We consider the problem of homotopy-type reconstruction of compact subsets $X\subset\R^N$ that have the Alexandrov curvature bounded above ($\leq$ $κ$) in the intrinsic length metric. The reconstructed spaces are in the form of Vietoris--Rips complexes computed from a compact sample $S$, Hausdorff--close to the unknown shape $X$. Instead of the Euclidean metric on the sample, our reconstruction technique leverages a path-based metric to compute these complexes. As naturally emerging in the framework of reconstruction, we also study the Gromov--Hausdorff topological stability and finiteness problem for general compact for subspaces of curvature bounded above. Our techniques provide novel sampling conditions as an alternative to the existing and commonly used techniques using weak feature size and $μ$--reach. To the best of our knowledge, this is the first work that establishes homotopy-type reconstruction guarantees for spaces with vanishing reach and $μ$--reach, a regime not covered by existing sampling conditions.

翻译：本文研究具有内在长度度量下Alexandrov上界曲率（$\leq$ $κ$）的紧致子集$X\subset\R^N$的同伦型重建问题。重建空间采用基于紧致样本点集$S$计算的Vietoris--Rips复形形式，该样本集与未知形状$X$具有Hausdorff逼近性。与在样本上使用欧氏度量的传统方法不同，我们的重建技术利用基于路径的度量来计算这些复形。作为重建框架中自然产生的问题，我们还研究了具有上界曲率的紧致子空间的一般Gromov--Hausdorff拓扑稳定性与有限性问题。我们的技术提供了新颖的采样条件，可作为现有常用弱特征尺寸与$μ$--可达性技术的替代方案。据我们所知，这是首个针对可达性与$μ$--可达性消失空间建立同伦型重建保证的研究，该区域未被现有采样条件所覆盖。