Discrete & Computational Geometry

, Volume 56, Issue 1, pp 93–113 | Cite as

Intrinsic Volumes of Random Cubical Complexes

  • Michael Werman
  • Matthew L. WrightEmail author


Intrinsic volumes, which generalize both Euler characteristic and Lebesgue volume, are important properties of d-dimensional sets. A random cubical complex is a union of unit cubes, each with vertices on a regular cubic lattice, constructed according to some probability model. We analyze and give exact polynomial formulae, dependent on a probability, for the expected value and variance of the intrinsic volumes of several models of random cubical complexes. We then prove a central limit theorem for these intrinsic volumes. For our primary model, we also prove an interleaving theorem for the zeros of the expected-value polynomials. The intrinsic volumes of cubical complexes are useful for understanding the shape of random d-dimensional sets and for characterizing noise in applications.


Intrinsic volume Cubical complex Random complex Euler characteristic 

Mathematics Subject Classification

60D05 52C99 



The authors gratefully acknowledge the support of the Institute for Mathematics and its Applications (IMA). This work was initiated at the IMA during the workshop on Topological Data Analysis in September 2013, and the second author was a postdoctoral fellow during the IMA’s annual program on Scientific and Engineering Applications of Algebraic Topology.


  1. 1.
    Adler, R.J., Taylor, J.E.: Random Fields and Geometry. Springer, New York (2007)zbMATHGoogle Scholar
  2. 2.
    Aizenman, M., Chayes, J.T., Chayes, L., Frhlich, J., Russo, L.: On a sharp transition from area law to a perimeter law in a system of random surfaces. Commun. Math. Phys. 92, 19–69 (1983)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Baryshnikov, Y., Ghrist, R., Wright, M.: Hadwiger’s theorem for definable functions. Adv. Math. 245, 573–586 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Cohen, D., Costa, A., Farber, M., Kappeler, T.: Topology of random 2-complexes. Discrete Comput. Geom. 47(1), 117–149 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Curry, J., Ghrist, R., Robinson, M.: Euler calculus and its applications to signals and sensing. In: Proceedings of Symposia in Applied Mathematics. AMS, Boston (2012)Google Scholar
  6. 6.
    Gray, S.B.: Local properties of binary images in two dimensions. IEEE Trans. Comput. 20(5), 551–561 (1971)CrossRefzbMATHGoogle Scholar
  7. 7.
    Grimmett, G.R., Holroyd, A.E.: Plaquettes, spheres, and entanglement. Electron. J. Probab. 15, 1415–1428 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Guderlei, R., Klenk, S., Mayer, J., Schmidt, V., Spodarev, E.: Algorithms for the computation of the Minkowski functionals of deterministic and random polyconvex sets. Image Vis. Comput. 25(4), 464–474 (2007)CrossRefGoogle Scholar
  9. 9.
    Kahle, M.: Random geometric complexes. Discrete Comput. Geom. 45(3), 553–573 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Kahle, M.: Topology of Random Simplicial Complexes: A Survey. AMS Contemporary Volumes in Mathematics. AMS, Boston (2014)Google Scholar
  11. 11.
    Kahle, M., Meckes, E.: Limit theorems for Betti numbers of random simplicial complexes. Homol. Homot. Appl. 15(1), 343–374 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  12. 12.
    Klain, D.A., Rota, G.-C.: Introduction to Geometric Probability. Cambridge University Press, New York (1997)zbMATHGoogle Scholar
  13. 13.
    Li, X., Mendonça, P.R.S, Bhotika, R.: Texture analysis using Minkowski functionals. In: Proceedings of SPIE Medical Imaging 2012: Image Processing, vol. 8314 (2012)Google Scholar
  14. 14.
    Linial, N., Meshulam, R.: Homological connectivity of random 2-complexes. Combinatorica 26, 475–487 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  15. 15.
    Meshulam, R., Wallach, N.: Homological connectivity of random \(k\)-dimensional complexes. Random Struct. Algorithms 34(3), 408–417 (2000)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Richardson, E., Werman, M.: Efficient classification using the Euler characteristic. Pattern Recognit. Lett. 49, 99–106 (2014)CrossRefGoogle Scholar
  17. 17.
    Ross, N.: Fundamentals of Stein’s method. Probab. Surv. 8, 210–293 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Schladitz, K., Ohser, J., Nagel, W.: Measurement of intrinsic volumes of sets observed on lattices. In: 13th International Conference on Discrete Geometry for Computer Imagery, pp. 247–258 (2006)Google Scholar
  19. 19.
    Schanuel, S.H.: What is the Length of a Potato? Lecture Notes in Mathematics. Springer, Berlin (1986)zbMATHGoogle Scholar
  20. 20.
    Schneider, R., Weil, W.: Stochastic and Integral Geometry. Springer, Berlin (2009)zbMATHGoogle Scholar
  21. 21.
    Svane, A.M.: Estimation of intrinsic volumes from digital grey-scale images. J. Math. Imaging Vis. 49(2), 352–376 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    van den Dries, L.: Tame Topology and O-Minimal Structures. Cambridge University Press, Cambridge (1998)CrossRefzbMATHGoogle Scholar
  23. 23.
    Wright, M.L.: Hadwiger integration of random fields. Topol. Methods Nonlinear Anal. 45(1), 117–128 (2015)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.The Hebrew University of JerusalemJerusalemIsrael
  2. 2.St. Olaf CollegeNorthfieldUSA

Personalised recommendations