We address the problem of homotopy-type reconstruction of compact shapes $X\subset {\mathbb{R}}^N$ that are $\mathrm{CAT}(\kappa )$ in the intrinsic length metric. The reconstructed spaces take the form of Vietoris–Rips complexes, computed from a compact sample $S$ that is Hausdorff-close to the unknown shape $X$. Instead of employing the Euclidean metric on the sample, our reconstruction technique utilizes a path-based metric to compute these complexes. Naturally emerging in the reconstruction framework, we also explore the Gromov–Hausdorff topological stability and the finiteness problem for general compact $\mathrm{CAT}(\kappa )$ spaces. Our techniques offer novel sampling conditions as alternatives to the existing and commonly used methods based on the weak feature size and $\mu$-reach.

Link to the preprint: https://arxiv.org/abs/2406.04259

Co-authors: Sushovan Majhi (s.majhi@gwu.edu) Will Tran (tranw@southwestern.edu)