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Geometric Covariograms, Indicator Variograms and Boundaries of Planar Closed Sets

Abstract : In the plane, consider a compact set with nonempty interior and piecewise C1 boundary such that any line segment crosses this boundary at only finitely many points. Then the second derivatives of the geometric covariogram of the compact set at the origin are related to geometrical characteristics of the set boundary. Specifically, the mean second derivative is zero when the boundary has no singular point, it is positive when the boundary has finitely many vertices but no cusp, and it is infinite when the boundary has at least one cusp. Similar results hold for a stationary random closed set with nonempty interior and piecewise C1 boundary such that any line segment crosses this boundary at almost surely finitely many points. For a set with almost surely no singular boundary point, the mean second derivative of its indicator variogram at the origin is zero. In contrast, when there is a positive probability that, in a given bounded domain of the plane, the boundary has finitely many singular points, this mean second derivative is negative, finite if all the singular points are vertices and infinite if some singular points are cusps.
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Contributor : Christian Lantuéjoul <>
Submitted on : Monday, June 18, 2012 - 11:06:48 AM
Last modification on : Thursday, September 24, 2020 - 4:34:05 PM

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Christian Lantuéjoul, Xavier Emery. Geometric Covariograms, Indicator Variograms and Boundaries of Planar Closed Sets. Mathematical Geology, Springer Verlag, 2011, 43, pp.905-927. ⟨10.1007/s11004-011-9364-3⟩. ⟨hal-00709210⟩

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