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Only distances are required to reconstruct submanifolds

Jean-Daniel Boissonnat 1 Ramsay Dyer 2 Arijit Ghosh 3 Steve Y. Oudot 1
1 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
3 Algorithms and Complexity
MPII - Max-Planck-Institut für Informatik
Abstract : In this paper, we give the first algorithm that outputs a faithful reconstruction of a submanifold of Euclidean space without maintaining or even constructing complicated data structures such as Voronoi diagrams or Delaunay complexes. Our algorithm uses the witness complex and relies on the stability of power protection, a notion introduced in this paper. The complexity of the algorithm depends exponentially on the intrinsic dimension of the manifold, rather than the dimension of ambient space, and linearly on the dimension of the ambient space. Another interesting feature of this work is that no explicit coordinates of the points in the point sample is needed. The algorithm only needs the distance matrix as input, i.e., only distance between points in the point sample as input.
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https://hal.inria.fr/hal-01096798
Contributor : Jean-Daniel Boissonnat <>
Submitted on : Thursday, December 18, 2014 - 11:39:47 AM
Last modification on : Tuesday, April 30, 2019 - 5:14:02 PM
Long-term archiving on: : Monday, March 23, 2015 - 4:36:50 PM

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Jean-Daniel Boissonnat, Ramsay Dyer, Arijit Ghosh, Steve Y. Oudot. Only distances are required to reconstruct submanifolds. [Research Report] INRIA Sophia Antipolis. 2014. ⟨hal-01096798⟩

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