{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T15:40:27Z","timestamp":1772466027296,"version":"3.50.1"},"reference-count":63,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T00:00:00Z","timestamp":1761782400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2301361"],"award-info":[{"award-number":["2301361"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["2145499"],"award-info":[{"award-number":["2145499"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["1910733"],"award-info":[{"award-number":["1910733"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0021015"],"award-info":[{"award-number":["DE-SC0021015"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/100000015","name":"U.S. Department of Energy","doi-asserted-by":"publisher","award":["DE-SC0023157"],"award-info":[{"award-number":["DE-SC0023157"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2026,3]]},"DOI":"10.1007\/s00454-025-00789-4","type":"journal-article","created":{"date-parts":[[2025,10,30]],"date-time":"2025-10-30T14:44:23Z","timestamp":1761835463000},"page":"511-536","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Measure-Theoretic Reeb Graphs and Reeb Spaces"],"prefix":"10.1007","volume":"75","author":[{"given":"Qingsong","family":"Wang","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanqun","family":"Ma","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Raghavendra","family":"Sridharamurthy","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9240-0700","authenticated-orcid":false,"given":"Bei","family":"Wang","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,10,30]]},"reference":[{"issue":"4","key":"789_CR1","doi-asserted-by":"publisher","first-page":"597","DOI":"10.1137\/17M1148025","volume":"2","author":"M Adamaszek","year":"2018","unstructured":"Adamaszek, M., Adams, H., Frick, F.: Metric reconstruction via optimal transport. SIAM Journal on Applied Algebra and Geometry 2(4), 597\u2013619 (2018)","journal-title":"SIAM Journal on Applied Algebra and Geometry"},{"issue":"1","key":"789_CR2","doi-asserted-by":"publisher","first-page":"393","DOI":"10.2140\/agt.2024.24.393","volume":"24","author":"H Adams","year":"2024","unstructured":"Adams, H., M\u00e9moli, F., Moy, M., Wang, Q.: The persistent topology of optimal transport based metric thickenings. Algebraic & Geometric Topology 24(1), 393\u2013447 (2024)","journal-title":"Algebraic & Geometric Topology"},{"key":"789_CR3","doi-asserted-by":"crossref","unstructured":"Alvarez-Melis, D., Jaakkola, T.: Gromov-Wasserstein alignment of word embedding spaces. In: Riloff, E., Chiang, D., Hockenmaier, J., Tsujii, J. (eds.) Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pp. 1881\u20131890. Association for Computational Linguistics, Brussels, Belgium (2018)","DOI":"10.18653\/v1\/D18-1214"},{"key":"789_CR4","unstructured":"Arjovsky, M., Chintala, S., Bottou, L.: Wasserstein generative adversarial networks. In: International Conference on Machine Learning, pp. 214\u2013223 (2017). PMLR"},{"key":"789_CR5","unstructured":"Babu, A.: Zigzag coarsenings, Mapper stability and gene-network analyses. PhD thesis, Stanford University (2013)"},{"issue":"1","key":"789_CR6","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1007\/s00454-021-00298-0","volume":"66","author":"H Bakke Bjerkevik","year":"2021","unstructured":"Bakke Bjerkevik, H.: On the stability of interval decomposable persistence modules. Discrete & Computational Geometry 66(1), 92\u2013121 (2021)","journal-title":"Discrete & Computational Geometry"},{"issue":"1","key":"789_CR7","doi-asserted-by":"publisher","first-page":"92","DOI":"10.1007\/s00454-021-00298-0","volume":"66","author":"H Bakke Bjerkevik","year":"2021","unstructured":"Bakke Bjerkevik, H.: On the stability of interval decomposable persistence modules. Discrete & Computational Geometry 66(1), 92\u2013121 (2021)","journal-title":"Discrete & Computational Geometry"},{"key":"789_CR8","doi-asserted-by":"crossref","unstructured":"Bauer, U., Ge, X., Wang, Y.: Measuring distance between Reeb graphs. In: Proceedings of the 30th International Symposium on Computational Geometry, pp. 464\u2013474 (2014)","DOI":"10.1145\/2582112.2582169"},{"key":"789_CR9","unstructured":"Bauer, U., Munch, E., Wang, Y.: Strong equivalence of the interleaving and functional distortion metrics for Reeb graphs. In: 31st International Symposium on Computational Geometry (SoCG 2015), vol. 34, pp. 461\u2013475 (2015). Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik"},{"key":"789_CR10","volume-title":"Eurographics Workshop on 3D Object Retrieval","author":"U Bauer","year":"2016","unstructured":"Bauer, U., Di Fabio, B., Landi, C.: An edit distance for reeb graphs. In: Ferreira, A., Giachetti, A., Giorgi, D. (eds.) Eurographics Workshop on 3D Object Retrieval. The Eurographics Association, Eindhoven, The Netherlands (2016)"},{"issue":"5","key":"789_CR11","doi-asserted-by":"publisher","first-page":"1441","DOI":"10.1007\/s10208-020-09488-3","volume":"21","author":"U Bauer","year":"2020","unstructured":"Bauer, U., Landi, C., Memoli, F.: The reeb graph edit distance is universal. Found. Comput. Math. 21(5), 1441\u20131464 (2020)","journal-title":"Found. Comput. Math."},{"key":"789_CR12","series-title":"Leibniz International Proceedings in Informatics (LIPIcs)","first-page":"14","volume-title":"38th International Symposium on Computational Geometry (SoCG 2022)","author":"U Bauer","year":"2022","unstructured":"Bauer, U., Bjerkevik, H.B., Fluhr, B.: Quasi-universality of reeb graph distances. In: 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), vol. 224, pp. 14\u201311418. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl, Germany (2022)"},{"issue":"1","key":"789_CR13","doi-asserted-by":"publisher","first-page":"7","DOI":"10.1007\/s41468-025-00203-1","volume":"9","author":"U Bauer","year":"2025","unstructured":"Bauer, U., Bjerkevik, H.B., Fluhr, B.: Tight quasi-universality of reeb graph distances. Journal of Applied and Computational Topology 9(1), 7 (2025)","journal-title":"Journal of Applied and Computational Topology"},{"issue":"1\u20133","key":"789_CR14","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1016\/j.tcs.2007.10.018","volume":"392","author":"S Biasotti","year":"2008","unstructured":"Biasotti, S., Giorgi, D., Spagnuolo, M., Falcidieno, B.: Reeb graphs for shape analysis and applications. Theoret. Comput. Sci. 392(1\u20133), 5\u201322 (2008)","journal-title":"Theoret. Comput. Sci."},{"key":"789_CR15","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1007\/s10208-014-9201-4","volume":"14","author":"AJ Blumberg","year":"2014","unstructured":"Blumberg, A.J., Gal, I., Mandell, M.A., Pancia, M.: Robust statistics, hypothesis testing, and confidence intervals for persistent homology on metric measure spaces. Found. Comput. Math. 14, 745\u2013789 (2014)","journal-title":"Found. Comput. Math."},{"key":"789_CR16","unstructured":"Bollen, B., Chambers, E., Levine, J.A., Munch, E.: Reeb graph metrics from the ground up. arXiv preprint arXiv:2110.05631 (2022)"},{"issue":"6","key":"789_CR17","doi-asserted-by":"publisher","first-page":"3133","DOI":"10.2140\/agt.2018.18.3133","volume":"18","author":"M Botnan","year":"2018","unstructured":"Botnan, M., Lesnick, M.: Algebraic stability of zigzag persistence modules. Algebraic & Geometric Topology 18(6), 3133\u20133204 (2018)","journal-title":"Algebraic & Geometric Topology"},{"key":"789_CR18","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1007\/s41468-020-00063-x","volume":"5","author":"A Brown","year":"2021","unstructured":"Brown, A., Bobrowski, O., Munch, E., Wang, B.: Probabilistic convergence and stability of random mapper graphs. Journal of Applied and Computational Topology 5, 99\u2013140 (2021)","journal-title":"Journal of Applied and Computational Topology"},{"key":"789_CR19","unstructured":"Bubenik, P., Silva, V., Scott, J.: Interleaving and Gromov-Hausdorff distance. arXiv preprint arXiv:1707.06288 (2018)"},{"key":"789_CR20","volume-title":"Topological inference from measures","author":"M Buchet","year":"2014","unstructured":"Buchet, M.: Topological inference from measures. Universit\u00e9 Paris Sud - Paris XI (December, Theses (2014)"},{"key":"789_CR21","doi-asserted-by":"publisher","first-page":"70","DOI":"10.1016\/j.comgeo.2016.07.001","volume":"58","author":"M Buchet","year":"2016","unstructured":"Buchet, M., Chazal, F., Oudot, S.Y., Sheehy, D.R.: Efficient and robust persistent homology for measures. Comput. Geom. 58, 70\u201396 (2016)","journal-title":"Comput. Geom."},{"issue":"1","key":"789_CR22","doi-asserted-by":"publisher","first-page":"42","DOI":"10.1016\/j.comgeo.2006.05.009","volume":"43","author":"H Carr","year":"2010","unstructured":"Carr, H., Snoeyink, J., Panne, M.: Flexible isosurfaces: simplifying and displaying scalar topology using the contour tree. Comput. Geom. 43(1), 42\u201358 (2010)","journal-title":"Comput. Geom."},{"key":"789_CR23","unstructured":"Carri\u00e8re, M., Oudot, S.: Local equivalence and intrinsic metrics between Reeb graphs. In: 33rd International Symposium on Computational Geometry (SoCG 2017), vol. 77, pp. 25\u201312515. Schloss Dagstuhl\u2013Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2017)"},{"issue":"6","key":"789_CR24","doi-asserted-by":"publisher","first-page":"1333","DOI":"10.1007\/s10208-017-9370-z","volume":"18","author":"M Carri\u00e9re","year":"2018","unstructured":"Carri\u00e9re, M., Oudot, S.: Structure and stability of the one-dimensional mapper. Found. Comput. Math. 18(6), 1333\u20131396 (2018)","journal-title":"Found. Comput. Math."},{"key":"789_CR25","first-page":"1","volume":"19","author":"M Carri\u00e9re","year":"2018","unstructured":"Carri\u00e9re, M., Michel, B., Oudot, S.: Statistical analysis and parameter selection for mapper. J. Mach. Learn. Res. 19, 1\u201339 (2018)","journal-title":"J. Mach. Learn. Res."},{"key":"789_CR26","series-title":"Leibniz International Proceedings in Informatics (LIPIcs)","first-page":"22","volume-title":"37th International Symposium on Computational Geometry (SoCG 2021)","author":"EW Chambers","year":"2021","unstructured":"Chambers, E.W., Munch, E., Ophelders, T.: A family of metrics from the truncated smoothing of reeb graphs. In: Buchin, K., Verdi\u00e8re, E. (eds.) 37th International Symposium on Computational Geometry (SoCG 2021). Leibniz International Proceedings in Informatics (LIPIcs), vol. 189, pp. 22\u201312217. Schloss Dagstuhl - Leibniz-Zentrum f\u00fcr Informatik, Dagstuhl, Germany (2021)"},{"key":"789_CR27","doi-asserted-by":"crossref","unstructured":"Chazal, F., Cohen-Steiner, D., Glisse, M., Guibas, L.J., Oudot, S.Y.: Proximity of persistence modules and their diagrams. In: 25th Annual Symposium on Computational Geometry (SoCG 2009), pp. 237\u2013246. Association for Computing Machinery, New York, NY, USA (2009)","DOI":"10.1145\/1542362.1542407"},{"key":"789_CR28","doi-asserted-by":"crossref","unstructured":"Chazal, F., Sun, J.: Gromov-hausdorff approximation of filament structure using Reeb-type graph. In: Proceedings of the Thirtieth Annual Symposium on Computational Geometry, pp. 491\u2013500 (2014)","DOI":"10.1145\/2582112.2582129"},{"key":"789_CR29","doi-asserted-by":"publisher","first-page":"733","DOI":"10.1007\/s10208-011-9098-0","volume":"11","author":"F Chazal","year":"2011","unstructured":"Chazal, F., Cohen-Steiner, D., M\u00e9rigot, Q.: Geometric inference for probability measures. Found. Comput. Math. 11, 733\u2013751 (2011)","journal-title":"Found. Comput. Math."},{"key":"789_CR30","unstructured":"Chen, S., Lim, S., M\u00e9moli, F., Wan, Z., Wang, Y.: Weisfeiler-Lehman meets Gromov-Wasserstein. In: International Conference on Machine Learning, pp. 3371\u20133416 (2022). PMLR"},{"issue":"6","key":"789_CR31","doi-asserted-by":"publisher","first-page":"557","DOI":"10.1016\/j.cagd.2012.03.019","volume":"30","author":"F Chen","year":"2013","unstructured":"Chen, F., Obermaier, H., Hagen, H., Hamann, B., Tierny, J., Pascucci, V.: Topology analysis of time-dependent multi-fluid data using the reeb graph. Computer Aided Geometric Design 30(6), 557\u2013566 (2013)","journal-title":"Computer Aided Geometric Design"},{"key":"789_CR32","unstructured":"Curry, J.: Sheaves, cosheaves and applications. PhD thesis, University of Pennsylvania (2014)"},{"issue":"4","key":"789_CR33","doi-asserted-by":"publisher","first-page":"854","DOI":"10.1007\/s00454-016-9763-9","volume":"55","author":"V De Silva","year":"2016","unstructured":"De Silva, V., Munch, E., Patel, A.: Categorified reeb graphs. Discrete & Computational Geometry 55(4), 854\u2013906 (2016)","journal-title":"Discrete & Computational Geometry"},{"issue":"4","key":"789_CR34","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/2461912.2462017","volume":"32","author":"TK Dey","year":"2013","unstructured":"Dey, T.K., Fan, F., Wang, Y.: An efficient computation of handle and tunnel loops via reeb graphs. ACM Transactions on Graphics (TOG) 32(4), 1\u201310 (2013)","journal-title":"ACM Transactions on Graphics (TOG)"},{"key":"789_CR35","series-title":"Leibniz International Proceedings in Informatics (LIPIcs)","first-page":"36","volume-title":"33rd International Symposium on Computational Geometry","author":"TK Dey","year":"2017","unstructured":"Dey, T.K., M\u00e9moli, F., Wang, Y.: Topological analysis of nerves, reeb spaces, mappers, and multiscale mappers. In: Aronov, B., Katz, M.J. (eds.) 33rd International Symposium on Computational Geometry. Leibniz International Proceedings in Informatics (LIPIcs), vol. 77, pp. 36\u201313616. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl, Germany (2017)"},{"issue":"2","key":"789_CR36","doi-asserted-by":"publisher","first-page":"423","DOI":"10.1007\/s00454-016-9758-6","volume":"55","author":"B Di Fabio","year":"2016","unstructured":"Di Fabio, B., Landi, C.: The edit distance for reeb graphs of surfaces. Discrete & Computational Geometry 55(2), 423\u2013461 (2016)","journal-title":"Discrete & Computational Geometry"},{"key":"789_CR37","doi-asserted-by":"crossref","unstructured":"Edelsbrunner, H., Harer, J., Patel, A.K.: Reeb spaces of piecewise linear mappings. In: Proceedings of the Twenty-fourth Annual Symposium on Computational Geometry, pp. 242\u2013250 (2008)","DOI":"10.1145\/1377676.1377720"},{"key":"789_CR38","volume-title":"Computational Topology: an Introduction","author":"H Edelsbrunner","year":"2010","unstructured":"Edelsbrunner, H., Harer, J.: Computational Topology: an Introduction. American Mathematical Society, Providence, Rhode Island (2010)"},{"key":"789_CR39","unstructured":"Engelking, R.: General topology. Sigma series in pure mathematics 6 (1989)"},{"key":"789_CR40","unstructured":"Ge, X., Safa, I., Belkin, M., Wang, Y.: Data skeletonization via Reeb graphs. Advances in Neural Information Processing Systems 24 (2011)"},{"issue":"4","key":"789_CR41","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s41468-019-00037-8","volume":"3","author":"H Hang","year":"2019","unstructured":"Hang, H., M\u00e9moli, F., Mio, W.: A topological study of functional data and fr\u00e9chet functions of metric measure spaces. Journal of Applied and Computational Topology 3(4), 359\u2013380 (2019)","journal-title":"Journal of Applied and Computational Topology"},{"issue":"3","key":"789_CR42","doi-asserted-by":"publisher","first-page":"643","DOI":"10.1111\/cgf.12933","volume":"35","author":"C Heine","year":"2016","unstructured":"Heine, C., Leitte, H., Hlawitschka, M., Iuricich, F., De Floriani, L., Scheuermann, G., Hagen, H., Garth, C.: A survey of topology-based methods in visualization. Computer Graphics Forum (CGF) 35(3), 643\u2013667 (2016)","journal-title":"Computer Graphics Forum (CGF)"},{"issue":"1\u20133","key":"789_CR43","doi-asserted-by":"publisher","first-page":"131","DOI":"10.1016\/S1524-0703(03)00005-5","volume":"65","author":"F H\u00e9troy","year":"2003","unstructured":"H\u00e9troy, F., Attali, D.: Topological quadrangulations of closed triangulated surfaces using the reeb graph. Graph. Models 65(1\u20133), 131\u2013148 (2003)","journal-title":"Graph. Models"},{"key":"789_CR44","doi-asserted-by":"crossref","unstructured":"Hilaga, M., Shinagawa, Y., Kohmura, T., Kunii, T.L.: Topology matching for fully automatic similarity estimation of 3D shapes. In: Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH 2001), pp. 203\u2013212 (2001)","DOI":"10.1145\/383259.383282"},{"key":"789_CR45","unstructured":"M\u00e9moli, F., Okutan, O.B., Wang, Q.: Metric graph approximations of geodesic spaces. arXiv preprint arXiv:1809.05566 (2018)"},{"key":"789_CR46","doi-asserted-by":"crossref","unstructured":"Morozov, D., Beketayev, K., Weber, G.: Interleaving distance between merge trees. Proceedings of Topology-Based Methods in Visualization (TopoInVis) (2013)","DOI":"10.1007\/978-3-319-04099-8_10"},{"key":"789_CR47","doi-asserted-by":"crossref","unstructured":"Munch, E., Stefanou, A.: The $$\\ell ^\\infty $$-cophenetic metric for phylogenetic trees as an interleaving distance. Association for Women in Mathematics Series, vol. 17, pp. 109\u2013127. Springer, Cham (2019)","DOI":"10.1007\/978-3-030-11566-1_5"},{"key":"789_CR48","unstructured":"Munch, E., Wang, B.: Convergence between categorical representations of Reeb space and Mapper. In: 32nd International Symposium on Computational Geometry (SoCG 2016) (2016). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik"},{"issue":"17","key":"789_CR49","doi-asserted-by":"publisher","first-page":"7265","DOI":"10.1073\/pnas.1102826108","volume":"108","author":"M Nicolau","year":"2011","unstructured":"Nicolau, M., Levine, A.J., Carlsson, G.: Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proc. Natl. Acad. Sci. 108(17), 7265\u20137270 (2011)","journal-title":"Proc. Natl. Acad. Sci."},{"key":"789_CR50","unstructured":"Phillips, J.M., Wang, B., Zheng, Y.: Geometric inference on kernel density estimates. In: Proceedings of the 31st International Symposium on Computational Geometry (2015). Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik"},{"key":"789_CR51","doi-asserted-by":"crossref","unstructured":"Phillips, J.M.: $$\\epsilon $$-samples for kernels. In: Proceedings of the 24th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1622\u20131632 (2013)","DOI":"10.1137\/1.9781611973105.116"},{"key":"789_CR52","first-page":"847","volume":"222","author":"G Reeb","year":"1946","unstructured":"Reeb, G.: Sur les points singuliers d\u2019une forme de pfaff completement integrable ou d\u2019une fonction numerique [on the singular points of a completely integrable pfaff form or of a numerical function]. Comptes Rendus Acad. Sciences Paris 222, 847\u2013849 (1946)","journal-title":"Comptes Rendus Acad. Sciences Paris"},{"key":"789_CR53","first-page":"91","volume-title":"4th Symposium on Point Based Graphics, 2007","author":"G Singh","year":"2007","unstructured":"Singh, G., M\u00e9moli, F., Carlsson, G.E.: Topological methods for the analysis of high dimensional data sets and 3d object recognition. In: Botsch, M., Pajarola, R., Chen, B., Zwicker, M. (eds.) 4th Symposium on Point Based Graphics, 2007, pp. 91\u2013100. Prague, Czech Republic, Eurographics Association (2007)"},{"issue":"3","key":"789_CR54","doi-asserted-by":"publisher","first-page":"1518","DOI":"10.1109\/TVCG.2018.2873612","volume":"26","author":"R Sridharamurthy","year":"2020","unstructured":"Sridharamurthy, R., Masood, T.B., Kamakshidasan, A., Natarajan, V.: Edit distance between merge trees. IEEE Transactions on Visualization and Computer Graphics (TVCG) 26(3), 1518\u20131531 (2020)","journal-title":"IEEE Transactions on Visualization and Computer Graphics (TVCG)"},{"key":"789_CR55","first-page":"1517","volume":"11","author":"BK Sriperumbudur","year":"2010","unstructured":"Sriperumbudur, B.K., Gretton, A., Fukumizu, K., Sch\u00f6lkopf, B., Lanckriet, G.R.: Hilbert space embeddings and metrics on probability measures. The Journal of Machine Learning Research 11, 1517\u20131561 (2010)","journal-title":"The Journal of Machine Learning Research"},{"issue":"1","key":"789_CR56","doi-asserted-by":"publisher","first-page":"65","DOI":"10.1007\/s11511-006-0002-8","volume":"196","author":"K-T Sturm","year":"2006","unstructured":"Sturm, K.-T.: On the geometry of metric measure spaces i. Acta Math. 196(1), 65\u2013131 (2006)","journal-title":"Acta Math."},{"issue":"1","key":"789_CR57","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1111\/j.1467-8659.2008.01190.x","volume":"28","author":"J Tierny","year":"2009","unstructured":"Tierny, J., Vandeborre, J.-P., Daoudi, M.: Partial 3d shape retrieval by reeb pattern unfolding. Computer Graphics Forum 28(1), 41\u201355 (2009)","journal-title":"Computer Graphics Forum"},{"issue":"6","key":"789_CR58","doi-asserted-by":"publisher","first-page":"1177","DOI":"10.1109\/TVCG.2009.163","volume":"15","author":"J Tierny","year":"2009","unstructured":"Tierny, J., Gyulassy, A., Simon, E., Pascucci, V.: Loop surgery for volumetric meshes: reeb graphs reduced to contour trees. IEEE Trans. Visual Comput. Graphics 15(6), 1177\u20131184 (2009)","journal-title":"IEEE Trans. Visual Comput. Graphics"},{"key":"789_CR59","unstructured":"Touli, E.F., Wang, Y.: FPT-algorithms for computing Gromov-Hausdorff and interleaving distances between trees. Proceedings of the 27th Annual European Symposium on Algorithms, 83\u201318314 (2019)"},{"issue":"01","key":"789_CR60","doi-asserted-by":"publisher","first-page":"91","DOI":"10.1142\/S0218654305000748","volume":"11","author":"T Tung","year":"2005","unstructured":"Tung, T., Schmitt, F.: The augmented multiresolution reeb graph approach for content-based retrieval of 3d shapes. Int. J. Shape Model. 11(01), 91\u2013120 (2005)","journal-title":"Int. J. Shape Model."},{"key":"789_CR61","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-540-71050-9","volume-title":"Optimal Transport: Old and New","author":"C Villani","year":"2009","unstructured":"Villani, C.: Optimal Transport: Old and New, vol. 338. Springer, Berlin, Heidelberg (2009)"},{"issue":"2","key":"789_CR62","doi-asserted-by":"publisher","first-page":"190","DOI":"10.1145\/990002.990007","volume":"23","author":"Z Wood","year":"2004","unstructured":"Wood, Z., Hoppe, H., Desbrun, M., Schr\u00f6der, P.: Removing excess topology from isosurfaces. ACM Transactions on Graphics (TOG) 23(2), 190\u2013208 (2004)","journal-title":"ACM Transactions on Graphics (TOG)"},{"issue":"3","key":"789_CR63","doi-asserted-by":"publisher","first-page":"599","DOI":"10.1111\/cgf.14331","volume":"40","author":"L Yan","year":"2021","unstructured":"Yan, L., Masood, T.B., Sridharamurthy, R., Rasheed, F., Natarajan, V., Hotz, I., Wang, B.: Scalar field comparison with topological descriptors: properties and applications for scientific visualization. Computer Graphics Forum (CGF) 40(3), 599\u2013633 (2021)","journal-title":"Computer Graphics Forum (CGF)"}],"container-title":["Discrete & Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00789-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-025-00789-4","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-025-00789-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,3,2]],"date-time":"2026-03-02T14:49:55Z","timestamp":1772462995000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-025-00789-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,10,30]]},"references-count":63,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["789"],"URL":"https:\/\/doi.org\/10.1007\/s00454-025-00789-4","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,10,30]]},"assertion":[{"value":"25 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 September 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 September 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"30 October 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}