Geometric Tomography With Topological Guarantees

Omid Amini (1, 2) , Jean-Daniel Boissonnat (1, 2) , Pooran Memari (1, 2)

1 DMA - Département de Mathématiques et Applications - ENS Paris
2 GEOMETRICA - Geometric computing

We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in R3 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. preserves the homotopy type of the original 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 shape reconstruction from cross-sections comes with such theoretical guarantees.

Keywords: Computational geometry; Computational topology; surface reconstruction; geometric tomography

Domains: Computational Geometry [cs.CG]

Omid Amini, Jean-Daniel Boissonnat, Pooran Memari. Geometric Tomography With Topological Guarantees. Symposium on Computational Geometry, Jun 2010, Snowbird, United States. pp.200. ⟨hal-00487884⟩