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,astroph...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.展开更多
Most existing applications of centroidal Voronoi tessellations(CVTs) lack consideration of the length of the cluster boundaries.In this paper we propose a new model and algorithms to produce segmentations which would ...Most existing applications of centroidal Voronoi tessellations(CVTs) lack consideration of the length of the cluster boundaries.In this paper we propose a new model and algorithms to produce segmentations which would minimize the total energy—a sum of the classic CVT energy and the weighted length of cluster boundaries.To distinguish it with the classic CVTs,we call it an Edge-Weighted CVT(EWCVT).The concept of EWCVT is expected to build a mathematical base for all CVT related data classifications with requirement of smoothness of the cluster boundaries.The EWCVT method is easy in implementation,fast in computation,and natural for any number of clusters.展开更多
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 co...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.展开更多
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...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.展开更多
基金supported by the US Department of Energy Office of Science Climate Change Prediction Program through grant numbers DE-FG02-07ER64431 and DE-FG02-07ER64432the US National Science Foundation under grant numbers DMS-0609575 and DMS-0913491
文摘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.
基金supported in part by the U.S.National Science Foundation under grant number DMS-0913491.
文摘Most existing applications of centroidal Voronoi tessellations(CVTs) lack consideration of the length of the cluster boundaries.In this paper we propose a new model and algorithms to produce segmentations which would minimize the total energy—a sum of the classic CVT energy and the weighted length of cluster boundaries.To distinguish it with the classic CVTs,we call it an Edge-Weighted CVT(EWCVT).The concept of EWCVT is expected to build a mathematical base for all CVT related data classifications with requirement of smoothness of the cluster boundaries.The EWCVT method is easy in implementation,fast in computation,and natural for any number of clusters.
基金supported by the grants DMS 0405343 and DMR 0520425.
文摘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.
基金supported by the National Natural Science Foundation of China (No. 60773179)the National Basic Research Program (973) of China (No. 2004CB318000)
文摘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.
