Centroidal Voronoi Tessellation Research Articles

On centroidal voronoi tessellation—energy smoothness and fast computation

Centroidal Voronoi tessellation (CVT) is a particular type of Voronoi tessellation that has many applications in computational sciences and engineering, including computer graphics. The prevailing method for computing CVT is Lloyd's method, which has linear convergence and is inefficient in practice. We develop new efficient methods for CVT computation and demonstrate the fast convergence of these methods. Specifically, we show that the CVT energy function has 2nd order smoothness for convex domains with smooth density, as well as in most situations encountered in optimization. Due to the 2nd order smoothness, it is possible to minimize the CVT energy functions using Newton-like optimization methods and expect fast convergence. We propose a quasi-Newton method to compute CVT and demonstrate its faster convergence than Lloyd's method with various numerical examples. It is also significantly faster and more robust than the Lloyd-Newton method, a previous attempt to accelerate CVT. We also demonstrate surface remeshing as a possible application.

Boundary Feedback Control of the Burgers Equations by a Reduced-Order Approach Using Centroidal Voronoi Tessellations

In this paper, we study a reduced-order modelling for boundary feedback control problem of Burgers equations. Brief review of the CVT (centroidal Voronoi tessellation) approaches to reduced-order basis are provided. In CVT-reduced order modelling, we start with a snapshot set just as is done in a SVD (Singular Value Decomposition)-based setting. A weighted (nonuniform density) CVT is introduced and low-order approximate solution and compensator-based control design of Burgers equation is discussed. Through CVT-nonuniform method, the low-order basis is obtained. Then this low-order basis is applied to low-order approximate solution and functional gains to design a low-order controller, and by using the low-order basis order of control modelling was reduced. Numerical experiments show that a solution of reduced-order controlled Burgers equation performs well in comparison with a solution of full order controlled Burgers equation.

Isotropic Remeshing with Fast and Exact Computation of Restricted Voronoi Diagram

AbstractWe propose a new isotropic remeshing method, based on Centroidal Voronoi Tessellation (CVT). Constructing CVT requires to repeatedly compute Restricted Voronoi Diagram (RVD), defined as the intersection between a 3D Voronoi diagram and an input mesh surface. Existing methods use some approximations of RVD. In this paper, we introduce an efficient algorithm that computes RVD exactly and robustly. As a consequence, we achieve better remeshing quality than approximation‐based approaches, without sacrificing efficiency. Our method for RVD computation uses a simple procedure and a kd‐tree to quickly identify and compute the intersection of each triangle face with its incident Voronoi cells. Its time complexity is O(mlog n), where n is the number of seed points and m is the number of triangles of the input mesh. Fast convergence of CVT is achieved using a quasi‐Newton method, which proved much faster than Lloyd's iteration. Examples are presented to demonstrate the better quality of remeshing results with our method than with the state‐of‐art approaches.

An Edge-Weighted Centroidal Voronoi Tessellation Model for Image Segmentation

Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations whose generators are also the centers of mass (centroids) of the Voronoi regions with respect to a given density function and CVT-based methodologies have been proven to be very useful in many diverse applications in science and engineering. In the context of image processing and its simplest form, CVT-based algorithms reduce to the well-known k -means clustering and are easy to implement. In this paper, we develop an edge-weighted centroidal Voronoi tessellation (EWCVT) model for image segmentation and propose some efficient algorithms for its construction. Our EWCVT model can overcome some deficiencies possessed by the basic CVT model; in particular, the new model appropriately combines the image intensity information together with the length of cluster boundaries, and can handle very sophisticated situations. We demonstrate through extensive examples the efficiency, effectiveness, robustness, and flexibility of the proposed method.

A 1D CELLULAR AUTOMATON THAT MOVES PARTICLES UNTIL REGULAR SPATIAL PLACEMENT

We consider a finite cellular automaton with particles where each site can host at most one particle. Starting from an arbitrary initial configuration, our goal is to move the particles between neighbor sites until the distance to the nearest particle is minimized globally. Such a configuration corresponds in fact to a regular placement. This problem is a cellular automata equivalent of load-balancing in parallel computing, where each task is a particle and each processor a connected set of sites. We present a cellular automata rule that solves this problem in the 1D case, and is convergent, i.e. once the regular placement is achieved, the configuration does not change anymore. The rule is inspired from the Lloyd algorithm, computing a centroidal Voronoi tessellation. The dynamic of the rule is described at a higher level, using self-explanatory space-time diagrams. They exhibit signals acting as quantity of movement carrying the energy of system. Each signal bounces or pass through particles, causing their movement, until it eventually reaches the border and vanishes. When signals have all vanished, particles are regularly placed.

Adaptive anisotropic meshing for steady convection-dominated problems

Obtaining accurate solutions for convection–diffusion equations is challenging due to the presence of layers when convection dominates the diffusion. To solve this problem, we design an adaptive meshing algorithm which optimizes the alignment of anisotropic meshes with the numerical solution. Three main ingredients are used. First, the streamline upwind Petrov–Galerkin method is used to produce a stabilized solution. Second, an adapted metric tensor is computed from the approximate solution. Third, optimized anisotropic meshes are generated from the computed metric tensor by an anisotropic centroidal Voronoi tessellation algorithm. Our algorithm is tested on a variety of two-dimensional examples and the results shows that the algorithm is robust in detecting layers and efficient in avoiding non-physical oscillations in the numerical approximation.

Halftoning with minimum spanning trees and its application to maze-like images

We propose a method for generating halftone images by using minimum spanning trees (MSTs). We generate a stippling from an input image by using centroidal Voronoi tessellation (CVT), and then connect all the points in the stippling with an MST. We derive a relationship of darkness between the input image and the MST halftone image by approximating CVT with the hexagonal lattice, and utilize it for correcting the brightness of the MST halftone images. We also propose methods for generating maze-like images on the basis of the MST halftone images. We embed user-specified solution paths in the MST halftone images in two ways. Experimental results show that the solution paths are successfully embedded and hidden in the maze-like images.

Adaptive triangular mesh coarsening with centroidal Voronoi tessellations

We present a novel algorithm for adaptive triangular mesh coarsening. The algorithm has two stages. First, the input triangular mesh is refined by iteratively applying the adaptive subdivision operator that performs a so-called red-green split. Second, the refined mesh is simplified by a clustering algorithm based on centroidal Voronoi tessellations (CVTs). The accuracy and good quality of the output triangular mesh are achieved by combining adaptive subdivision and the CVTs technique. Test results showed the mesh coarsening scheme to be robust and effective. Examples are shown that validate the method.

Sampling strategies for incorporating uncertainty and turbulent scattering into outdoor sound propagation calculations.

The accuracy of mean sound‐level estimates derived from different sampling methods is studied. The methods considered are Monte Carlo sampling (MCS), Latin Hypercube sampling (LHS), and Latinized Centroidal Voronoi Tessellation sampling (LCVTS). The goal is to determine which sampling method converges most rapidly to the actual mean sound level. The approach involves a model acoustic atmosphere, based on surface‐layer similarity theory and a relaxation model for the ground, with sound fields calculated numerically with a parabolic equation solver. The range of consequences of epistemic uncertainty due to imperfect knowledge of the atmospheric and ground variables is examined through a simplified probabilistic model. Random (aleatory) uncertainty due to turbulent scattering also is considered. The samples are drawn randomly within the domain of uncertainty of the environmental variables. When only epistemic uncertainty is present owing to imprecise knowledge of the environmental variables, LCVTS is found to converge to an accurate estimate with the fewest number of samples, followed by LHS and then MCS. When random turbulent scattering is present, however, the sampling method has little effect on convergence.

Formations with Decentralized Centroidal Voronoi Tessellation Algorithm

Centroidal Voronoi tessellations (CVTs) have recently attracted attention as a method for generating emergent behavior and swarm intelligence, for instance efficiently locating a source and creating evenly distributed formations. Previous research has focused on a centralized algorithm which depends on a base station computer to perform calculations and give the robotic agents direct movement commands. This research attempts to create an adaptive decentralized procedure for autonomous agents to create formations using CVTs. This decentralized approach will allow swarm systems a greater level of autonomy by removing the dependence on a centralized base station for calculating common agent tasks.

コンピュータグラフィックス 加法的重み付きべき乗重心ボロノイ分割によるノンフォトリアリスティックハーフトーニング

We developed additively weighted power centroidal Voronoi tessellation (APCVT) in which each generator of additively weighted power Voronoi tessellation coincides with the centroid of the corresponding Voronoi region and apply APCVT to non-photorealistic halftoning. The tone reproduction capability of the developed method is demonstrated on monochromatic and color images.

Comparison of descriptive statistics for multidimensional point sets

This work produces a statistical description of several sample-based techniques for numerical integration probabilistic assessment. The sampling techniques include pseudo-random, quasi-random, and centroidal Voronoi tessellation methods. The compared statistics are the star-discrepancy, minimum distance between points, and independence of sample sets. These statistics are selected because they influence the size of confidence intervals generated by repeated analyses. The optimal distance between points for two, three and four dimensions is derived as well as the general form for m dimensions. The results show that for problems with few random variables the complex methods would produce smaller confidence intervals. However, the benefits of more complex techniques are marginalized when there are more random variables.

Equilibrium Configurations for a Territorial Model

We consider a territorial model based on Voronoi tessellations, which form a partition of a planar region by enclosing each agent in a polygon such that every point within the polygon is closest to that agent instead of any other. Equilibria for this model correspond to centroidal Voronoi tessellations. For rectangular domains and for small population sizes, we show that there can be distinct coexisting stable equilibrium configurations, including the possibility of stable equilibria that are not related by symmetry. By considering randomly distributed initial positions, we give a statistical characterization of the basins of attraction for these equilibria in the case of a square domain. Furthermore, we show that the final territory that an agent occupies can have a wide range of sizes, which suggests that an individual can obtain a competitive advantage or disadvantage due entirely to its initial position. By treating the ratio of the length of the shorter side to the length of the longer side of the rectangle as a bifurcation parameter, we also numerically explore how stable and unstable equilibrium configurations are related to each other. Results for three agents are verified through experiments using robots which move according to a related territorial algorithm.

A multiresolution method for climate system modeling: application of spherical centroidal Voronoi tessellations

During the next decade and beyond, climate system models will be challenged to resolve scales and processes that are far beyond their current scope. Each climate system component has its prototypical example of an unresolved process that may strongly influence the global climate system, ranging from eddy activity within ocean models, to ice streams within ice sheet models, to surface hydrological processes within land system models, to cloud processes within atmosphere models. These new demands will almost certainly result in the develop of multiresolution schemes that are able, at least regionally, to faithfully simulate these fine-scale processes. Spherical centroidal Voronoi tessellations (SCVTs) offer one potential path toward the development of a robust, multiresolution climate system model components. SCVTs allow for the generation of high-quality Voronoi diagrams and Delaunay triangulations through the use of an intuitive, user-defined density function. In each of the examples provided, this method results in high-quality meshes where the quality measures are guaranteed to improve as the number of nodes is increased. Real-world examples are developed for the Greenland ice sheet and the North Atlantic ocean. Idealized examples are developed for ocean–ice shelf interaction and for regional atmospheric modeling. In addition to defining, developing, and exhibiting SCVTs, we pair this mesh generation technique with a previously developed finite-volume method. Our numerical example is based on the nonlinear, shallow-water equations spanning the entire surface of the sphere. This example is used to elucidate both the po tential benefits of this multiresolution method and the challenges ahead.

Centroidal Voronoi tessellation‐based finite element superconvergence

AbstractIn this article, a finding on finite element superconvergence is reported. The Laplacian operator with Dirichlet boundary condition is considered. The linear finite element solutions have an O(h2+α)(α≈0.5)‐superconvergence in l2 norm at nodes on an almost equilateral triangular mesh generated based on centroidal Voronoi tessellation, for an arbitrary 2D bounded domain. Extensive numerical examples are presented to demonstrate the superconvergence property. Copyright © 2008 John Wiley & Sons, Ltd.

Construction of positive definite cubature formulae and approximation of functions via Voronoi tessellations

Let Ω ⊂ ℝd be a compact convex set of positive measure. In a recent paper, we established a definiteness theory for cubature formulae of order two on Ω. Here we study extremal properties of those positive definite formulae that can be generated by a centroidal Voronoi tessellation of Ω. In this connection we come across a class of operators of the form \(L_n[f](\boldsymbol{x}):= \sum_{i=1}^n \phi_i(\boldsymbol{x})(f(\boldsymbol{y}_i) + \langle\nabla f(\boldsymbol{y}_i), \boldsymbol{x}-\boldsymbol{y}_i\rangle)\), where \(\boldsymbol{y}_1,\dots, \boldsymbol{y}_n\) are distinct points in Ω and {ϕ1, ..., ϕn} is a partition of unity on Ω. We present best possible pointwise error estimates and describe operators Ln with a smallest constant in an Lp error estimate for 1 ≤ p < ∞ . For a generalization, we introduce a new type of Voronoi tessellation in terms of a twice continuously differentiable and strictly convex function f. It allows us to describe a best operator Ln for approximating f by Ln[f] with respect to the Lp norm.

Constrained CVT meshes and a comparison of triangular mesh generators

Mesh generation in regions in Euclidean space is a central task in computational science, and especially for commonly used numerical methods for the solution of partial differential equations, e.g., finite element and finite volume methods. We focus on the uniform Delaunay triangulation of planar regions and, in particular, on how one selects the positions of the vertices of the triangulation. We discuss a recently developed method, based on the centroidal Voronoi tessellation (CVT) concept, for effecting such triangulations and present two algorithms, including one new one, for CVT-based grid generation. We also compare several methods, including CVT-based methods, for triangulating planar domains. To this end, we define several quantitative measures of the quality of uniform grids. We then generate triangulations of several planar regions, including some having complexities that are representative of what one may encounter in practice. We subject the resulting grids to visual and quantitative comparisons and conclude that all the methods considered produce high-quality uniform grids and that the CVT-based grids are at least as good as any of the others.

Geometric Properties of Optimal Photonic Crystals

Photonic crystals can be designed to control and confine light. Since the introduction of the concept by Yablonovitch and John two decades ago, there has been a quest for the optimal structure, i.e., the periodic arrangement of dielectric and air that maximizes the photonic band gap. Based on numerical optimization studies, we have discovered some surprisingly simple geometric properties of optimal planar band gap structures. We conjecture that optimal structures for gaps between bands n and n+1 correspond to n elliptic rods with centers defined by the generators of an optimal centroidal Voronoi tessellation (transverse magnetic polarization) and to the walls of this tessellation (transverse electric polarization).

Nondegeneracy and Weak Global Convergence of the Lloyd Algorithm in $\mathbb{R}^d$

The Lloyd algorithm originated in the context of optimal quantization and represents a fixed point iteration for computing an optimal quantizer. Reducing average distortion at every step, it constructs a Voronoi partition of the domain and replaces each generator with the centroid of the corresponding Voronoi cell. Optimal quantization is obtained in the case of a centroidal Voronoi tessellation (CVT), which is a special Voronoi tessellation of a domain $\Omega\in\mathbb{R}^d$ having the property that the generators of the Voronoi diagram are also the centers of mass, with respect to a given density function $\rho\geq 0$, of the corresponding Voronoi cells. The Lloyd iteration is currently the most popular and elegant algorithm for computing CVTs and optimal quantizers, but many questions remain about its convergence, especially in $d$-dimensional spaces $(d>1)$. In this paper, we prove that any limit point of the Lloyd iteration in any dimensional spaces is nondegenerate provided that $\Omega$ is a convex and bounded set and $\rho$ belongs to $L^{1}(\Omega)$ and is positive almost everywhere. This ensures that the fixed point map remains closed and hence the standard theory of descent methods guarantees weak global convergence of the Lloyd iteration to the set of nondegenerate fixed point quantizers. While previously only conjectured, the convergence properties of the Lloyd iteration are rigorously justified under such minimal regularity assumptions on the density functional. The results presented in this paper go beyond existing convergence theories for CVTs and optimal quantization related algorithms and should be of interest to both the mathematical and engineering communities.

Uniform Convergence of a Nonlinear Energy-Based Multilevel Quantization Scheme

A popular vector quantization scheme can be constructed by centroidal Voronoi tessellations (CVTs) which also have many other applications in diverse areas of science and engineering. The development of efficient algorithms for their construction is a key to the successful applications of CVTs in practice. This paper studies the details of a new optimization-based multilevel algorithm for the numerical computation of CVTs. The rigorous proof of its uniform convergence in one space dimension and the results of computational simulations are provided. They substantiate recent claims on the significant speedup demonstrated by the new scheme in comparison with traditional methods.