The paper introduces a Morse-graph formalism that yields approximate, one-parameter-fitted formulas for expected Betti numbers of 2D Gaussian random fields and shows these homology statistics are sensitive to local non-Gaussianity.
Betti number signatures of homogeneous Poisson point processes
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abstract
The Betti numbers are fundamental topological quantities that describe the k-dimensional connectivity of an object: B_0 is the number of connected components and B_k effectively counts the number of k-dimensional holes. Although they are appealing natural descriptors of shape, the higher-order Betti numbers are more difficult to compute than other measures and so have not previously been studied per se in the context of stochastic geometry or statistical physics. As a mathematically tractable model, we consider the expected Betti numbers per unit volume of Poisson-centred spheres with radius alpha. We present results from simulations and derive analytic expressions for the low intensity, small radius limits of Betti numbers in one, two, and three dimensions. The algorithms and analysis depend on alpha-shapes, a construction from computational geometry that deserves to be more widely known in the physics community.
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2019 1verdicts
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Stochastic Homology of Gaussian vs. non-Gaussian Random Fields: Graphs towards Betti Numbers and Persistence Diagrams
The paper introduces a Morse-graph formalism that yields approximate, one-parameter-fitted formulas for expected Betti numbers of 2D Gaussian random fields and shows these homology statistics are sensitive to local non-Gaussianity.