On the reconstruction of convex sets from random normal measurements

Hiba Abdallah (1) , Quentin Mérigot (2, 1)

1 MGMI - Modélisation Géométrique & Multirésolution pour l'Image
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision

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.

Mots clés: Minkowski problem; surface area measure; random sampling

Domaines: Géométrie algorithmique [cs.CG]

Hiba Abdallah, Quentin Mérigot. On the reconstruction of convex sets from random normal measurements. Discrete and Computational Geometry, 2015, 53 (3), pp.569-586. ⟨10.1007/s00454-015-9673-2⟩. ⟨hal-00950000⟩