Construction Of Voronoi Diagrams Research Articles

Vector-Based Realization of Area-Weight Proportional Multiplicatively Weighted Voronoi Diagrams With the ArcGIS Engine

In the geographical field, the studies have been on service areas using the Voronoi diagram and its derived models is extensive, but there is a lack of effective methods to achieve a good area-weight proportionality between generators and their exclusive regions. As a famous visualizing method, adaptive multiplicatively weighted Voronoi diagrams are able to achieve it, but are limited to displaying non-spatial data. The approach of the area-weight proportional multiplicatively weighted Voronoi diagram is proposed to solve these problems by allowing for spatial division with a point-fixed iteration approach and a vector-based multiplicatively weighted Voronoi diagram construction method from point features with spatial coordinates and references in GIS environments. It enables one to create a set of regions that is proportional to the weights of the generators. The method is successfully tested on a series of cases. The approach aims to establish a kind of spatial data model to represent demand and supply situations in real life.

A digital geometric approach for discrete Voronoi diagram construction using GPU

SummaryVoronoi diagram is a versatile geometric structure used in solving many computational geometry problems in the field of image analysis, computer vision, engineering, medicine, molecular biology and various other disciplines. As a result it is a popular research domain. The literature boasts of many elegant and efficient methods for the construction of the Voronoi diagram in the real space. But these algorithms often fail to perform well when applied to the discrete domain. We present in this paper a novel algorithm for the construction of discrete Voronoi diagram using digital geometric concepts that takes care of the geometric perspective of the Voronoi diagram in the discrete domain. Although our technique is more suited for the CPU but by careful use of some cut‐off heuristics we are able to design a parallel programming approach for the construction of the Voronoi diagram making use of the GPU computational power. The speedup of our algorithm is compatible with the popular GPU technique for Voronoi diagram construction, namely, Jump flood algorithm. But since our algorithm uses the geometric perspective for the construction of Voronoi diagram it gives more accurate output and hence requires no correction step as is the case for the Jump flood algorithm. Furthermore, our approach focuses on extensibility, allowing for easy amenability for construction of higher dimensional Voronoi diagram. This ensures that the Voronoi diagram generation process can be adapted and expanded upon to meet specific requirements or incorporate future enhancements. We present an implementation of the proposed algorithm on a GPU using TAICHI language and describe its performance benefits on multiple input sites.

Accelerating surface remeshing through GPU-based computation of the restricted tangent face

High-quality mesh surfaces are crucial for geometric processing in a variety of applications. To generate these meshes, polyhedral remeshing techniques truncate Voronoi cells of the original surface and yield precise intersections, but the calculation is complicated. Some methods apply auxiliary points to construct Voronoi diagrams to simplify these techniques, thereby extracting co-planar facets to approximate the original surface. However, extracting these approximate facets from the constructed Voronoi diagram makes it inefficient and non-parallelizable. To this end, we propose an efficient GPU method for manifold surface remeshing, where the restricted tangent face (RTF) is utilized to approximate the original surface. By intersecting the pre-clipped Voronoi cell with the tangent plane, this method directly calculates the RTF of each point without any auxiliary points or traversing Voronoi cells. Moreover, to restrict the movement of points, we introduce a projection method based on the KNN strategy, where each point is projected onto the triangular facet in the original surface. Owing to the independence and non-interference of the RTF computation and projection of each point, our method is implemented in parallel on the GPU. Experimental results on various mesh surfaces demonstrate the superior performance of our method in the viability, effectiveness, and efficiency.

Sensitivity analysis and tailored design of minimization diagrams

Minimization diagrams encompass a large class of diagrams of interest in the literature, such as generalized Voronoi diagrams. We develop an abstract perturbation theory in two dimensions and perform a sensitivity analysis for functions depending on sets defined through intersections of smooth sublevel sets, and formulate precise conditions to avoid singular situations. This allows us to define a general framework for solving optimization problems depending on two-dimensional minimization diagrams. The particular case of Voronoi diagrams is discussed to illustrate the general theory. A variety of numerical experiments is presented. The experiments include constructing Voronoi diagrams with cells of equal size, cells satisfying conditions on the relative size of their edges or their internal angles, cells with the midpoints of pairs of Voronoi and Delaunay edges as close as possible, or cells of varying sizes governed by a given function. Overall, the experiments show that the proposed methodology allows the construction of customized Voronoi diagrams using off-the-shelf well-established optimization algorithms.

An efficient algorithm for approximate Voronoi diagram construction on triangulated surfaces

Voronoi diagrams on triangulated surfaces based on the geodesic metric play a key role in many applications of computer graphics. Previous methods of constructing such Voronoi diagrams generally depended on having an exact geodesic metric. However, exact geodesic computation is time-consuming and has high memory usage, limiting wider application of geodesic Voronoi diagrams (GVDs). In order to overcome this issue, instead of using exact methods, we reformulate a graph method based on Steiner point insertion, as an effective way to obtain geodesic distances. Further, since a bisector comprises hyperbolic and line segments, we utilize Apollonius diagrams to encode complicated structures, enabling Voronoi diagrams to encode a medial-axis surface for a dense set of boundary samples. Based on these strategies, we present an approximation algorithm for efficient Voronoi diagram construction on triangulated surfaces. We also suggest a measure for evaluating similarity of our results to the exact GVD. Although our GVD results are constructed using approximate geodesic distances, we can get GVD results similar to exact results by inserting Steiner points on triangle edges. Experimental results on many 3D models indicate the improved speed and memory requirements compared to previous leading methods.

Robust Construction of Voronoi Diagrams of Spherical Balls in Three-Dimensional Space

Voronoi diagrams are useful for spatial reasoning among particles and there are many prior studies on their construction. However, most prior works were for the ordinary Voronoi diagrams of points in R 2 and R 3 . Here we propose a robust algorithm for constructing the Voronoi diagram of spherical balls in R 3 , where the output is guaranteed to be at least topologically consistent. This topology-oriented incremental algorithm constructs the Voronoi diagram in O ( n 3 ) time in the worst case, whereas its empirical time behavior shows a strong linear fashion for all data we tested. The proposed algorithm is the three-dimensional generalization of its counterpart in the plane. It is implemented, thoroughly tested, and compared with two well-known programs. Current implementation processes approximately 350 balls per second using one core of ordinary desktop computer. This paper also contains an extensive review on the Voronoi diagrams of 2D circular disks and 3D spherical balls. We anticipate the algorithm will be widely used to solve application problems from many disciplines in science and engineering. The library is freely available from the github repository. • Topology-oriented incremental construction of the Voronoi diagram of 3D spheres. • The first robust algorithm for constructing the Voronoi diagram of 3D spheres. • Verification and validation through a complete implementation and experiment.

Energy storage optimization of ferroelectric ceramics during phase-transition process of amorphous/nanocrystalline and polycrystalline by using a phase-field model for dielectric breakdown

Ferroelectric ceramics have the potential to be widely applied in the modern industry and military power systems due to their ultrafast charging/discharging speed and high energy density. Considering the structural design and electrical properties of ferroelectric capacitor, it is still a challenge to find out the optimal energy storage of ferroelectric ceramics during the phase-transition process of amorphous/nanocrystalline and polycrystalline. In this work, a finite element model suitable for the multiphase ceramic system is constructed based on the phase field breakdown theory. The nonlinear coupling relationship of multiple physical fields between multiphase ceramics was taken into account in this model. The basic structures of multiphase ceramics are generated by using the Voronoi diagram construction method. The specified structure of multiphase ceramics in the phase- transition process of amorphous/nanocrystalline and polycrystalline was further obtained through the grain boundary diffusion equation. The simulation results show that the multiphase ceramics have an optimal energy storage in the process of amorphous polycrystalline transformation, and the energy storage density reaches the maximum when the crystallinity is 13.96% and the volume fraction of grain is 2.08%. It provides a research plan and idea for revealing the correlation between microstructure and breakdown characteristics of multiphase ceramics. This simulation model realizes the nonlinear coupling of the multiphase ceramic mesoscopic structure and the phase field breakdown. It provides a reference scheme for the structural design and performance optimization of ferroelectric ceramics.

Algorithm for Constructing Voronoi Diagrams with Optimal Placement of Generator Points Based on Theory of Optimal Set Partitioning

An algorithm is proposed for constructing a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of n-dimensional Euclidean space. The algorithm is based on the formulation of the corresponding continuous problem of optimal set partitioning with a partition quality criterion providing the corresponding form of Voronoi diagram, and on the application of the apparatus of the optimal partitioning theory to solve this problem. Herewith, the efficient method of non-differentiable optimization is used for the numerical solution of an auxiliary finite-dimensional optimization problem arising in the development of the method for solving the mentioned infinite-dimensional problem of optimal set partitioning. Namely, that is one of the variants of the generalized gradient descent method with space expansion in the direction of the difference of two successive generalized antigradients (Shor's r-algorithm). The proposed algorithm for constructing a generalized Voronoi diagram with optimal placement of a finite number of generator points in a bounded set of n-dimensional Euclidean space has some advantages compared to those known in a scientific literature. It does not depend on a dimension of Euclidean space containing the initial bounded set; it is applicable for the large-scale problems (over 300 generator points); it works not only for Euclidean metrics, but also for Chebyshev, Manhattan and other ones; the complexity of the algorithm implementation for constructing Voronoi diagram based on the proposed approach does not increase with increasing number of generator points. The results of software implementation of the developed algorithm are presented for constructing a standard Voronoi diagram with an optimal placement of generator points as well as some of its generalizations such as additive, multiplicative and additive-multiplicative Voronoi diagrams with optimal placement of generator points.

Оптимальний алгоритм динамічної підтримки Діаграми Вороного для множини точок

The article is devoted to the development of a dynamic data structure for solving proximity problems based on the dynamic Voronoi Diagram. This data structure can be used as the core of the common algorithmic space model for solving a set of visualization and computer modeling problems. The data structure is based on the strategy of "divide and rule" for Voronoi diagram construction. Similar to the original algorithm, we store a binary tree that represents the Voronoi diagram, but define three new operations: insert, delete, and balance. To ensure the efficiency of operations, it is proposed to use red-black tree. In general, the proposed data structure shows much better results than the original static algorithm. Compared to existing algorithms, this data structure is both simple and efficient.

Applying the Whitening Transformation to Echo Signals for Reducing Pattern Noise in Ultrasonic Testing

In the ultrasonic testing of industrial facilities, it is often necessary to deal with metals having an increased level of pattern noise formed by a multitude of pulses rescattered by the structural inhomogeneities of the material. Extracting a useful signal from the pattern noise background is a nontrivial task. The fact is that the spectra and spatial distribution of pattern noise pulses and of the echo signal due to a reflector are close to each other, thus making the common methods of reducing the level of interference inapplicable. Therefore, special methods have been developed to reduce the level of pattern noise. One of such methods is the whitening (decorrelation) transformation of echo signals, originally developed for radar. This technique consists in converting correlated pattern noise pulses into white noise, which is well suppressed by conventional coherent methods such as C-SAFT, for reconstructing reflector images. In numerical experiments, echoes with pattern noise were calculated using the method of finite differences in a time domain (FDTD) when forming domains according to the rule of constructing Voronoi diagrams. As an alternative, pattern noise was calculated taking account of multiple rescattering of the probing pulse by point reflectors. It is shown that pattern noise reduction is possible when the pattern-noise correlation matrix is known with an accuracy of at least $$ \approx $$10%. In model experiments, the method of whitening transformation was successfully applied to eliminate reverberation noise.

Robust Construction of Voronoi Diagram of Circles by Region-Expansion Algorithm

This paper presents a numerically robust algorithm to construct a Voronoi diagram of circles in the plane. The circles are allowed to have intersections among them, but one circle cannot fully contain another circle. The Voronoi diagram is a tessellation of the plane into Voronoi regions of given circles. Each circle has its Voronoi region which is defined by a set of points in the plane closer to the circle than any other circles. The distance from a point p to a circle ci of center pi and radius ri is ||p-pi||-ri, which is the closest Euclidean distance from p to the circle boundary. The proposed algorithm first constructs the point Voronoi diagram of centers of given circles, then it enlarges each point to the circle and expands its Voronoi region accordingly. This region-expansion process is done by local modifications and after completing this process for the whole circles the desired circle Voronoi diagram can be obtained. The proposed algorithm is numerically robust and we provide with a few examples to show its robustness. The algorithm runs in O(n2) time in the worst case and O(n) time on average where n is the number of the circles. The experiment shows that the region-expansion algorithm is robust and runs fast with strong linear time behavior.

Constrained Construction of Planar Delaunay Triangulations without Flipping

The construction of Voronoi diagrams and Delaunay triangulations finds wide application in many branches of science. Delaunay triangulations have properties which make them more desirable than other triangulations for the same node set. Delaunay has characterized his triangulations by the empty circle property. The partitioning and flipping methods which have been developed for digital construction of Voronoi diagrams and Delaunay triangulations only make indirect use of this property. A novel method of construction is proposed, which is based directly on the empty circle property of Delaunay. The geometry of the steps of the algorithm is simple and can be grasped intuitively. The method can be applied to constrained triangulations, in which a triangulation domain and some of the edges are prescribed. A data structure for triangulations of concave and multiply-connected domains is presented which permits convenient specification of the constraints and the triangulation. The method is readily implemented, efficient and robust.

Abstract Voronoi Diagrams from Closed Bisecting Curves

We present the first algorithm for constructing abstract Voronoi diagrams from bisectors that are unbounded or closed Jordan curves. It runs in expected [Formula: see text] many steps and [Formula: see text] space, where [Formula: see text] is the number of sites, [Formula: see text] denotes the average number of faces (connected components) per Voronoi region in any diagram of a subset of [Formula: see text] sites, and [Formula: see text] is the maximum number of intersection points between any two related bisectors.

Fast and Memory‐Efficient Voronoi Diagram Construction on Triangle Meshes

AbstractGeodesic based Voronoi diagrams play an important role in many applications of computer graphics. Constructing such Voronoi diagrams usually resorts to exact geodesics. However, exact geodesic computation always consumes lots of time and memory, which has become the bottleneck of constructing geodesic based Voronoi diagrams. In this paper, we propose the window‐VTP algorithm, which can effectively reduce redundant computation and save memory. As a result, constructing Voronoi diagrams using the proposed window‐VTP algorithm runs 3–8 times faster than Liu et al.'s method [LCT11], 1.2 times faster than its FWP‐MMP variant and more importantly uses 10–70 times less memory than both of them.

Constructing Voronoi Diagrams from Hollow Spheres Using Conformal Geometric Algebra

This paper proposes a new procedure for the construction of Voronoi diagrams using spheres. The auxiliary spheres that assist in the diagram construction are defined following the concept of a “hollow sphere”, and their geometrical properties, within the Conformal Geometric Algebra model, turn out to be crucial elements for increasing the global efficiency procedure with complexity \(O(n \log n)\). Several examples are presented, using the CLUCalc software, that show the effectiveness and ability of the proposed procedure. Also, we argue that the Conformal Geometric Algebra model provides straightforward and intuitive concepts to the field of Computational Geometry, that prove invaluable for allowing the incremental definition and construction of Voronoi diagrams.

Construction of Voronoi Diagram using the Hollow Sphere Concept

This paper implements an algorithm for constructing Voronoi regions using the method we call "hollow sphere". This principle uses the circle, sphere or hyper-sphere as a geometric structure taking into account an Euclidean space of an arbitrary dimension. Boris Deloné used the property of the empty circle to build the Delaunay triangulation; in our case, the same property is used to perform the validations of the hollow spheres, but without using triangles as a fundamental structure. For convenience sake, the hollow sphere as a circle so as to work in two dimensions will be explained and illustrated. However, there is no impediment to working with spheres to take the principles to a three-dimensional space. Furthermore, the properties of the hollow sphere are detailed and an algorithm of incremental construction is used with O(nlogn) time.

A refinement for rational functions of Pólya's method to construct Voronoi diagrams

Given a complex polynomial P with zeroes z1,…,zd, we show that the asymptotic zero-counting measure of the iterated derivatives Q(n), n=1,2,…, where Q=R/P is any irreducible rational function, converges to an explicitly constructed probability measure supported by the Voronoi diagram associated with z1,…,zd. This refines Pólya's Shire theorem for these functions. In addition, we prove a similar result, using currents, for Voronoi diagrams associated with generic hyperplane configurations in Cm.

Topology-Oriented Incremental Algorithm for the Robust Construction of the Voronoi Diagrams of Disks

Voronoi diagrams are useful for spatial reasoning, and the robust and efficient construction of the ordinary Voronoi diagram of points is well known. However, its counterpart for circular disks in R 2 and spherical balls in R 3 remains a challenge. In this article, we propose a topology-oriented incremental algorithm which robustly and efficiently computes a Voronoi diagram by incrementing a new disk generator to an existing one. The key idea is to enforce the convexity of the Voronoi cell corresponding to the incrementing disk so that a simple variation of the algorithm for points proposed by Sugihara in 1992 can be applied. A benchmark using both random and degenerate disks shows that the proposed algorithm is superior to CGAL in both computational efficiency and algorithmic robustness.

A randomized divide and conquer algorithm for higher-order abstract Voronoi diagrams

Given a set of n sites in the plane, their order-k Voronoi diagram partitions the plane into regions such that all points within one region have the same k nearest sites. The order-k abstract Voronoi diagram offers a unifying framework that represents a wide range of concrete order-k Voronoi diagrams. It is defined in terms of bisecting curves satisfying some simple combinatorial properties, rather than the geometric notions of sites and distance.In this paper we develop a randomized divide-and-conquer algorithm to compute the order-k abstract Voronoi diagram in expected O(kn1+ε) operations. For solving small sub-instances in the divide-and-conquer process, we also give two auxiliary algorithms with expected O(k2nlog⁡n) and O(n22α(n)log⁡n) time, respectively, where α(⋅) is the inverse of the Ackermann function. Our approach directly implies an O(kn1+ε)-time algorithm for several concrete order-k instances such as points in any convex distance, disjoint line segments or convex polygons of constant size in the Lp norms, and others. It also provides basic techniques that can enable the application of well-known random sampling techniques to the construction of Voronoi diagrams in the abstract setting and for non-point sites.

Efficient Voronoi diagram construction for planar freeform spiral curves

We present a real-time algorithm for computing the Voronoi diagram of planar freeform piecewise-spiral curves. The efficiency and robustness of our algorithm is based on a simple topological structure of Voronoi cells for spirals, which also enables us a direct construction of Voronoi structure without relying on intermediate polygonal or biarc approximations to the given planar curves. Using a Möbius transformation, we provide an efficient search for maximal disks. The correct topology of Voronoi diagram is computed by sampling maximal disks systematically, which entails subdividing spirals until each belongs to a pair/triple of spirals under a certain matching condition. The matching pairs and triples serve as the basic building blocks for bisectors and bifurcations, and their connectivity implies the Voronoi structure. We demonstrate a real-time performance of our algorithm using experimental results including the medial axis computation for planar regions under deformation with non-trivial self-intersections and the Voronoi diagram construction for disconnected planar freeform curves.