Surface area and volume of excursion sets ...
Document type :
Pré-publication ou Document de travail
Title :
Surface area and volume of excursion sets observed on point cloud based polytopic tessellations
Author(s) :
Cotsakis, Ryan [Auteur]
Laboratoire Jean Alexandre Dieudonné [LJAD]
Bernardino, Elena [Auteur]
Laboratoire Jean Alexandre Dieudonné [LJAD]
Duval, Céline [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
Laboratoire Jean Alexandre Dieudonné [LJAD]
Bernardino, Elena [Auteur]
Laboratoire Jean Alexandre Dieudonné [LJAD]
Duval, Céline [Auteur]
Laboratoire Paul Painlevé - UMR 8524 [LPP]
English keyword(s) :
Bias correction
Crofton formula
Crossing probabilities
Excursion sets
Geometric inference
Joint Central Limit Theorem
Lipschitz-Killing curvatures
Surface area
Voronoi tessellations
Crofton formula
Crossing probabilities
Excursion sets
Geometric inference
Joint Central Limit Theorem
Lipschitz-Killing curvatures
Surface area
Voronoi tessellations
HAL domain(s) :
Mathématiques [math]/Probabilités [math.PR]
Mathématiques [math]/Statistiques [math.ST]
Mathématiques [math]/Statistiques [math.ST]
English abstract : [en]
The excursion set of a C2 smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but ...
Show more >The excursion set of a C2 smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension d. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in Rd , including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices.Show less >
Show more >The excursion set of a C2 smooth random field carries relevant information in its various geometric measures. From a computational viewpoint, one never has access to the continuous observation of the excursion set, but rather to observations at discrete points in space. It has been reported that for specific regular lattices of points in dimensions 2 and 3, the usual estimate of the surface area of the excursions remains biased even when the lattice becomes dense in the domain of observation. In the present work, under the key assumptions of stationarity and isotropy, we demonstrate that this limiting bias is invariant to the locations of the observation points. Indeed, we identify an explicit formula for the bias, showing that it only depends on the spatial dimension d. This enables us to define an unbiased estimator for the surface area of excursion sets that are approximated by general tessellations of polytopes in Rd , including Poisson-Voronoi tessellations. We also establish a joint central limit theorem for the surface area and volume estimates of excursion sets observed over hypercubic lattices.Show less >
Language :
Anglais
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