Abstract
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.
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Notes
Intrinsic volumes known by various names, including Hadwiger measures, Minkowski functionals, and quermassintegrale. These concepts are equivalent up to normalization.
Valuations are not measures, though the concepts are similar. In particular, the intrinsic volumes may take on negative values and are not monotonic.
The combinatorial Euler characteristic is not a homotopy invariant, but on compact sets it agrees with the topological Euler characteristic (which has value 1 on any contractible set).
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Acknowledgments
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.
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Werman, M., Wright, M.L. Intrinsic Volumes of Random Cubical Complexes. Discrete Comput Geom 56, 93–113 (2016). https://doi.org/10.1007/s00454-016-9789-z
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DOI: https://doi.org/10.1007/s00454-016-9789-z