Stability of Reeb Ordering by Interleaving Distance

Tomoki Uda

doi:10.51094/jxiv.519

##article.authors##

Tomoki Uda Advanced Institute for Materials Research, Tohoku University https://orcid.org/0000-0002-6073-1572 https://researchmap.jp/t_uda

DOI:

https://doi.org/10.51094/jxiv.519

キーワード:

Reeb graph、 Topological Data Analysis、 interleaving distance、 partially ordered space

抄録

The Reeb graph is instrumental in extracting topological features from contour plots. In this context, the Reeb ordering method offers both a natural discretisation and an algorithmic approach to compute a Reeb tree. Our main theorem establishes stability within the interleaving distance among order-compatible topological spaces. Our contributions are fourfold: we construct the reflector functor for quotient structures in ordered spaces, introduce generalised trees in poset terms, define branch completeness for graph-like posets, and prove a strong normalisation theorem for posets. Furthermore, our interleaving distance metric makes our stability estimate much finer than the preceding study.

利益相反に関する開示

I have no conflicts of interest to declare with regards to financial or non-financial matters related to this project or topic.

ダウンロード *前日までの集計結果を表示します

ダウンロード実績データは、公開の翌日以降に作成されます。

著者の経歴

Tomoki Uda、Advanced Institute for Materials Research, Tohoku University

January 2018 - Present: Assistant Professor, Mathematics Group, AIMR, Tohoku University
April 2017 - December 2017: JSPS Postdoctoral Fellow (PD), Department of Mathematics, Kyoto University
April 2016 - March 2017: JSPS Research Fellow (DC), Department of Mathematics, Kyoto University

引用文献

Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoretical Computer Science 392(1), 5–22 (2008) https://doi.org/10.1016/j.tcs.2007.10.018 . Computational Algebraic Geometry and Applications

Silva, V., Munch, E., Petal, A.: Categorified reeb graphs. Discrete and Computational Geometry 55(4), 854–906 (2016) https://doi.org/10.1007/s00454-016-9763-9

Brown, A., Bobrowski, O., Munch, E., Wang, B.: Probabilistic convergence and sta- bility of random mapper graphs. Journal of Applied and Computational Topology 5(1), 99–140 (2021) https://doi.org/10.1007/s41468-020-00063-x

Yokoyama, T., Yokoyama, T.: Cot representations of 2D Hamiltonian flows and their computable applications. To submit, 1–68 (2020)

Uda, T., Yokoyama, T., Sakajo, T.: Algorithms converting streamline topologies for 2D Hamiltonian vector fields using reeb graphs and persistent homology. TJSIAM 29(2), 187–224 (2019) https://doi.org/10.11540/jsiamt.29.2 187

Doraiswamy, H., Natarajan, V.: Computing reeb graphs as a union of contour trees. IEEE Transactions on Visualization and Computer Graphics 19(2), 249–262 (2013) https://doi.org/10.1109/TVCG.2012.115

Birkhoff, G.: Lattice Theory. American Mathematical Society colloquium publications, vol. 25. American Mathematical Society, Rhode Island (1940). https://books.google. co.jp/books?id=ePqVAwAAQBAJ

Wolk, E.S.: On partially ordered sets possessing a unique order-compatible topology. Proceedings of the American Mathematical Society 11(3), 487–492 (1960). Accessed 2023-09-03

Munster, B.: The Hausdorff quotient. Bachelor’s thesis, Mathematical Institute, Leiden University (2014). https://math.leidenuniv.nl/scripties/BachVanMunster.pdf

Munster, B.: Hausdorffization and homotopy. The American Mathematical Monthly 124(1), 81–82 (2017) https://doi.org/10.4169/amer.math.monthly.124.1. 81 https://www.tandfonline.com/doi/pdf/10.4169/amer.math.monthly.124.1.81

Osborne, M.S.: Hausdorffization and such. Amer. Math. Monthly 121(8), 727–734 (2014) https://doi.org/10.4169/amer.math.monthly.121.08.727

Wolk, E.S.: Order-compatible topologies on a partially ordered set. Proceedings of the American Mathematical Society 9(4), 524–529 (1958) https://doi.org/10.2307/ 2033201

Schr ̈oder, B.: Ordered Sets, 2nd edn. Birkh ̈auser Cham, Hattiesburg (2016). https: //doi.org/10.1007/978-3-319-29788-0

Kashiwara, M., Schapira, P.: Categories and Sheaves. Springer, Heidelberg (2010). https://doi.org/10.1007/3-540-27950-4