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

Efficient Irregular Wavefront Propagation Algorithms on Hybrid CPU-GPU Machines

In this paper, we address the problem of efficient execution of a computation pattern, referred to here as the irregular wavefront propagation pattern (IWPP), on hybrid systems with multiple CPUs and GPUs. The IWPP is common in several image processing operations. In the IWPP, data elements in the wavefront propagate waves to their neighboring elements on a grid if a propagation condition is satisfied. Elements receiving the propagated waves become part of the wavefront. This pattern results in irregular data accesses and computations. We develop and evaluate strategies for efficient computation and propagation of wavefronts using a multi-level queue structure. This queue structure improves the utilization of fast memories in a GPU and reduces synchronization overheads. We also develop a tile-based parallelization strategy to support execution on multiple CPUs and GPUs. We evaluate our approaches on a state-of-the-art GPU accelerated machine (equipped with 3 GPUs and 2 multicore CPUs) using the IWPP implementations of two widely used image processing operations: morphological reconstruction and euclidean distance transform. Our results show significant performance improvements on GPUs. The use of multiple CPUs and GPUs cooperatively attains speedups of 50x and 85x with respect to single core CPU executions for morphological reconstruction and euclidean distance transform, respectively.Comment: 37 pages, 16 figure

Exact Generalized Voronoi Diagram Computation using a Sweepline Algorithm

Voronoi Diagrams can provide useful spatial information. Little work has been done on computing exact Voronoi Diagrams when the sites are more complex than a point. We introduce a technique that measures the exact Generalized Voronoi Diagram from points, line segments and, connected lines including lines that connect to form simple polygons. Our technique is an extension of Fortune’s method. Our approach treats connected lines (or polygons) as a single site

Three dimensional extension of Bresenham’s algorithm with Voronoi diagram

Bresenham’s algorithm for plotting a two-dimensional line segment is elegant and efficient in its deployment of mid-point comparison and integer arithmetic. It is natural to investigate its three-dimensional extensions. In so doing, this paper uncovers the reason for little prior work. The concept of the mid-point in a unit interval generalizes to that of nearest neighbours involving a Voronoi diagram. Algorithmically, there are challenges. While a unit interval in two-dimension becomes a unit square in three-dimension, “squaring” the number of choices in Bresenham’s algorithm is shown to have difficulties. In this paper, the three-dimensional extension is based on the main idea of Bresenham’s algorithm of minimum distance between the line and the grid points. The structure of the Voronoi diagram is presented for grid points to which the line may be approximated. The deployment of integer arithmetic and symmetry for the three-dimensional extension of the algorithm to raise the computation efficiency are also investigated