Showing papers on "Voronoi diagram published in 2015"

Cascading Divide-and-conquer: A Technique for Designing Parallel Algorithms

Abstract: We present techniques for parallel divide-and-conquer, resulting in improved parallel algorithms for a number of problems. The problems for which we give improved algorithms include intersection detection, trapezoidal decomposition (hence, polygon triangulation), and planar point location (hence, Voronoi diagram construction). We also give efficient parallel algorithms for fractional cascading, 3-dimensional maxima, 2-set dominance counting, and visibility from a point. All of our algorithms run in O(log n) time with either a linear or sub-linear number of processors in the CREW PRAM model.

Voronoi dipole moments for the simulation of bulk phase vibrational spectra

Abstract: We present the Voronoi tessellation of electron density data to obtain molecular dipole moments in bulk phase ab initio molecular dynamics simulations for the calculation of vibrational spectra. Opposed to the established scheme of maximally localized Wannier functions, this approach does not rely on computationally demanding localization procedures. Nevertheless, we show at the examples of methanol, benzene, and phenol that it provides infrared and Raman spectra of similar quality and is even superior in specific cases like the Raman spectra of benzene and phenol. We have also applied the Voronoi method to a mixture of the ionic liquid 1-ethyl-3-methylimidazolium acetate with water, and show that it is advantageous in systems with significant charge transfer.

Data-driven spatial b value estimation with applications to California seismicity: To b or not to b

Abstract: In this paper we present a penalized likelihood-based method for spatial estimation of Gutenberg-Richter's b value. Our method incorporates a nonarbitrary partitioning scheme based on Voronoi tessellation, which allows for the optimal partitioning of space using a minimum number of free parameters. By random placement of an increasing number of Voronoi nodes, we are able to explore the whole solution space in terms of model complexity. We obtain an overall likelihood for each model by estimating the b values in all Voronoi regions and calculating its joint likelihood using Aki's formula. Accounting for the number of free parameters, we then calculate the Bayesian Information Criterion for all random realizations. We investigate the ensemble of the best performing models and demonstrate the robustness and validity of our method through extensive synthetic tests. We apply our method to the seismicity of California using two different time spans of the Advanced National Seismic System catalog (1984–2014 and 2004–2014). The results show that for the last decade, the b value variation in the well-instrumented parts of mainland California is limited to the range of (0.94 ± 0.04–1.15 ± 0.06). Apart from the Geysers region, the observed variation can be explained by network-related discrepancies in the magnitude estimations. Our results suggest that previously reported spatial b value variations obtained using classical fixed radius or nearest neighbor methods are likely to have been overestimated, mainly due to subjective parameter choices. We envision that the likelihood-based model selection criteria used in this study can be a useful tool for generating improved earthquake forecasting models.

Automatic Optical-to-SAR Image Registration by Iterative Line Extraction and Voronoi Integrated Spectral Point Matching

Abstract: Automatic optical-to-SAR image registration is considered as a challenging problem because of the inconsistency of radiometric and geometric properties. Feature-based methods have proven to be effective; however, common features are difficult to extract and match, and the robustness of those methods strongly depends on feature extraction results. In this paper, a new method based on iterative line extraction and Voronoi integrated spectral point matching is developed. The core idea consists of three aspects: 1) An iterative procedure that combines line segment extraction and line intersections matching is proposed to avoid registration failure caused by poor feature extraction. 2) A multilevel strategy of coarse-to-fine registration is presented. The coarse registration aims to preserve main linear structures while reducing data redundancy, thus providing robust feature matching results for fine registration. 3) Voronoi diagram is introduced into spectral point matching to further enhance the matching accuracy between two sets of line intersection. Experimental results show that the proposed method improves the matching performance. Compared with previous methods, the proposed algorithm can effectively and robustly generate sufficient reliable point pairs and provide accurate registration.

Image partitioning into convex polygons

Abstract: The over-segmentation of images into atomic regions has become a standard and powerful tool in Vision. Traditional superpixel methods, that operate at the pixel level, cannot directly capture the geometric information disseminated into the images. We propose an alternative to these methods by operating at the level of geometric shapes. Our algorithm partitions images into convex polygons. It presents several interesting properties in terms of geometric guarantees, region compactness and scalability. The overall strategy consists in building a Voronoi diagram that conforms to preliminarily detected line-segments, before homogenizing the partition by spatial point process distributed over the image gradient. Our method is particularly adapted to images with strong geometric signatures, typically man-made objects and environments. We show the potential of our approach with experiments on large-scale images and comparisons with state-of-the-art superpixel methods.

Intrinsic computation of centroidal Voronoi tessellation (CVT) on meshes

Abstract: Centroidal Voronoi tessellation (CVT) is a special type of Voronoi diagram such that the generating point of each Voronoi cell is also its center of mass. The CVT has broad applications in computer graphics, such as meshing, stippling, sampling, etc. The existing methods for computing CVTs on meshes either require a global parameterization or compute it in the restricted sense (that is, intersecting a 3D CVT with the surface). Therefore, these approaches often fail on models with complicated geometry and/or topology. This paper presents two intrinsic algorithms for computing CVT on triangle meshes. The first algorithm adopts the Lloyd framework, which iteratively moves the generator of each geodesic Voronoi diagram to its mass center. Based on the discrete exponential map, our method can efficiently compute the Riemannian center and the center of mass for any geodesic Voronoi diagram. The second algorithm uses the L-BFGS method to accelerate the intrinsic CVT computation. Thanks to the intrinsic feature, our methods are independent of the embedding space, and work well for models with arbitrary topology and complicated geometry, where the existing extrinsic approaches often fail. The promising experimental results show the advantages of our method. We propose two intrinsic methods for computing centroidal Voronoi tessellation (CVT) on triangle meshes.Thanks to their intrinsic nature, our methods compute CVT using metric only.Our results are independent of the embedding space.

Non-universal Voronoi cell shapes in amorphous ellipsoid packs

Abstract: In particulate systems with short-range interactions, such as granular matter or simple fluids, local structure determines the macroscopic physical properties. We analyse local structure metrics derived from the Voronoi diagram of oblate ellipsoids, for various aspect ratios and global packing fractions φg. We focus on jammed static configurations of frictional ellipsoids, obtained by tomographic imaging and by discrete element method simulations. The rescaled distribution of local packing fractions φl, defined as the ratio of particle volume and its Voronoi cell volume, is found to be independent of the particle aspect ratio, and coincide with results for sphere packs. By contrast, the typical Voronoi cell shape, quantified by the Minkowski tensor anisotropy index β = β02,0, points towards a difference between random packings of spheres and those of oblate ellipsoids. While the average cell shape β of all cells with a given value of is similar in dense and loose jammed sphere packings, the structure of dense and loose ellipsoid packings differs substantially such that this does not hold true.

Distributed Collaborative Coverage-Control Schemes for Non-Convex Domains

Abstract: The coverage control problem for mobile sensor networks in non-convex environments is addressed in this article. The sensing model of each node is assumed identical and radial either in the geodesic or the Euclidean sense. Depending on the selected metric, the partitioning of the non-convex domain can result either in the geodesic or the Euclidean Voronoi diagram. The coverage problem is examined from a two-fold aspect: based on a) proper selection of the sensing model, and b) the partitioning scheme, as directed by the application itself. Distributed collaborative control schemes are developed, further validated via extensive numerical studies.

A generic regional spatio-temporal co-occurrence pattern mining model: a case study for air pollution

Abstract: Spatio-temporal co-occurrence patterns represent subsets of object types which are located together in both space and time. Existing algorithms for co-occurrence pattern mining cannot handle complex applications such as air pollution in several ways. First, the existing models assume that spatial relationships between objects are explicitly represented in the input data, while the new method allows extracting implicitly contained spatial relationships algorithmically. Second, instead of extracting co-occurrence patterns of only point data, the proposed method deals with different feature types that is with point, line and polygon data. Thus, it becomes relevant for a wider range of real applications. Third, it also allows mining a spatio-temporal co-occurrence pattern simultaneously in space and time so that it illustrates the evolution of patterns over space and time. Furthermore, the proposed algorithm uses a Voronoi tessellation to improve efficiency. To evaluate the proposed method, it was applied on a real case study for air pollution where the objective is to find correspondences of air pollution with other parameters which affect this phenomenon. The results of evaluation confirm not only the capability of this method for co-occurrence pattern mining of complex applications, but also it exhibits an efficient computational performance.

Segmentation of subcellular compartments combining superpixel representation with Voronoi diagrams

Abstract: We propose and implement a computer vision algorithm for detecting individual cells, including the ability to distinguish subcellular compartments, such as nucleus and cytoplasm. Our approach consists of three main steps: (a) cellular mass estimation, and (b) nuclei detection through superpixel representation and, (c) cytoplasm detection through nuclear narrow-band seeding, graph-based region growing and Voronoi diagrams. We test our implementation on both real and simulated cervical cell images, containing an assortment of cells and configurations that often present occlusion and/or poor contrast. Our results show both qualitative and quantitative assessment of the datasets, using a completely automated computer program. The quantitative performance presents average Dice Coefficient equals to 0.85.

A Distributed Deployment Strategy for a Network of Cooperative Autonomous Vehicles

Abstract: This brief presents a distributed deployment algorithm for a network of heterogeneous mobile agents to minimize a prescribed cost function. This function is concerned with the cost of serving the entire field by all agents, where the so called operation cost of different agents are not necessarily the same. The problem is investigated for the case where agents have different types of dynamics. Using a multiplicatively-weighted Voronoi diagram, the field is partitioned to smaller regions (one for each agent). A distributed coverage control law is then provided that guarantees the convergence of agents to the optimal configuration with respect to the above-mentioned cost function. The effectiveness of the proposed algorithm is demonstrated by simulations and experiments on a testbed with two types of unmanned vehicles (aerial and ground).

Generating Optimal Initial Conditions for Smoothed Particle Hydrodynamics Simulations

Abstract: We review existing smoothed particle hydrodynamics setup methods and outline their advantages, limitations, and drawbacks. We present a new method for constructing initial conditions for smoothed particle hydrodynamics simulations, which may also be of interest for N-body simulations, and demonstrate this method on a number of applications. This new method is inspired by adaptive binning techniques using weighted Voronoi tessellations. Particles are placed and iteratively moved based on their proximity to neighbouring particles and the desired spatial resolution. This new method can satisfy arbitrarily complex spatial resolution requirements.

Non-universal Voronoi cell shapes in amorphous ellipsoid packings

Abstract: In particulate systems with short-range interactions, such as granular matter or simple fluids, local structure plays a pivotal role in determining the macroscopic physical properties. Here, we analyse local structure metrics derived from the Voronoi diagram of configurations of oblate ellipsoids, for various aspect ratios $\alpha$ and global volume fractions $\phi_g$. We focus on jammed static configurations of frictional ellipsoids, obtained by tomographic imaging and by discrete element method simulations. In particular, we consider the local packing fraction $\phi_l$, defined as the particle's volume divided by its Voronoi cell volume. We find that the probability $P(\phi_l)$ for a Voronoi cell to have a given local packing fraction shows the same scaling behaviour as function of $\phi_g$ as observed for random sphere packs. Surprisingly, this scaling behaviour is further found to be independent of the particle aspect ratio. By contrast, the typical Voronoi cell shape, quantified by the Minkowski tensor anisotropy index $\beta=\beta_0^{2,0}$, points towards a significant difference between random packings of spheres and those of oblate ellipsoids. While the average cell shape $\beta$ of all cells with a given value of $\phi_l$ is very similar in dense and loose jammed sphere packings, the structure of dense and loose ellipsoid packings differs substantially such that this does not hold true. This non-universality has implications for our understanding of jamming of aspherical particles.

Nonuniform Coverage Control With Stochastic Intermittent Communication

Abstract: We address the nonuniform coverage of a planar region by a platoon of autonomous mobile agents when communications among them are stochastically intermittent. Given a set of generating points and a suitable metric, the solution of the optimal coverage problem is the well known Voronoi tessellation, with a set of mobile agents converging to the centroids of the corresponding Voronoi cells. In the framework of decentralized motion control, this implementation requires that all agents have knowledge of the state of other agents in the platoon. Here we generalize this scenario by considering the optimal area coverage when a group of agents share information in a time varying, stochastically intermittent fashion. We embed on board of each agent a full state estimator that relies on local estimates and on information received by others, when available. We show that under appropriate conditions on the communication network, all agents' estimates asymptotically converge to true states while maximizing the coverage metric despite intermittent communications. The current work has applications in military and civilian domains including harbor protection, perimeter surveillance, and search and rescue missions. Theoretical results are illustrated through computer simulations.

Analysis of spatio-temporal mobile phone data: a case study in the metropolitan area of Milan

Abstract: We analyze geo-referenced high-dimensional data describing the use over time of the mobile-phone network in the urban area of Milan, Italy. Aim of the analysis is to identify subregions of the metropolitan area of Milan sharing a similar pattern along time, and possibly related to activities taking place in specific locations and/or times within the city. To tackle this problem, we develop a non-parametric method for the analysis of spatially dependent functional data, named Bagging Voronoi Treelet analysis. This novel approach integrates the treelet decomposition with a proper treatment of spatial dependence, obtained through a Bagging Voronoi strategy. The latter relies on the aggregation of different replicates of the analysis, each involving a set of functional local representatives associated to random Voronoi-based neighborhoods covering the investigated area. Results clearly point out some interesting temporal patterns interpretable in terms of population density mobility (e.g., daily work activities in the tertiary district, leisure activities in residential areas in the evenings and in the weekend, commuters movements along the highways during rush hours, and localized mob concentrations related to occasional events). Moreover we perform simulation studies, aimed at investigating the properties and performances of the method, and whose description is available online as Supplementary material.

Complex absorbing potentials with Voronoi isosurfaces wrapping perfectly around molecules.

Abstract: Complex absorbing potentials (CAPs) are imaginary potentials that are added to a Hamiltonian to change the boundary conditions of the problem from scattering to square-integrable. In other words, with a CAP, standard bound-state methods can be used in problems involving unbound states such as identifying resonance states and predicting their energies and lifetimes. Although in wave packet dynamics, many CAP forms are used, in electronic structure theory, the so-called box-CAP is used almost exclusively, because of the ease of evaluating its integrals in a Gaussian basis set. However, the box-CAP does has certain disadvantages. First, it will, e.g., break the symmetry of Cnv molecules if n is odd and the main axis is placed along the z-axis by the "standard orientation" of the electronic structure code. Second, it provides a CAP starting at the smallest box around the entire molecular system. For larger molecules or clusters, which do not fill the space efficiently, that implies that much "dead space" within the molecule will be left, where there is neither a CAP nor a sufficient description with basis functions. Here, two new CAP forms are introduced and systematically explored: first, a Voronoi-CAP (that is, a CAP defined in each atom's Voronoi cell), and second, a smooth Voronoi-CAP (which is similar to the Voronoi-CAP; however, the noncontinuously differentiable behavior at the surfaces between the Voronoi cells is smoothed out). Both have isosurfaces that are similar to the cavities used in solvation modeling. An obvious disadvantage of these two CAPs is that the integrals cannot be obtained analytically, but must be computed numerically. However, Voronoi-CAPs share the advantage of having the same symmetry as the molecular system, and, more importantly, considerably facilitate the treatment of larger molecules with asymmetric side chains and of molecular clusters.

BetaCavityWeb: a webserver for molecular voids and channels

Abstract: Molecular cavities, which include voids and channels, are critical for molecular function. We present a webserver, BetaCavityWeb, which computes these cavities for a given molecular structure and a given spherical probe, and reports their geometrical properties: volume, boundary area, buried area, etc. The server's algorithms are based on the Voronoi diagram of atoms and its derivative construct: the beta-complex. The correctness of the computed result and computational efficiency are both mathematically guaranteed. BetaCavityWeb is freely accessible at the Voronoi Diagram Research Center (VDRC) (http://voronoi.hanyang.ac.kr/betacavityweb).

Pressure-field extraction on unstructured flow data using a Voronoi tessellation-based networking algorithm: a proof-of-principle study

Abstract: A novel technique is described for pressure extraction from Lagrangian particle-tracking data. The technique uses a Poisson solver to extract the pressure field on a network of data nodes, which is constructed using the Voronoi tessellation and the Delaunay triangulation. The technique is demonstrated on two cases: synthetic Lagrangian data generated for the analytical case of Hill’s spherical vortex, and the flow in the wake behind a NACA 0012 which was impulsively accelerated to $$Re = 7{,}500$$ . The experimental data were collected using four-camera, three-dimensional particle-tracking velocimetry. For both the analytical case and the experimental case, the dependence of pressure-field error or sensitivity on the normalized spatial particle density was found to follow similar power-law relationships. It was shown that in order to resolve the salient flow structures from experimental data, the required particle density was an order of magnitude greater than for the analytical case. Furthermore, additional sub-structures continued to be identified in the experimental data as the particle density was increased. The increased density requirements of the experimental data were assumed to be due to a combination of phase-averaging error and the presence of turbulent coherent structures in the flow. Additionally, the computational requirements of the technique were assessed. It was found that in the current implementation, the computational requirements are slightly nonlinear with respect to the number of particles. However, the technique will remain feasible even as advancements in particle-tracking techniques in the future increase the density of Lagrangian data.

Rich: open-source hydrodynamic simulation on a moving voronoi mesh

Abstract: We present here RICH, a state-of-the-art two-dimensional hydrodynamic code based on Godunov's method, on an unstructured moving mesh (the acronym stands for Racah Institute Computational Hydrodynamics). This code is largely based on the code AREPO. It differs from AREPO in the interpolation and time-advancement schemeS as well as a novel parallelization scheme based on Voronoi tessellation. Using our code, we study the pros and cons of a moving mesh (in comparison to a static mesh). We also compare its accuracy to other codes. Specifically, we show that our implementation of external sources and time-advancement scheme is more accurate and robust than is AREPO when the mesh is allowed to move. We performed a parameter study of the cell rounding mechanism (Lloyd iterations) and its effects. We find that in most cases a moving mesh gives better results than a static mesh, but it is not universally true. In the case where matter moves in one way and a sound wave is traveling in the other way (such that relative to the grid the wave is not moving) a static mesh gives better results than a moving mesh. We perform an analytic analysis for finite difference schemes that reveals that a Lagrangian simulation is better than a Eulerian simulation in the case of a highly supersonic flow. Moreover, we show that Voronoi-based moving mesh schemes suffer from an error, which is resolution independent, due to inconsistencies between the flux calculation and the change in the area of a cell. Our code is publicly available as open source and designed in an object-oriented, user-friendly way that facilitates incorporation of new algorithms and physical processes.

Multi-Region Active Contours with a Single Level Set Function

Abstract: Segmenting an image into an arbitrary number of coherent regions is at the core of image understanding. Many formulations of the segmentation problem have been suggested over the past years. These formulations include, among others, axiomatic functionals, which are hard to implement and analyze, and graph-based alternatives, which impose a non-geometric metric on the problem. We propose a novel method for segmenting an image into an arbitrary number of regions using an axiomatic variational approach. The proposed method allows to incorporate various generic region appearance models, while avoiding metrication errors. In the suggested framework, the segmentation is performed by level set evolution. Yet, contrarily to most existing methods, here, multiple regions are represented by a single non-negative level set function. The level set function evolution is efficiently executed through the Voronoi Implicit Interface Method for multi-phase interface evolution. The proposed approach is shown to obtain accurate segmentation results for various natural 2D and 3D images, comparable to state-of-the-art image segmentation algorithms.

An Energy-Efficient and Distributed Cooperation Mechanism for k-Coverage Hole Detection and Healing in WSNs

Abstract: Present approaches to achieve k-coverage for Wireless Sensor Networks still rely on centralized techniques. In this paper, we devise a distributed method for this problem, namely Distributed VOronoi based Cooperation scheme (DVOC), where nodes cooperate in hole detection and recovery. In previous Voronoi based schemes, each node only monitors its own critical points. Such methods are inefficient for k-coverage because the critical points are far away from their generating nodes in k-order Voronoi diagram, causing high cost for transmission and computing. As a solution, DVOC enables nodes to monitor others' critical points around themselves by building local Voronoi diagrams (LVDs). Further, DVOC constrains the movement of every node to avoid generating new holes. If a node cannot reach its destination due to the constraint, its hole healing responsibility will fall to other cooperating nodes. The experimental results from the real world testbed demonstrate that DVOC outperforms the previous schemes.

Cell packing structures

Abstract: This paper is an overview of architectural structures which are either composed of polyhedral cells or closely related to them. We introduce the concept of a support structure of such a polyhedral cell packing. It is formed by planar quads and obtained by connecting corresponding vertices in two combinatorially equivalent meshes whose corresponding edges are coplanar and thus determine planar quads. Since corresponding triangle meshes only yield trivial structures, we focus on support structures associated with quad meshes or hex-dominant meshes. For the quadrilateral case, we provide a short survey of recent research which reveals beautiful relations to discrete differential geometry. Those are essential for successfully initializing numerical optimization schemes for the computation of quad-based support structures. Hex-dominant structures may be designed via Voronoi tessellations, power diagrams, sphere packings and various extensions of these concepts. Apart from the obvious application as load-bearing structures, we illustrate here a new application to shading and indirect lighting. On a higher level, our work emphasizes the interplay between geometry, optimization, statics, and manufacturing, with the overall aim of combining form, function and fabrication into novel integrated design tools. Recent and ongoing research in architectural geometry.Links between cell packing structures and discrete differential geometry.Applications, e.g. to shading and indirect lighting.Interplay of geometry, optimization, statics, manufacturing.Combining form, function and fabrication into novel design tools.

An enhanced grain-boundary framework for computational homogenization and micro-cracking simulations of polycrystalline materials

Abstract: An enhanced three-dimensional (3D) framework for computational homogenization and intergranular cracking of polycrystalline materials is presented The framework is aimed at reducing the computational cost of polycrystalline micro simulations, with an aim towards effective multiscale modelling The scheme is based on a recently developed Voronoi cohesive-frictional grain-boundary formulation A regularization scheme is used to avoid excessive mesh refinements often induced by the presence of small edges and surfaces in mathematically exact 3D Voronoi morphologies For homogenization purposes, periodic boundary conditions are enforced on non-prismatic periodic micro representative volume elements ($$\mu $$μRVEs), eliminating pathological grains generally induced by the procedures used to generate prismatic periodic $$\mu $$μRVEs An original meshing strategy is adopted to retain mesh effectiveness without inducing numerical complexities at grain edges and vertices The proposed methodology offers remarkable computational savings and high robustness, both highly desirable in a multiscale perspective The determination of the effective properties of several polycrystalline materials demonstrate the accuracy of the technique Several microcracking simulations complete the study and confirm the performance of the method

Optimal parameterized algorithms for planar facility location problems using Voronoi diagrams

Abstract: We study a general family of facility location problems defined on planar graphs and on the 2-dimensional plane. In these problems, a subset of $k$ objects has to be selected, satisfying certain packing (disjointness) and covering constraints. Our main result is showing that, for each of these problems, the $n^{O(k)}$ time brute force algorithm of selecting $k$ objects can be improved to $n^{O(\sqrt{k})}$ time. The algorithm is based on an idea that was introduced recently in the design of geometric QPTASs, but was not yet used for exact algorithms and for planar graphs. We focus on the Voronoi diagram of a hypothetical solution of $k$ objects, guess a balanced separator cycle of this Voronoi diagram to obtain a set that separates the solution in a balanced way, and then recurse on the resulting subproblems. We complement our study by giving evidence that packing problems have $n^{O(\sqrt{k})}$ time algorithms for a much more general class of objects than covering problems have.

A Robust Delaunay Triangulation Matching for Multispectral/Multidate Remote Sensing Image Registration

Abstract: A novel dual-graph-based matching method is proposed in this letter particularly for the multispectral/multidate images with low overlapping areas, similar patterns, or large transformations. First, scale invariant feature transform based matching is improved by normalizing gradient orientations and maximizing the scale ratio similarity of all corresponding points. Next, Delaunay graphs are generated for outlier removal, and the candidate outliers are selected by comparing the distinction of Delaunay graph structures. In order to bring back the inliers removed in Delaunay triangulation matching iterations and to exclude the remaining outliers, the recovery strategy equipped with the dual graph of Delaunay is explored. Inliers located in the corresponding Voronoi cells are recovered to the residual sets. The experimental results demonstrate the accuracy and robustness of the proposed algorithm for various representative remote sensing images.

Centroidal Power Diagrams, Lloyd's Algorithm, and Applications to Optimal Location Problems

Abstract: In this paper we develop a numerical method for solving a class of optimization problems known as optimal location or quantization problems. The target energy can be written either in terms of atomic measures and the Wasserstein distance or in terms of weighted points and power diagrams (generalized Voronoi diagrams). The latter formulation is more suitable for computation. We show that critical points of the energy are centroidal power diagrams, which are generalizations of centroidal Voronoi tessellations, and that they can be approximated by a generalization of Lloyd's algorithm (Lloyd's algorithm is a common method for finding centroidal Voronoi tessellations). We prove that the algorithm is energy decreasing and prove a convergence theorem. Numerical experiments suggest that the algorithm converges linearly. We illustrate the algorithm in two and three dimensions using simple models of optimal location and crystallization (see online supplementary material).