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On the reconstruction of convex sets from random normal measurements

Abdallah, Hiba; Mérigot, Quentin (2015), On the reconstruction of convex sets from random normal measurements, Discrete and Computational Geometry, 53, 3, p. 569-586. 10.1007/s00454-015-9673-2

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Type
Article accepté pour publication ou publié
Date
2015
Journal name
Discrete and Computational Geometry
Volume
53
Number
3
Pages
569-586
Publication identifier
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Author(s)
Abdallah, Hiba
Laboratoire Jean Kuntzmann [LJK]
Mérigot, Quentin
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
We study the problem of reconstructing a convex body using only a finite number of measurements of outer normal vectors. More precisely, we suppose that the normal vectors are measured at independent random locations uniformly distributed along the boundary of our convex set. Given a desired Hausdorff error $\eta$, we provide an upper bounds on the number of probes that one has to perform in order to obtain an $\eta$-approximation of this convex set with high probability. Our result rely on the stability theory related to Minkowski's theorem.
Subjects / Keywords
Minkowski problem; surface area measure; random sampling