On the Reconstruction of Convex Sets from Random Normal Measurements

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 Hausdor...

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Bibliographic Details
Published in:Discrete & computational geometry 2015-04, Vol.53 (3), p.569-586
Main Authors: Abdallah, Hiba, Mérigot, Quentin
Format: Article
Language:English
Subjects:
Boundaries
Combinatorics
Computational Geometry
Computational mathematics
Computational Mathematics and Numerical Analysis
Computer Science
Errors
Mathematical analysis
Mathematics
Mathematics and Statistics
Reconstruction
Stability
Texts
Upper bounds
Vector space
Vectors (mathematics)
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Summary: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 η , we provide upper bounds on the number of probes that one has to perform in order to obtain an η -approximation of this convex set with high probability. Our result relies on the stability theory related to Minkowski’s theorem.
ISSN:0179-5376
1432-0444
DOI:10.1007/s00454-015-9673-2