We show that reconstructing a curve in ℝ^d for d ≥ 2 from a 0.66-sample is always possible using an algorithm similar to the classical NN-Crust algorithm. Previously, this was only known to be possible for 0.47-samples in ℝ² and 1/3-samples in ℝ^d for d ≥ 3. In addition, we show that there is not always a unique way to reconstruct a curve from a 0.72-sample; this was previously only known for 1-samples. We also extend this non-uniqueness result to hypersurfaces in all higher dimensions.
@InProceedings{bakkebjerkevik:LIPIcs.SoCG.2022.9, author = {Bakke Bjerkevik, H\r{a}vard}, title = {{Tighter Bounds for Reconstruction from \epsilon-Samples}}, booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)}, pages = {9:1--9:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-227-3}, ISSN = {1868-8969}, year = {2022}, volume = {224}, editor = {Goaoc, Xavier and Kerber, Michael}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2022.9}, URN = {urn:nbn:de:0030-drops-160170}, doi = {10.4230/LIPIcs.SoCG.2022.9}, annote = {Keywords: Curve reconstruction, surface reconstruction, \epsilon-sampling} }
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