Skip to main content
SpringerLink
Log in

Second Order Analysis of Geometric Functionals of Boolean Models

  • Chapter
  • First Online:
Tensor Valuations and Their Applications in Stochastic Geometry and Imaging

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2177))

Abstract

This chapter presents asymptotic covariance formulae and central limit theorems for geometric functionals, including volume, surface area, and all Minkowski functionals and translation invariant Minkowski tensors as prominent examples, of stationary Boolean models. Special focus is put on the anisotropic case. In the (anisotropic) example of aligned rectangles, we provide explicit analytic formulae and compare them with simulation results. We discuss which information about the grain distribution second moments add to the mean values.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. C. Arns, M. Knackstedt, K. Mecke, Reconstructing complex materials via effective grain shapes. Phys. Rev. Lett. 91, 1–4 (2003)

    Article  Google Scholar 

  2. A. Baddeley, A limit theorem for statistics of spatial data. Adv. Appl. Probab. 12, 447–461 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Baryshnikov, J.E. Yukich, Gaussian limits for random measures in geometric probability. Ann. Appl. Probab. 15, 213–253 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  4. U. Brodatzki, K. Mecke, Simulating stochastic geometries: morphology of overlapping grains. Comput. Phys. Commun. 147, 218–221 (2002)

    Article  MATH  Google Scholar 

  5. CGAL, Computational Geometry Algorithms Library. http://www.cgal.org

  6. S.N. Chiu, D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and its Applications, 3rd edn. (Wiley, Chichester, 2013)

    Book  MATH  Google Scholar 

  7. P. Davy, Projected thick sections through multi-dimensional particle aggregates. J. Appl. Probab. 13, 714–722 (1976); Correction: J. Appl. Probab. 15, 456 (1978)

    Google Scholar 

  8. D. Göring, M.A. Klatt, C. Stegmann, K. Mecke, Morphometric analysis in gamma-ray astronomy using Minkowski functionals. Astron. Astrophys. 555, A38 (2013)

    Article  Google Scholar 

  9. P. Hall, Introduction to the Theory of Coverage Processes (Wiley, New York, 1988)

    MATH  Google Scholar 

  10. L. Heinrich, Large deviations of the empirical volume fraction for stationary Poisson grain models. Ann. Appl. Probab. 15, 392–420 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. L. Heinrich, I.S. Molchanov, Central limit theorems for a class of random measures associated with germ-grain models. Adv. Appl. Probab. 31, 283–314 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Hörrmann, The method of densities for non-isotropic Boolean models. Ph.D Thesis, Karlsruhe Institute of Technology (2014)

    Google Scholar 

  13. J. Hörrmann, D. Hug, M.A. Klatt, K. Mecke, Minkowski tensor density formulas for Boolean models. Adv. Appl. Math. 55, 48–85 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  14. D. Hug, Measures, curvatures and currents in convex geometry. Habilitationsschrift (Albert-Ludwigs-Universität Freiburg, Freiburg, 1999)

    Google Scholar 

  15. D. Hug, J. Rataj, Mixed curvature measures of translative integral geometry. (to appear). arXiv:1606.04224

    Google Scholar 

  16. D. Hug, G. Last, M. Schulte, Second order properties and central limit theorems for geometric functionals of Boolean. Ann. Appl. Probab. 26, 73–135 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Kerscher, K. Mecke, J. Schmalzing, C. Beisbart, T. Buchert, H. Wagner, Morphological fluctuations of large-scale structure: the PSCz survey. Astron. Astrophys. 373, 1–11 (2001)

    Article  Google Scholar 

  18. M.A. Klatt, Morphometry of random spatial structures in physics. Ph.D Thesis, FAU Erlangen-Nürnberg (2016)

    Google Scholar 

  19. M.A. Klatt, D. Göring, C. Stegmann, K. Mecke, Shape analysis of counts maps. AIP Conf. Proc. 1505, 737–740 (2012)

    Article  Google Scholar 

  20. G. Last, M.D. Penrose, Fock space representation, chaos expansion and covariance inequalities for general Poisson processes. Probab. Theory Relat. Fields 150, 663–690 (2011)

    Article  MATH  Google Scholar 

  21. G. Last, M.D. Penrose, Lectures on the Poisson Process (Cambridge University Press, Cambridge, to appear). http://www.math.kit.edu/stoch/last/seite/lehrbuch_poissonp/de

  22. G. Last, G. Peccati, M. Schulte, Normal approximation on Poisson spaces: Mehler’s formula, second order Poincaré inequalities and stabilization. Probab. Theory Relat. Fields 165, 667–723 (2016)

    Article  MATH  Google Scholar 

  23. S. Mase, Asymptotic properties of stereological estimators for stationary random sets. J. Appl. Probab. 19, 111–126 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  24. K.R. Mecke, Additivity, convexity, and beyond: applications of Minkowski functionals in statistical physics, in Statistical Physics and Spatial Statistics, ed. by K.R. Mecke, D. Stoyan. Lecture Notes in Physics, vol. 554 (Springer, Berlin/Heidelberg, 2000), pp. 111–184

    Google Scholar 

  25. K.R. Mecke, Exact moments of curvature measures in the Boolean model. J. Stat. Phys. 102, 1343–1381 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  26. K.R. Mecke, T. Buchert, H. Wagner, Robust morphological measures for large-scale structure in the universe. Astron. Astrophys. 288, 697–704 (1994)

    Google Scholar 

  27. K.R. Mecke, A. Seyfried, Strong dependence of percolation thresholds on polydispersity. Europhys. Lett. 58, 28–34 (2002)

    Article  Google Scholar 

  28. K.R. Mecke, D. Stoyan, Morphological characterization of point patterns. Biometrical J. 47, 473–488 (2005)

    Article  MathSciNet  Google Scholar 

  29. K. Mecke, H. Wagner, Euler characteristic and related measures for random geometric sets. J. Stat. Phys. 64, 843–850 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. R. Meester, R. Roy, Continuum Percolation (Cambridge University Press, Cambridge, 1996)

    Book  MATH  Google Scholar 

  31. R.E. Miles, Estimating aggregate and overall characteristics from thick sections by transmission microscopy. J. Microsc. 107, 227–233 (1976)

    Article  Google Scholar 

  32. I.S. Molchanov, Statistics of the Boolean model: from the estimation of means to the estimation of distributions. Adv. Appl. Probab. 21, 63–86 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  33. I.S. Molchanov, Statistics of the Boolean Model for Practitioners and Mathematicians (Wiley, Chichester, 1996)

    MATH  Google Scholar 

  34. G. Peccati, J.L. Solé, M.S. Taqqu, F. Utzet, Stein’s method and normal approximation of Poisson functionals. Ann. Probab. 38, 443–478 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. M.D. Penrose, Gaussian limits for random geometric measures. Electron. J. Probab. 12, 989–1035 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  36. R. Schneider, Convex Bodies: The Brunn-Minkowski Theory. Encyclopedia of Mathematics and Its Applications, vol. 151, 2nd edn. (Cambridge University Press, Cambridge, 2014)

    Google Scholar 

  37. R. Schneider, W. Weil, Stochastic and Integral Geometry (Springer, Berlin, 2008)

    Book  MATH  Google Scholar 

  38. C. Scholz, F. Wirner, M.A. Klatt, D. Hirneise, G.E. Schröder-Turk, K. Mecke, C. Bechinger, Direct relations between morphology and transport in Boolean models. Phys. Rev. E 92, 043023 (2015)

    Article  Google Scholar 

  39. G.E. Schröder-Turk, S.C. Kapfer, B. Breidenbach, C. Beisbart, K. Mecke, Tensorial Minkowski functionals and anisotropy measures for planar patterns. J. Microsc. 238, 57–74 (2010)

    Article  MathSciNet  Google Scholar 

  40. G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, M.A. Klatt, F.M. Schaller, M.J.F. Hoffmann, N. Kleppmann, P. Armstrong, A. Inayat, D. Hug, M. Reichelsdorfer, W. Peukert, W. Schwieger, K. Mecke, Minkowski tensor shape analysis of cellular, granular and porous structures. Adv. Mater. 23, 2535–2553 (2011)

    Article  Google Scholar 

  41. G.E. Schröder-Turk, W. Mickel, S.C. Kapfer, F.M. Schaller, B. Breidenbach, D. Hug, K. Mecke, Minkowski tensors of anisotropic spatial structure. New J. Phys. 15, 083028 (2013)

    Article  MathSciNet  Google Scholar 

  42. S. Torquato, Random Heterogeneous Materials (Springer, Heidelberg, 2002)

    Book  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank Julia Schulte and Klaus Mecke for some helpful remarks and discussions. The authors were supported by the German Research Foundation (DFG) through the research unit “Geometry and Physics of Spatial Random Systems”.

Author information

Authors and Affiliations

  1. Department of Mathematics, Karlsruhe Institute of Technology, Englerstraße 2, 76131, Karlsruhe, Germany

    Daniel Hug, Michael A. Klatt & Günter Last

  2. Department of Mathematics and Statistics, Institute of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, 3012, Bern, Switzerland

    Matthias Schulte

Authors
  1. Daniel Hug
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Michael A. Klatt
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Günter Last
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Matthias Schulte
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel Hug .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Aarhus University, Aarhus C, Denmark

    Eva B. Vedel Jensen

  2. Department of Mathematics, Aarhus University, Aarhus C, Denmark

    Markus Kiderlen

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Hug, D., Klatt, M.A., Last, G., Schulte, M. (2017). Second Order Analysis of Geometric Functionals of Boolean Models. In: Jensen, E., Kiderlen, M. (eds) Tensor Valuations and Their Applications in Stochastic Geometry and Imaging. Lecture Notes in Mathematics, vol 2177. Springer, Cham. https://doi.org/10.1007/978-3-319-51951-7_12

Download citation

Publish with us

Policies and ethics

Search

Navigation