Methodology and Computing in Applied Probability

, Volume 15, Issue 3, pp 643–654 | Cite as

On the Arrangement of Cells in Planar STIT and Poisson Line Tessellations

  • Claudia RedenbachEmail author
  • Christoph Thäle


It is well known that the distributions of the interiors of the typical cells of a Poisson line tessellation and a STIT tessellation with the same parameters coincide. In this paper, differences in the arrangement of the cells in these two tessellation models are investigated. In particular, characteristics of the set of cells neighbouring the typical cell are studied, mainly by simulation. Furthermore, the pair-correlation function and several mark correlation functions of the point processes of cell centres are estimated and compared.


Mark-correlation function Neighbourhood of typical cell Pair-correlation function Poisson line tessellation Random tessellation Spatial statistics STIT tessellation Stochastic geometry 

AMS 2000 Subject Classification

60D05 60G55 62M30 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Baddeley A, Turner R (2005) Spatstat: an R package for analyzing spatial point patterns. J Stat Softw 12:1–42Google Scholar
  2. Chiu SN (1994) Mean-value formulae for the neighbourhood of the typical cell of a random tessellation. Adv Appl Probab 26:565–576MathSciNetzbMATHCrossRefGoogle Scholar
  3. Favis W, Weiss V (1998) Mean values of weighted cells of stationary Poisson hyperplane tessellations of \({\Bbb R}^d\). Math Nachr 193:37–48MathSciNetzbMATHCrossRefGoogle Scholar
  4. Møller J (1989) Random tessellations in \({\Bbb R}^d\). Adv Appl Probab 21:37–73CrossRefGoogle Scholar
  5. Nagel W, Weiss V (2005) Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv Appl Probab 37:859–883MathSciNetzbMATHCrossRefGoogle Scholar
  6. Nagel W, Weiss V (2006) STIT tessellations in the plane. Rend Circ Mat Palermo (2) Suppl 77:441–458MathSciNetGoogle Scholar
  7. Redenbach C, Thäle C (2011) Second-order comparison of three fundamental tessellation models. Statistics. doi: 10.1080/02331888.2011.586458 zbMATHGoogle Scholar
  8. Schneider R, Weil W (2008) Stochastic and integral geometry. Springer, BerlinzbMATHCrossRefGoogle Scholar
  9. Schreiber T, Thäle C (2010) Second-order properties and central limit theory for the vertex process of iteration infinitely divisible and iteration stable random tessellations in the plane. Adv Appl Probab 42:913–935zbMATHCrossRefGoogle Scholar
  10. Stoyan D, Kendall WS, Mecke J (1995) Stochastic geometry and its applications, 2nd edn. Wiley, ChichesterzbMATHGoogle Scholar
  11. Voloshin VP, Medvedev NN, Geiger A, Stoyan D (2010) Hydration shells in Voronoi tessellations. In: 2010 international symposium on Voronoi diagrams in science and engineering. IEEE Computer Society, pp 254–259Google Scholar
  12. Weiss V (1995) Second-order quantities for random tessellations of \({\Bbb R}^d\). Stoch Stoch Rep 55:195–205zbMATHCrossRefGoogle Scholar
  13. Weiss V, Ohser J, Nagel W (2010) Second moment measure and K-function for planar STIT tessellations. Image Anal Stereol 29:121–131MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of KaiserslauternKaiserslauternGermany
  2. 2.Department of Mathematics and Computer ScienceUniversity of OsnabrückOsnabrückGermany

Personalised recommendations