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Geometric Tomography With Topological Guarantees

Omid Amini 1 Jean-Daniel Boissonnat 2 Pooran Memari 2, *
* Corresponding author
2 GEOMETRICA - Geometric computing
CRISAM - Inria Sophia Antipolis - Méditerranée , Inria Saclay - Ile de France
Abstract : We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in $\mathbb{R}^3$ from its cross-sections with a given set of cutting planes having arbitrary orientations. Under appropriate sampling conditions that are satisfied when the set of cutting planes is dense enough, we prove that the algorithm presented by Liu et al. in [1] preserves the homotopy type of the object. Using the homotopy equivalence, we also show that the reconstructed object is homeomorphic (and isotopic) to the original object. This is the first time that 3D shape reconstruction from cross-sections comes with such theoretical guarantees. [1] L. Liu, C.L. Bajaj, J.O. Deasy, D.A. Low, and T. Ju. Surface reconstruction from non-parallel curve networks. Computer Graphics Forum, 27:155-163, 2008.
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Submitted on : Thursday, December 10, 2009 - 12:07:17 PM
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Omid Amini, Jean-Daniel Boissonnat, Pooran Memari. Geometric Tomography With Topological Guarantees. [Research Report] RR-7147, INRIA. 2009, pp.26. ⟨inria-00440322⟩

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