithm for moving a convex polygonal robot among a set of polygonal obstacles with translations and rotations is presented. The running time is O(c((n + k)N + nlogn)), where c is a parameter controlling the precision of the results, n is the total number of obstacle vertices, k is the number of intersections of configuration space obstacles, and N is the number of obstacles, decomposed into convex objects. This dissertation exploits a simple 3D passage-network to incorporate robot rotations as an alternative to complex cell decomposition techniques or building passage networks on approximated 3D C-space obstacles;A common approach in path planning is to compute the Minkowski difference of a polygonal robot model with the polygonal obstacle environment. However such a configuration space is valid only for a single robot orientation. In this research, multiple configuration spaces are computed between the obstacle environment and the robot at successive angular orientations spanning [pi] . Although the obstacles do not intersect, each configuration space may contain intersecting configuration space obstacles (C-space obstacles). For each configuration space, the algorithm finds the contour of the intersected C-space obstacles and the associated passage network by slabbing the collision-free space. The individual configuration spaces are then related to one another by a heuristic called proper links that exploit spatial coherence. Thus, each level is connected to the adjacent levels by proper links to construct a 3D network. Dijkstra\u27s algorithm is used to search for the shortest path in the 3D network. Finally, the path is projected onto the plane to show the final locus of the path