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
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.
