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On the reconstruction of convex sets from random normal measurements

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minkowski.pdf (366.2Kb)
Date
2015
Dewey
Modèles mathématiques. Algorithmes
Sujet
Minkowski problem; surface area measure; random sampling
Journal issue
Discrete and Computational Geometry
Volume
53
Number
3
Publication date
04-2015
Article pages
569-586
DOI
http://dx.doi.org/10.1007/s00454-015-9673-2
URI
https://basepub.dauphine.fr/handle/123456789/17253
Collections
  • CEREMADE : Publications
Metadata
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Author
Abdallah, Hiba
24474 Laboratoire Jean Kuntzmann [LJK]
Mérigot, Quentin
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Article accepté pour publication ou publié
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.

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