Intrinsic Volumes of Random Cubical Complexes
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.
KeywordsIntrinsic volume Cubical complex Random complex Euler characteristic
Mathematics Subject Classification60D05 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.
- 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
- 10.Kahle, M.: Topology of Random Simplicial Complexes: A Survey. AMS Contemporary Volumes in Mathematics. AMS, Boston (2014)Google Scholar
- 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
- 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