Inverting Dirichlet Tessellations

Given a collection of points in the plane, one may draw a cell around each point in such a way that each point’s cell is the portion of the plane consisting of all locations closer to that point than to any of the other points. The resulting geometric figure is called a Dirichlet tessellation. An algorithm for obtaining the boundaries of the cells given the points was derived by Green and Sibson in 1978. Here, methods are described for obtaining the locations of the points, given only the cell boundaries.

[1]  T. Mattfeldt.  Stochastic Geometry and Its Applications , 1996 .

[2]  Adrian Bowyer,et al.  Computing Dirichlet Tessellations , 1981, Comput. J..

[3]  Shmuel Peleg,et al.  Image representation using Voronoi tessellation: adaptive and secure , 1988, Proceedings CVPR '88: The Computer Society Conference on Computer Vision and Pattern Recognition.

[4]  Franz Aurenhammer,et al.  Recognising Polytopical Cell Complexes and Constructing Projection Polyhedra , 1987, J. Symb. Comput..

[5]  D. G. Evans,et al.  Detecting Voronoi (area-of-influence) polygons , 1987 .

[6]  C. Q. Lee,et al.  The Computer Journal , 1958, Nature.

[7]  Robin Sibson,et al.  Computing Dirichlet Tessellations in the Plane , 1978, Comput. J..

[8]  David Hartvigsen,et al.  Recognizing Voronoi Diagrams with Linear Programming , 1992, INFORMS J. Comput..

[9]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[10]  Azriel Rosenfeld,et al.  Hierarchical Image Analysis Using Irregular Tessellations , 1991, IEEE Trans. Pattern Anal. Mach. Intell..

[11]  Ethan D. Bolker,et al.  Recognizing Dirichlet tessellations , 1985 .