SCG '05 Proceedings of the twenty-first annual symposium on Computational geometry
Graphical Models
Inferring Local Homology from Sampled Stratified Spaces
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Finding the Homology of Submanifolds with High Confidence from Random Samples
Discrete & Computational Geometry
Smooth manifold reconstruction from noisy and non-uniform approximation with guarantees
Computational Geometry: Theory and Applications
A Sampling Theory for Compact Sets in Euclidean Space
Discrete & Computational Geometry
Incremental construction of the delaunay triangulation and the delaunay graph in medium dimension
Proceedings of the twenty-fifth annual symposium on Computational geometry
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
The power crust, unions of balls, and the medial axis transform
Computational Geometry: Theory and Applications
Optimal reconstruction might be hard
Proceedings of the twenty-sixth annual symposium on Computational geometry
Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes
Proceedings of the twenty-seventh annual symposium on Computational geometry
Efficient data structure for representing and simplifying simplicial complexes in high dimensions
Proceedings of the twenty-seventh annual symposium on Computational geometry
Preserving geometric properties in reconstructing regions from internal and nearby points
Computational Geometry: Theory and Applications
Vietoris-Rips complexes also provide topologically correct reconstructions of sampled shapes
Computational Geometry: Theory and Applications
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A simple way to reconstruct a shape A from a sample P is to output an r-offset P + r B, where B = {x ∈ RN x ≤ 1} designates the unit Euclidean ball centered at the origin. Recently, it has been proved that the output P + r B is homotopy equivalent to the shape A, for a dense enough sample P of A and for a suitable value of the parameter r. In this paper, we extend this result and find convex sets C ⊂ RN, besides the unit Euclidean ball B, for which P + rC reconstructs the topology of A. This class of convex sets includes in particular N-dimensional cubes in RN. We proceed in two steps. First, we establish the result when P is an ε-offset of A. Building on this first result, we then consider the case when P is a finite noisy sample of A.