Advances in Studies and Applications of Centroidal Voronoi Tessellations

<正>Centroidal Voronoi tessellations(CVTs) have become a useful tool in many applications ranging from geometric modeling,image and data analysis,and numerical partial differential equations,to problems in physics,astrophysics,chemistry,and biology. In this paper,we briefly review the CVT concept and a few of its generalizations and well-known properties.We then present an overview of recent advances in both mathematical and computational studies and in practical applications of CVTs.Whenever possible,we point out some outstanding issues that still need investigating.     

2D Centroidal Voronoi Tessellations with Constraints

<正>We tackle the problem of constructing 2D centroidal Voronoi tessellations with constraints through an efficient and robust construction of bounded Voronoi diagrams, the pseudo-dual of the constrained Delaunay triangulation.We exploit the fact that the cells of the bounded Voronoi diagram can be obtained by clipping the ordinary ones against the constrained Delaunay edges.The clipping itself is efficiently computed by identifying for each constrained edge the(connected) set of triangles whose dual Voronoi vertices are hidden by the constraint.The resulting construction is amenable to Lloyd relaxation so as to obtain a centroidal tessellation with constraints.     

Cooperative Sensing and Distributed Control of a Diffusion Process Using Centroidal Voronoi Tessellations

<正>This paper considers how to use a group of robots to sense and control a diffusion process.The diffusion process is modeled by a partial differential equation (PDE),which is a both spatially and temporally variant system.The robots can serve as mobile sensors,actuators,or both.Centroidal Voronoi Tessellations based coverage control algorithm is proposed for the cooperative sensing task.For the diffusion control problem,this paper considers spraying control via a group of networked mobile robots equipped with chemical neutralizers,known as smart mobile sprayers or actuators,in a domain of interest having static mesh sensor network for concentration sensing.This paper also introduces the information sharing and consensus strategy when using centroidal Voronoi tessellations algorithm to control a diffusion process.The information is shared not only on where to spray but also on how much to spray among the mobile actuators.Benefits from using CVT and information consensus seeking for sensing and control of a diffusion process are demonstrated in simulation results.     

Truncated Newton-Based Multigrid Algorithm for Centroidal Voronoi Diagram Calculation

In a variety of modern applications there arises a need to tessellate the domain into representative regions, called Voronoi cells. A particular type of such tessellations, called centroidal Voronoi tessellations or CVTs, are in big demand due to their optimality properties important for many applications. The availability of fast and reliable algorithms for their construction is crucial for their successful use in practical settings. This paper introduces a new multigrid algorithm for constructing CVTs that is based on the MG/Opt algorithm that was originally designed to solve large nonlinear optimization problems. Uniform convergence of the new method and its speedup comparing to existing techniques are demonstrated for linear and nonlinear densities for several 1d and 2d problems, and $O(k)$ complexity estimation is provided for a problem with $k$ generators.     

Bernoulli Cluster Field: Voronoi Tessellations

A new point process is proposed which can be viewed either as a Boolean cluster model with two cluster modes or as a p-thinned Neyman-Scott cluster process with the retention of the original parent point. Voronoi tessellation generated by such a point process has extremely high coefficients of variation of cell volumes as well as of profile areas and lengths in the planar and line induced tessellations. An approximate numerical model of tessellation characteristics is developed for the case of small cluster size; its predictions are compared with the results of computer simulations. Tessellations of this type can be used as models of grain structures in steels.     

Asymptotics for Statistical Distances Based on Voronoi Tessellations

We obtain an information-type inequality and a strong law for a wide class of statistical distances between empirical estimates and random measures based on Voronoi tessellations. This extends some basic results in the asymptotic theory of sample spacings, when the cells of the Voronoi tessellation are interpreted as d-dimensional spacings.     

Fast Multilevel CVT-Based Adaptive Data Visualization Algorithm

<正>Efficient data visualization techniques are critical for many scientific applications. Centroidal Voronoi tessellation(CVT) based algorithms offer a convenient vehicle for performing image analysis,segmentation and compression while allowing to optimize retained image quality with respect to a given metric.In experimental science with data counts following Poisson distributions,several CVT-based data tessellation algorithms have been recently developed.Although they surpass their predecessors in robustness and quality of reconstructed data,time consumption remains to be an issue due to heavy utilization of the slowly converging Lloyd iteration.This paper discusses one possible approach to accelerating data visualization algorithms.It relies on a multidimensional generalization of the optimization based multilevel algorithm for the numerical computation of the CVTs introduced in[1],where a rigorous proof of its uniform convergence has been presented in 1-dimensional setting.The multidimensional implementation employs barycentric coordinate based interpolation and maximal independent set coarsening procedures.It is shown that when coupled with bin accretion algorithm accounting for the discrete nature of the data,the algorithm outperforms Lloyd-based schemes and preserves uniform convergence with respect to the problem size.Although numerical demonstrations provided are limited to spectroscopy data analysis,the method has a context-independent setup and can potentially deliver significant speedup to other scientific and engineering applications.     

Approximations of a Ginzburg-Landau model for superconducting hollow spheres based on spherical centroidal Voronoi tessellations

In this paper the numerical approximations of the Ginzburg- Landau model for a superconducting hollow spheres are constructed using a gauge invariant discretization on spherical centroidal Voronoi tessellations. A reduced model equation is used on the surface of the sphere which is valid in the thin spherical shell limit. We present the numerical algorithms and their theoretical convergence as well as interesting numerical results on the vortex configurations. Properties of the spherical centroidal Voronoi tessellations are also utilized to provide a high resolution scheme for computing the supercurrent and the induced magnetic field.

     

On the spectral function of the Poisson–Voronoi cells

Denote by (t)=∑_n1e^−λ_nt, t>0, the spectral function related to the Dirichlet Laplacian for the typical cell of a standard Poisson–Voronoi tessellation in . We show that the expectation E(t), t>0, is a functional of the convex hull of a standard d-dimensional Brownian bridge. This enables us to study the asymptotic behaviour of E(t), when t→0⁺,+∞. In particular, we prove that the law of the first eigenvalue λ₁ of satisfies the asymptotic relation lnP{λ₁t}−2^dω_dj_(d−2)/2^d·t^−d/2 when t→0⁺, where ω_d and j_(d−2)/2 are respectively the Lebesgue measure of the unit ball in and the first zero of the Bessel function J_(d−2)/2.     

一类非线性方程解的存在和唯一性及在图像中的应用

本文我们给出一个修正的非线性扩散方程模型，与Cotte Lions和Morel的模型相比该模型有许多实质上的优点。主要的想法是把原来去噪声部分：卷积Gauss过程替代为解一个有界区域上的线性抛物方程问题，因此避开了对初始数值如何全平面延拓的问题。我们从数学上的证明该问题解的存在性和适定性，同时给出对矩形域情况的解的级数形式。最后我们给基于本模型的数值计算差分模型，并且给出几个具体图像在该模型下处理结果。     

基于三角隶属函数和模糊熵的图像增强方法及其相关应用研究

鉴于图像增强技术在生活应用中的重要性,模糊技术在图像应用中的实用性和广泛性,提出了一种基于三角隶属函数和模糊熵的新的图像增强算法(T-FE增强算法),使用三角函数作为隶属函数,重构参数型对比增强算子,运用模糊熵最大原则选取阈值,计算快速,简单.并且将T-FE算法运用于图像分割,边缘检测.通过实验仿真表明,T-FE算法在进行图像处理时有较好效果.     

Reconstruction of signals from multiscale edges

We present a method for signal reconstruction based upon the location of its singularities which are identified by a wavelet transform technique. The reconstructed signal is an infconvolution spline approximant. The data for the interpolation problem which we solve comes from a compact signal coding procedure. Error bounds and convergence results for the calculated signal are given.     

基于Canny算子边缘检测的小波阈值去噪方法

本文研究了信号处理中图像去噪的问题.利用小波变换理论提出了一种基于Canny算子边缘检测的小波阈值去噪方法,实验结果表明,该方法在有效去除噪声的同时能够更好地保留图像的边缘.     

On a Variational Model for Selective Image Segmentation of Features with Infinite Perimeter

Variational models provide reliable formulation for segmentation of features and their boundaries in an image, following the seminal work of Mumford-Shah (1989, Commun. Pure Appl. Math.) on dividing a general surface into piecewise smooth sub-surfaces. A central idea of models based on this work is to minimize the length of feature’s boundaries (i.e., H1 Hausdorff measure). However there exist problems with irregular and oscillatory object boundaries, where minimizing such a length is not appropriate, as noted by Barchiesi et al. (2010, SIAM J. Multiscale Model. Simu.) who proposed to miminize L2 Lebesgue measure of the γ-neighborhood of the boundaries. This paper presents a dual level set selective segmentation model based on Barchiesi et al. (2010) to automatically select a local feature instead of all global features. Our model uses two level set functions: a global level set which segments all boundaries, and the local level set which evolves and finds the boundary of the object closest to the geometric constraints. Using real life images with oscillatory boundaries, we show qualitative results demonstrating the effectiveness of the proposed method.     

连续小波在医学图像处理中的一种应用

本文构造了一类具有紧支撑的无限次可导的对称小波，提出了适应医学图像处理的边缘检测算法，并给出了计算实例。     

基于上下文光谱角映射的无监督图像变化检测方法

对传统遥感图像变化检测方法未充分利用像素上下文信息的问题,提出一种无需关于像素的概率分布假设、基于上下文光谱角映射的无监督图像变化检测方法.方法避免了在图像分析过程中将像素看作独立单元,通过引入图像的空间上下文信息特征,在对像素变化类别的判别测试阶段加以利用,从而达到提高变化检测精确度的效果.对卫星图像的实施过程首先利用上下文光谱角映射创建一幅相似图像,然后用K均值聚类算法将其分为有变化和无变化的两类来生成映射图像.实验从定量和定性的两方面与最大似然估计法(MLC)比较,结果显示所提方法比已有方法有所改进,对于二时刻图像和多光谱图像的变化检测问题有更好的适用性.     

On the \Gamma-limit of the Mumford-Shah functional

We study by means of -convergence the asymptotics of the rescaled Mumford-Shah functional