{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/delaunay-bifiltrations-of-functions-on-point","title":"Delaunay Bifiltrations of Functions on Point Clouds","arxiv_id":"2310.15902","date":"2023-10-24","proceeding":null,"authors":["Ángel Javier Alonso","Michael Kerber","Tung Lam","Michael Lesnick"],"abstract":"The Delaunay filtration $\\mathcal{D}_{\\bullet}(X)$ of a point cloud $X\\subset \\mathbb{R}^d$ is a central tool of computational topology. Its use is justified by the topological equivalence of $\\mathcal{D}_{\\bullet}(X)$ and the offset (i.e., union-of-balls) filtration of $X$. Given a function $\\gamma: X \\to \\mathbb{R}$, we introduce a Delaunay bifiltration $\\mathcal{DC}_{\\bullet}(\\gamma)$ that satisfies an analogous topological equivalence, ensuring that $\\mathcal{DC}_{\\bullet}(\\gamma)$ topologically encodes the offset filtrations of all sublevel sets of $\\gamma$, as well as the topological relations between them. $\\mathcal{DC}_{\\bullet}(\\gamma)$ is of size $O(|X|^{\\lceil\\frac{d+1}{2}\\rceil})$, which for $d$ odd matches the worst-case size of $\\mathcal{D}_{\\bullet}(X)$. Adapting the Bowyer-Watson algorithm for computing Delaunay triangulations, we give a simple, practical algorithm to compute $\\mathcal{DC}_{\\bullet}(\\gamma)$ in time $O(|X|^{\\lceil \\frac{d}{2}\\rceil +1})$. Our implementation, based on CGAL, computes $\\mathcal{DC}_{\\bullet}(\\gamma)$ with modest overhead compared to computing $\\mathcal{D}_{\\bullet}(X)$, and handles tens of thousands of points in $\\mathbb{R}^3$ within seconds.","url_abs":"https://arxiv.org/abs/2310.15902v1","url_pdf":"https://arxiv.org/pdf/2310.15902v1.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"delaunay-bifiltrations-of-functions-on-point","repo_url":"https://github.com/davidlapous/multipers","is_official":0,"mentioned_in_paper":0,"mentioned_in_github":1,"framework":"pytorch","reach":{"status":"ok","spdx":"GPL-3.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"syntology_url":null,"atlas_url":"https://app.syntology.ai/?focus=2310.15902","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}