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Bicovariograms and Euler characteristic of random fields excursions

Abstract : Let f be a C1 bivariate function with Lipschitz derivatives, and F = {x ∈ R2 : f(x) λ} an upper level set of f, with λ ∈ R. We present a new identity giving the Euler charac- teristic of F in terms of its three-points indicator functions. A bound on the number of connected components of F in terms of the values of f and its gradient, valid in higher dimensions, is also derived. In dimension 2, if f is a random field, this bound allows to pass the former identity to expectations if f’s partial derivatives have Lipschitz constants with finite moments of sufficiently high order, without requiring bounded conditional den- sities. This approach provides an expression of the mean Euler characteristic in terms of the field’s third order marginal. Sufficient conditions and explicit formulas are given for Gaussian fields, relaxing the usual C2 Morse hypothesis.
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Preprints, Working Papers, ...
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Contributor : Raphael Lachieze-Rey <>
Submitted on : Wednesday, November 15, 2017 - 2:21:01 PM
Last modification on : Tuesday, March 24, 2020 - 4:08:29 PM
Long-term archiving on: : Friday, February 16, 2018 - 12:51:22 PM


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  • HAL Id : hal-01207503, version 3
  • ARXIV : 1510.00502


Raphaël Lachièze-Rey. Bicovariograms and Euler characteristic of random fields excursions. 2015. ⟨hal-01207503v3⟩



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