Voronoi diagram generation on the ellipsoidal earth

Voronoi diagram on the earth surface is a powerful tool to study spatial proximity at continental or global scale. However, its computation remains challenging because geospatial features have complex shapes. This paper presents a raster-based algorithm to generate Voronoi diagrams on earth's surface. The algorithm approximates the exact point-to-point geographical distances using the cell-to-cell geographical distances calculated by a geographical distance transform. The result is a distance image on which Voronoi diagram is delineated. Compared to existing methods, the proposed algorithm calculates geographical distances based on an earth ellipsoid and allows Voronoi generators to take complex shapes. Most importantly, its approximation error is bounded thus enabling users to control the accuracy of the Voronoi diagram through grid resolution. We developed a raster-based method to generate Voronoi diagrams on the earth.The method is based on geographical distance transform.Voronoi generators are allowed to have any arbitrary shape.The accuracy of the Voronoi diagram can be controllable through grid resolution.

[1]  Kevin Q. Brown.  Geometric transforms for fast geometric algorithms , 1979 .

[2]  Michael Andrew Christie,et al.  Prediction under uncertainty in reservoir modeling , 2002 .

[3]  Gunilla Borgefors,et al.  Distance transformations in digital images , 1986, Comput. Vis. Graph. Image Process..

[4]  Clifford H. Thurber,et al.  Adaptive mesh seismic tomography based on tetrahedral and Voronoi diagrams: Application to Parkfield, California , 2005 .

[5]  Kazuo Murota,et al.  A fast Voronoi-diagram algorithm with applications to geographical optimization problems , 1984 .

[6]  Jun Chen,et al.  An Algorithm for the Generation of Voronoi Diagrams on the Sphere Based on QTM , 2003 .

[7]  M. Gahegan,et al.  Data structures and algorithms to support interactive spatial analysis using dynamic Voronoi diagrams , 2000 .

[8]  Jun Chen,et al.  A raster-based method for computing Voronoi diagrams of spatial objects using dynamic distance transformation , 1999, Int. J. Geogr. Inf. Sci..

[9]  Yuan Tian,et al.  A vector-based algorithm to generate and update multiplicatively weighted Voronoi diagrams for points, polylines, and polygons , 2012, Comput. Geosci..

[10]  Frank Y. Shih,et al.  Fast Euclidean distance transformation in two scans using a 3 × 3 neighborhood , 2004, Comput. Vis. Image Underst..

[11]  M. Sambridge,et al.  On entropy and clustering in earthquake hypocentre distributions , 2000 .

[12]  Nicole M. Gasparini,et al.  An object-oriented framework for distributed hydrologic and geomorphic modeling using triangulated irregular networks , 2001 .

[13]  Franz Aurenhammer,et al.  Voronoi diagrams—a survey of a fundamental geometric data structure , 1991, CSUR.

[14]  Yu Liu,et al.  Probability issues in locality descriptions based on Voronoi neighbor relationship , 2012, J. Vis. Lang. Comput..

[15]  S. Subbeya,et al.  Prediction under uncertainty in reservoir modeling , 2004 .

[16]  Xiao Wang,et al.  Population landscape: a geometric approach to studying spatial patterns of the US urban hierarchy , 2006, Int. J. Geogr. Inf. Sci..

[17]  Pinliang Dong,et al.  Generating and updating multiplicatively weighted Voronoi diagrams for point, line and polygon features in GIS , 2008, Comput. Geosci..

[18]  Robert J. Renka,et al.  Algorithm 772: STRIPACK: Delaunay triangulation and Voronoi diagram on the surface of a sphere , 1997, TOMS.

[19]  Charles S. Peskin,et al.  On the construction of the Voronoi mesh on a sphere , 1985 .

[20]  Benoit M. Macq,et al.  Fast Euclidean Distance Transformation by Propagation Using Multiple Neighborhoods , 1999, Comput. Vis. Image Underst..

[21]  K. Sugihara.  Laguerre Voronoi Diagram on the Sphere , 2002 .

[22]  Atsuyuki Okabe,et al.  Spatial Tessellations: Concepts and Applications of Voronoi Diagrams , 1992, Wiley Series in Probability and Mathematical Statistics.

[23]  J. Boissonnat,et al.  3D volumetric modelling of Cadomian terranes (Northern Brittany, France): an automatic method using Voronoı̈ diagrams , 2001 .

[24]  K. Sahr,et al.  Geodesic Discrete Global Grid Systems , 2003 .

[25]  Barry Boots,et al.  Techniques for accuracy assessment of tree locations extracted from remotely sensed imagery. , 2005, Journal of environmental management.

[26]  W. Featherstone,et al.  Validation of Vincenty's Formulas for the Geodesic Using a New Fourth-Order Extension of Kivioja's Formula , 2005 .

[27]  Marina L. Gavrilova,et al.  Toward 3D spatial dynamic field simulation within GIS using kinetic Voronoi diagram and Delaunay tetrahedralization , 2011, Int. J. Geogr. Inf. Sci..

[28]  Otfried Cheong,et al.  Voronoi diagrams on the spher , 2002, Comput. Geom..

[29]  Todd D. Ringler,et al.  A multiresolution method for climate system modeling: application of spherical centroidal Voronoi tessellations , 2008 .

[30]  C. Gold.  Problems with handling spatial data ― the Voronoi approach , 1991 .

[31]  Luciano da Fontoura Costa,et al.  2D Euclidean distance transform algorithms: A comparative survey , 2008, CSUR.

[32]  T. Vincenty.  DIRECT AND INVERSE SOLUTIONS OF GEODESICS ON THE ELLIPSOID WITH APPLICATION OF NESTED EQUATIONS , 1975 .

[33]  Qiang Du,et al.  Constrained Centroidal Voronoi Tessellations for Surfaces , 2002, SIAM J. Sci. Comput..