Geometric Tomography With Topological Guarantees

TL;DR

This paper provides the first theoretical guarantees for 3D shape reconstruction from cross-sections, showing that under certain conditions, the reconstructed object preserves the original's topological features.

Contribution

It proves that a natural reconstruction algorithm preserves homotopy type, homeomorphism, and isotopy of the original 3-manifold under appropriate sampling conditions.

Findings

Reconstruction preserves homotopy type of the original object.

Reconstructed object is homeomorphic and isotopic to the original.

First theoretical guarantees for 3D shape reconstruction from cross-sections.

Abstract

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.