Three-dimensional Voronoi Diagram Research Articles

Triangular mesh generation on free-form surfaces based on bubble dynamics simulation

Purpose Modern CAD systems facilitate the creation of any surface geometry imaginable, and complex surfaces for free-form grid shells are often represented by a set of Non-Uniform Rational B-Splines surface patches. But it remains an intractable issue how to generate high-quality grids on complex surfaces efficiently. To solve this issue, an automatic triangular mesh generation method is presented, based on bubble dynamics simulation and a modified Delaunay method. Design/methodology/approach A moderate amount of points are first distributed on a given surface. Next, by regarding the points as elastic bubbles with the same size and introducing the forces acting on bubbles, the motion control equations of bubbles are established. The equilibrium state of the bubble system is found by Verlet algorithm. Then, the Voronoi diagram on the surface is obtained by calculating the intersection between the surface and the three-dimensional (3D) Voronoi diagram of the centers of bubbles. Finally, a triangular mesh, Delaunay triangulation on the surface, is determined based on the dual change of the Voronoi diagram. Findings This method generates meshes on the surface directly, unlike mapping-based methods, avoiding the mapping distortion. Examples are given to demonstrate the successful execution of this method. The result also illustrates that this method is applicable to various surfaces in high automation level and resultant meshes are highly uniform and well-shaped. Originality/value Thus, this method provides the convenience for the geometry design of complex free-form grid structure.

Three-dimensional Voronoi Diagram–based Self-deployment Algorithm in IoT Sensor Networks

With the rapid development of 4G/5G technology and the Internet of Things (IoT), data security and privacy problems are becoming more serious. Wireless sensor networks (WSNs), as the main data source of IoT, are an important stage to ensure data availability and data privacy protection. In this paper, a novel deployment algorithm for 3D WSNs based on the Voronoi diagram is proposed. The algorithm uses the characteristics of adjacency and fast partition of the Voronoi diagram to realize fast division of the 3D monitoring area, calculates the center of each Voronoi area as the latest position of node, repeatedly builds the Voronoi diagram to maximize the coverage of the monitoring area, and maximizes the availability and integrity of data. At the same time, the 4G/5G communication technology is used to realize communication between nodes, and data encryption is used to improve data security. An improved algorithm is also proposed to adapt to different deployment conditions. In this paper, data and privacy security are protected from data sources, and the effectiveness of the algorithm is tested by computer simulation.

Data-driven spatio-temporal discretization for pedestrian flow characterization

We propose a novel approach to pedestrian flow characterization. The definitions of density, flow and velocity existing in the literature are extended through a data-driven spatio-temporal discretization framework. The framework is based on three-dimensional Voronoi diagrams. Synthetic data is used to empirically investigate the performance of the approach and to illustrate its advantages. Our approach outperforms the considered approaches from the literature in terms of the robustness with respect to the simulation noise and with respect to the sampling frequency. Additionally, the proposed approach is by design (i) independent from an arbitrarily chosen discretization; (ii) appropriate for the multidirectional composition of pedestrian traffic; (iii) able to reflect the heterogeneity of the pedestrian population; and (iv) applicable to pedestrian trajectories described either analytically or as a sample of points.

Data-driven spatio-temporal discretization for pedestrian flow characterization

Abstract: We propose a novel approach to pedestrian flow characterization. The definitions of density, flow and velocity existing in the literature are extended through a data-driven spatio-temporal discretization framework. The framework is based on three-dimensional Voronoi diagrams. Synthetic data is used to empirically investigate the performance of the approach and to illustrate its advantages. Our approach outperforms the considered approaches from the literature in terms of the robustness with respect to the simulation noise and with respect to the sampling frequency. Additionally, the proposed approach is by design (i) independent from an arbitrarily chosen discretization; (ii) appropriate for the multidirectional composition of pedestrian traffic; (iii) able to reflect the heterogeneity of the pedestrian population; and (iv) applicable to pedestrian trajectories described either analytically or as a sample of points.

Fuzzy Three-Dimensional Voronoi Diagram and its Application to Geographical Data Analysis

A concept of fuzzy three-dimensional Voronoi Diagram is presented for spatial relations analysis of real world three-dimensional geographical data, where it is an extension of well known two-dimensional Voronoi Diagram to three-dimensional representation with uncertain spatial relation information in terms of fuzzy set. It makes possible to analyze quantitatively complex boundaries of geographically intricate areas, to give human friendly fuzzy explanation of determining three-dimensional directions, and to express uncertain spatial relations by precise unified fuzzy description. It is applied to decide spatial direction relations of artificial geographicalmountain data, which includes 8 spatial directions with at most 60 relative direction relations, and it leads to detect threedimensional directions whereas the expression of traditional 4 directions and 12 relative directions indicate two-dimensional directions only. The proposed concept aims to discriminate neighbors’ class relations and spatial-temporal changes of specially appointed objects, and also aims to be a tool to achieve the intellective extraction and analysis of geographical data of a mountainous area located in northeast China.

Delaunay meshing of isosurfaces

We present an isosurface meshing algorithm, DelIso, based on the Delaunay refinement paradigm. This paradigm has been successfully applied to mesh a variety of domains with guarantees for topology, geometry, mesh gradedness, and triangle shape. A restricted Delaunay triangulation, dual of the intersection between the surface and the three-dimensional Voronoi diagram, is often the main ingredient in Delaunay refinement. Computing and storing three-dimensional Voronoi/Delaunay diagrams become bottlenecks for Delaunay refinement techniques since isosurface computations generally have large input datasets and output meshes. A highlight of our algorithm is that we find a simple way to recover the restricted Delaunay triangulation of the surface without computing the full 3D structure. We employ techniques for efficient ray tracing of isosurfaces to generate surface sample points, and demonstrate the effectiveness of our implementation using a variety of volume datasets.

La modélisation de données océanographiques à l'aide du diagramme de Voronoï tridimensionnel

plusieurs applications en oceanographie. ABSTRACT. Managing oceanographic data with traditional geographical information sy stems (GIS) is a difficult task because these systems have been primarily desig ned for land-based ap- plications. The main problem is that the nature of objects at sea is completely d ifferent from the nature of objects found on the land. To solve this problem, we propose usin g a new spatial model to handle topological relationships between objects in oceanographic datas ets. This model is based on the three-dimensional Voronoi diagram (VD) and its dual the D triangulation (DT). We present in this paper the main properties of the VD and the DT, alg orithms to construct and modify them, and show how some 3D GIS operations are greatly simplifie d when a spatial model is built upon them. We also show some examples of applications in oce anography.

Constructing three-dimensional discrete Voronoi diagrams by the incremental method and application to self-organizing maps

Constructing three-dimensional discrete Voronoi diagrams by the incremental method and application to self-organizing maps

Constructing three-dimensional discrete Voronoi diagrams by the incremental method and application to self-organizing maps

We propose an algorithm to construct a three-dimensional (3D) discrete Voronoi diagram using the incremental method on a digitized space and apply it to measure the topology preservation rate of se...

Three‐dimensional Identification and Reconstruction of Galaxy Systems within Flux‐limited Redshift Surveys

We have developed a new geometrical method for identifying and reconstructing a homogeneous and highly complete set of galaxy groups within flux-limited redshift surveys. Our method combines information from the three-dimensional Voronoi diagram and its dual, the Delaunay triangulation, to obtain group and cluster catalogs that are remarkably robust over wide ranges in redshift and degree of density enhancement. As free by-products, this Voronoi-Delaunay method (VDM) provides a nonparametric measurement of the galaxy density around each object observed and a quantitative measure of the distribution of cosmological voids in the survey volume. In this paper, we describe the VDM algorithm in detail and test its effectiveness using a family of mock catalogs that simulate the Deep Extragalactic Evolutionary Probe (DEEP2) Redshift Survey, which should present at least as much challenge to cluster reconstruction methods as any other near-future survey that is capable of resolving their velocity dispersions. Using these mock DEEP2 catalogs, we demonstrate that the VDM algorithm can be used to identify a homogeneous set of groups in a magnitude-limited sample throughout the survey redshift window 0.7 < z < 1.2. The actual group membership can be effectively reconstructed even in the distorted redshift space environment for systems with line-of-sight velocity dispersion σlos greater than ≈200 km s-1. By applying the sampling rate and the instrument-imposed target selection biases expected for DEEP2, we show that even in a worst-case scenario our VDM method can construct a homogeneous sample of systems that reproduces major properties of the cluster parent population down to ≈200 km s-1 for systems with at least five members (and down to ≈400 km s-1 for clusters as a whole). In a Λ cold dark matter cosmology, that limit translates into an identification rate of ~270 systems per square degree with 0.7 < z < 1.2 and a total of more than 1000 groups within the full DEEP2 survey volume. By comparing the galaxy cluster catalogs derived from mock observations to the underlying distribution of clusters as found in real space with much fainter galaxies included (which should more closely trace mass in the cluster), we can assess completeness in velocity dispersion directly. We conclude that if our VDM algorithm is used, the recovered DEEP2 group and cluster sample should be statistically complete for σlos 400 km s-1. Finally, we argue that the bivariate distribution of systems as a function of redshift and velocity dispersion reconstructed with these techniques reproduces with high fidelity the underlying real space distribution and can thus be used robustly to constrain cosmological parameters. We expect that the VDM algorithm, which has performed so well when faced with the challenges posed by the DEEP2 survey, should only be more effective when applied to the better sampled, larger surveys of the local universe now underway.

Towards a singularity-based shape language: ridges, ravines, and skeletons for polygonal surfaces

High demands on digital contents have posing strong needs on visual languages on three-dimensional (3D) shapes for improved human communication. For a visual language to effectively communicate essential 3D shape information, shape features defined in terms of singularity signs have been recognized as key shape descriptors. In this paper, we study salient shape features defined via distance function singularities: ridges, ravines, and a skeleton. We propose a method for robust extraction of the 3D skeleton of a polygonal surface and detection of salient surface features, ridges and ravines, corresponding to the skeletal edges. The method adapts the three-dimensional Voronoi diagram technique for skeleton extraction, explores singularity theory for ridge and ravine detection, and combines several filtering methods for skeleton denoising and for selecting perceptually salient ridges and ravines. We demonstrate that the ridges and ravines convey important shape information and, in particular, can be used for face recognition purposes.

Primal Dividing and Dual Pruning: Output-Sensitive Construction of Four-Dimensional Polytopes and Three-Dimensional Voronoi Diagrams

In this paper, we give an algorithm for output-sensitive construction of an f-face convex hull of a set of n points in general position in E 4 . Our algorithm runs in \(O((n+f)\log^2 f)\) time and uses O(n+f) space. This is the first algorithm within a polylogarithmic factor of optimal \(O(n \log f + f)\) time over the whole range of f. By a standard lifting map, we obtain output-sensitive algorithms for the Voronoi diagram or Delaunay triangulation in E 3 and for the portion of a Voronoi diagram that is clipped to a convex polytope. Our approach simplifies the ``ultimate convex hull algorithm'' of Kirkpatrick and Seidel in E 2 and also leads to improved output-sensitive results on constructing convex hulls in E d for any even constant d > 4.

Improving Accuracy in a Robust Algorithm for Three-Dimensional Voronoi Diagrams

Abstract This paper describes extensions to a previous algorithm that robustly builds three-dimensional Voronoi diagrams in the presence of inexact numerical computations. The extensions improve the algorithm's accuracy, making its results more nearly represent the proximity properties of an ideal Voronoi diagram. In empirical tests, these extensions have improved accuracy by more than eight orders of magnitude. Complete pseudocode for the algorithm appears in an appendix of this paper.