Geometric Tomography With Topological Guarantees

Details Geometric Tomography With Topological Guarantees Geometric Tomography With Topological Guarantees

2010-07-14

arxiv.org/abs/1007.2452v1

Computer Science

Computational Geometry

25 Pages, 21 Figure; Extended abstract appeared in the proceedings of the 26th Symposium on Computational Geometry 2010

Scientific paper

We consider the problem of reconstructing a compact 3-manifold (with boundary) embedded in $\mathbb{R}^3$ from its cross-sections $\mathcal S$ with a given set of cutting planes $\mathcal P$ having arbitrary orientations. Using the obvious fact that a point $x \in \mathcal P$ belongs to the original object if and only if it belongs to $\mathcal S$, we follow a very natural reconstruction strategy: we say that a point $x \in \mathbb{R}^3$ belongs to the reconstructed object if (at least one of) its nearest point(s) in $\mathcal P$ belongs to $\mathcal S$. This coincides with the algorithm presented by Liu et al. in \cite{LB+08}. In the present paper, we prove that under appropriate sampling conditions, the output of this algorithm 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 3-dimensional shape reconstruction from cross-sections comes with theoretical guarantees.

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