Journal of Statistical Physics

, 144:1329 | Cite as

Arak-Clifford-Surgailis Tessellations. Basic Properties and Variance of the Total Edge Length

Article

Abstract

Non-homogeneous random tessellations in the plane with T-shaped vertices are considered, which are defined as Gibbsian modifications of a Poisson line process with arbitrary locally finite intensity measure. In the homogeneous set-up they can be regarded as a specific case of the general Arak-Surgailis polygonal fields in the plane and share some properties with the iteration stable (STIT) tessellations. In the non-homogeneous and anisotropic environment an explicit expression for the partition function is provided and first- and second-order properties of the random length measure of the tessellations restricted to a convex sampling window are derived. In the more special isotropic regime a closed formula for the pair-correlation function and the variance of the total edge length is obtained.

Keywords

Gibbs modification Pair-correlation function Polygonal Markov field Random tessellation Stochastic geometry 

References

  1. 1.
    Arak, T.: On Markovian random fields with finite number of values. In: 4th USSR-Japan Symposium on Probability Theory and Mathematical Statistics. Abstracts of Communications, Tbilisi (1982) Google Scholar
  2. 2.
    Arak, T., Surgailis, D.: Markov fields with polygonal realizations. Probab. Theory Relat. Fields 80, 543–579 (1989) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Arak, T., Surgailis, D.: Consistent polygonal fields. Probab. Theory Relat. Fields 89, 319–346 (1991) MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arak, T., Clifford, P., Surgailis, D.: Point-based polygonal models for random graphs. Adv. Appl. Probab. 25, 348–372 (1993) MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Clifford, P., Nicholls, G.: A Metropolis sampler for polygonal image reconstruction. Available online at http://www.stats.ox.ac.uk/clifford/papers/met_poly.html (1994)
  6. 6.
    Georgii, H.-O.: Gibbs Measures and Phase Transitions. de Gruyter, Berlin (1988) MATHCrossRefGoogle Scholar
  7. 7.
    Kluszczyński, R., van Lieshout, M.N.M., Schreiber, T.: Image segmentation by polygonal Markov fields. Ann. Inst. Stat. Math. 59, 615–625 (2007) Google Scholar
  8. 8.
    Miles, R.E., Mackisack, M.S.: A large class of random tessellations with the classic Poisson polygon distributions. Forma 17, 1–17 (2002) MathSciNetGoogle Scholar
  9. 9.
    Nagel, W., Weiss, V.: Crack STIT tessellations: characterization of stationary random tessellations stable with respect to iteration. Adv. Appl. Probab. 37, 859–883 (2005) MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Schreiber, T.: Non-homogeneous polygonal Markov fields in the plane: graphical representations and geometry of higher order correlations. J. Stat. Phys. 132, 669–705 (2008) MathSciNetADSMATHCrossRefGoogle Scholar
  11. 11.
    Schreiber, T., Thäle, C.: 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, 913–935 (2010) MATHCrossRefGoogle Scholar
  12. 12.
    Schreiber, T., Thäle, C.: Second-order theory for iteration stable tessellations. arXiv:1103.3959 [math.PR] (2011)
  13. 13.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2008) MATHCrossRefGoogle Scholar
  14. 14.
    Stoyan, D., Kendall, D.G., Mecke, J.: Stochastic Geometry and Its Applications, 2nd edn. Wiley, Chichester (1995) MATHGoogle Scholar
  15. 15.
    Surgailis, D.: The thermodynamic limit of polygonal models. Acta Appl. Math. 2, 77–102 (1991) MathSciNetGoogle Scholar
  16. 16.
    Weiss, V., Ohser, J., Nagel, W.: Second moment measure and K-function for planar STIT tessellations. Image Anal. Stereol. 29, 121–131 (2010) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2011

Authors and Affiliations

  1. 1.Institut für Mathematik der Universität OsnabrückOsnabrückGermany

Personalised recommendations