Geometric tomography with topological guarantees

We consider the problem of reconstructing a compact 3manifold (with boundary) embedded in R3 from its crosssections 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 [LBD+ 08] 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. Categories and Subject Descriptors F.2.2 [Nonnumerical Algorithms and Problems]: Geometrical problems and computations; I.3.5 [Computational Geometry and Object Modeling]: Curve, surface, solid, and object representations General Terms Algorithms, Reliability, Theory.

• Type: Conference
• Year: 2010
• Where: COMPGEOM
• Authors: Omid Amini, Jean-Daniel Boissonnat, Pooran Memari
• Keywords: Appropriate Sampling Conditions, COMPGEOM 2010, Liu Et Al, Original Object