[{"department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"creator":"dernst","file_id":"5953","relation":"main_file","content_type":"application/pdf","access_level":"open_access","date_updated":"2020-07-14T12:46:38Z","checksum":"1c8d58cd489a66cd3e2064c1141c8c5e","date_created":"2019-02-12T06:47:52Z","file_size":708357,"file_name":"2018_Edelsbrunner.pdf"}],"date_created":"2018-12-11T11:46:59Z","date_published":"2018-03-01T00:00:00Z","month":"03","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Elsevier","page":"119 - 133","file_date_updated":"2020-07-14T12:46:38Z","publication":"Computational Geometry: Theory and Applications","type":"journal_article","day":"01","status":"public","intvolume":"        68","isi":1,"ddc":["000"],"ec_funded":1,"doi":"10.1016/j.comgeo.2017.06.014","year":"2018","external_id":{"isi":["000415778300010"]},"title":"Multiple covers with balls I: Inclusion–exclusion","article_processing_charge":"No","volume":68,"publist_id":"7289","date_updated":"2023-09-13T08:59:00Z","oa":1,"oa_version":"Preprint","project":[{"call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493"}],"quality_controlled":"1","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"530","citation":{"ama":"Edelsbrunner H, Iglesias Ham M. Multiple covers with balls I: Inclusion–exclusion. <i>Computational Geometry: Theory and Applications</i>. 2018;68:119-133. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">10.1016/j.comgeo.2017.06.014</a>","mla":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” <i>Computational Geometry: Theory and Applications</i>, vol. 68, Elsevier, 2018, pp. 119–33, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">10.1016/j.comgeo.2017.06.014</a>.","ista":"Edelsbrunner H, Iglesias Ham M. 2018. Multiple covers with balls I: Inclusion–exclusion. Computational Geometry: Theory and Applications. 68, 119–133.","short":"H. Edelsbrunner, M. Iglesias Ham, Computational Geometry: Theory and Applications 68 (2018) 119–133.","ieee":"H. Edelsbrunner and M. Iglesias Ham, “Multiple covers with balls I: Inclusion–exclusion,” <i>Computational Geometry: Theory and Applications</i>, vol. 68. Elsevier, pp. 119–133, 2018.","apa":"Edelsbrunner, H., &#38; Iglesias Ham, M. (2018). Multiple covers with balls I: Inclusion–exclusion. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">https://doi.org/10.1016/j.comgeo.2017.06.014</a>","chicago":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls I: Inclusion–Exclusion.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2018. <a href=\"https://doi.org/10.1016/j.comgeo.2017.06.014\">https://doi.org/10.1016/j.comgeo.2017.06.014</a>."},"publication_status":"published","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","first_name":"Mabel","full_name":"Iglesias Ham, Mabel","last_name":"Iglesias Ham"}],"abstract":[{"lang":"eng","text":"Inclusion–exclusion is an effective method for computing the volume of a union of measurable sets. We extend it to multiple coverings, proving short inclusion–exclusion formulas for the subset of Rn covered by at least k balls in a finite set. We implement two of the formulas in dimension n=3 and report on results obtained with our software."}]},{"date_published":"2017-05-01T00:00:00Z","article_type":"original","month":"05","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"American Mathematical Society","department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:49:59Z","type":"journal_article","day":"01","status":"public","intvolume":"       369","page":"3741 - 3762","publication":"Transactions of the American Mathematical Society","issue":"5","ec_funded":1,"doi":"10.1090/tran/6991","year":"2017","title":"The Morse theory of Čech and delaunay complexes","external_id":{"arxiv":["1312.1231"],"isi":["000398030400024"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1312.1231"}],"isi":1,"citation":{"chicago":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” <i>Transactions of the American Mathematical Society</i>. American Mathematical Society, 2017. <a href=\"https://doi.org/10.1090/tran/6991\">https://doi.org/10.1090/tran/6991</a>.","ieee":"U. Bauer and H. Edelsbrunner, “The Morse theory of Čech and delaunay complexes,” <i>Transactions of the American Mathematical Society</i>, vol. 369, no. 5. American Mathematical Society, pp. 3741–3762, 2017.","apa":"Bauer, U., &#38; Edelsbrunner, H. (2017). The Morse theory of Čech and delaunay complexes. <i>Transactions of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/tran/6991\">https://doi.org/10.1090/tran/6991</a>","ista":"Bauer U, Edelsbrunner H. 2017. The Morse theory of Čech and delaunay complexes. Transactions of the American Mathematical Society. 369(5), 3741–3762.","short":"U. Bauer, H. Edelsbrunner, Transactions of the American Mathematical Society 369 (2017) 3741–3762.","mla":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Complexes.” <i>Transactions of the American Mathematical Society</i>, vol. 369, no. 5, American Mathematical Society, 2017, pp. 3741–62, doi:<a href=\"https://doi.org/10.1090/tran/6991\">10.1090/tran/6991</a>.","ama":"Bauer U, Edelsbrunner H. The Morse theory of Čech and delaunay complexes. <i>Transactions of the American Mathematical Society</i>. 2017;369(5):3741-3762. doi:<a href=\"https://doi.org/10.1090/tran/6991\">10.1090/tran/6991</a>"},"publication_status":"published","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","full_name":"Bauer, Ulrich","last_name":"Bauer","orcid":"0000-0002-9683-0724","first_name":"Ulrich"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"}],"abstract":[{"text":"Given a finite set of points in Rn and a radius parameter, we study the Čech, Delaunay–Čech, Delaunay (or alpha), and Wrap complexes in the light of generalized discrete Morse theory. Establishing the Čech and Delaunay complexes as sublevel sets of generalized discrete Morse functions, we prove that the four complexes are simple-homotopy equivalent by a sequence of simplicial collapses, which are explicitly described by a single discrete gradient field.","lang":"eng"}],"article_processing_charge":"No","publist_id":"6311","date_updated":"2023-09-20T12:05:56Z","oa":1,"volume":369,"arxiv":1,"quality_controlled":"1","oa_version":"Preprint","project":[{"name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493"}],"acknowledgement":"This research has been supported by the EU project Toposys(FP7-ICT-318493-STREP), by ESF under the ACAT Research Network Programme, by the Russian Government under mega project 11.G34.31.0053, and by the DFG Collaborative Research Center SFB/TRR 109 “Discretization in Geometry and Dynamics”.","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","_id":"1072"},{"intvolume":"        37","status":"public","day":"01","type":"journal_article","publication":"Combinatorica","issue":"5","page":"887 - 910","scopus_import":"1","publisher":"Springer","language":[{"iso":"eng"}],"month":"10","date_published":"2017-10-01T00:00:00Z","date_created":"2018-12-11T11:50:32Z","department":[{"_id":"HeEd"}],"abstract":[{"lang":"eng","text":"We introduce the Voronoi functional of a triangulation of a finite set of points in the Euclidean plane and prove that among all geometric triangulations of the point set, the Delaunay triangulation maximizes the functional. This result neither extends to topological triangulations in the plane nor to geometric triangulations in three and higher dimensions."}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"last_name":"Glazyrin","full_name":"Glazyrin, Alexey","first_name":"Alexey"},{"full_name":"Musin, Oleg","last_name":"Musin","first_name":"Oleg"},{"first_name":"Anton","orcid":"0000-0002-0659-3201","last_name":"Nikitenko","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"citation":{"short":"H. Edelsbrunner, A. Glazyrin, O. Musin, A. Nikitenko, Combinatorica 37 (2017) 887–910.","ista":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. 2017. The Voronoi functional is maximized by the Delaunay triangulation in the plane. Combinatorica. 37(5), 887–910.","mla":"Edelsbrunner, Herbert, et al. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” <i>Combinatorica</i>, vol. 37, no. 5, Springer, 2017, pp. 887–910, doi:<a href=\"https://doi.org/10.1007/s00493-016-3308-y\">10.1007/s00493-016-3308-y</a>.","ama":"Edelsbrunner H, Glazyrin A, Musin O, Nikitenko A. The Voronoi functional is maximized by the Delaunay triangulation in the plane. <i>Combinatorica</i>. 2017;37(5):887-910. doi:<a href=\"https://doi.org/10.1007/s00493-016-3308-y\">10.1007/s00493-016-3308-y</a>","chicago":"Edelsbrunner, Herbert, Alexey Glazyrin, Oleg Musin, and Anton Nikitenko. “The Voronoi Functional Is Maximized by the Delaunay Triangulation in the Plane.” <i>Combinatorica</i>. Springer, 2017. <a href=\"https://doi.org/10.1007/s00493-016-3308-y\">https://doi.org/10.1007/s00493-016-3308-y</a>.","apa":"Edelsbrunner, H., Glazyrin, A., Musin, O., &#38; Nikitenko, A. (2017). The Voronoi functional is maximized by the Delaunay triangulation in the plane. <i>Combinatorica</i>. Springer. <a href=\"https://doi.org/10.1007/s00493-016-3308-y\">https://doi.org/10.1007/s00493-016-3308-y</a>","ieee":"H. Edelsbrunner, A. Glazyrin, O. Musin, and A. Nikitenko, “The Voronoi functional is maximized by the Delaunay triangulation in the plane,” <i>Combinatorica</i>, vol. 37, no. 5. Springer, pp. 887–910, 2017."},"publication_status":"published","publication_identifier":{"issn":["02099683"]},"_id":"1173","quality_controlled":"1","oa_version":"Submitted Version","project":[{"grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","acknowledgement":"This research is partially supported by the Russian Government under the Mega Project 11.G34.31.0053, by the Toposys project FP7-ICT-318493-STREP, by ESF under the ACAT Research Network Programme, by RFBR grant 11-01-00735, and by NSF grants DMS-1101688, DMS-1400876.","article_processing_charge":"No","date_updated":"2023-09-20T11:23:53Z","oa":1,"publist_id":"6182","volume":37,"title":"The Voronoi functional is maximized by the Delaunay triangulation in the plane","external_id":{"isi":["000418056000005"]},"doi":"10.1007/s00493-016-3308-y","year":"2017","ec_funded":1,"isi":1,"main_file_link":[{"url":"https://arxiv.org/abs/1411.6337","open_access":"1"}]},{"isi":1,"alternative_title":["PROMS"],"title":"Finding eigenvalues of self-maps with the Kronecker canonical form","external_id":{"isi":["000434088200008"]},"ec_funded":1,"doi":"10.1007/978-3-319-56932-1_8","year":"2017","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","project":[{"grant_number":"318493","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"quality_controlled":"1","oa_version":"None","_id":"836","publication_identifier":{"isbn":["978-331956930-7"]},"volume":198,"publist_id":"6812","date_updated":"2023-09-26T15:50:52Z","article_processing_charge":"No","author":[{"first_name":"Marc","full_name":"Ethier, Marc","last_name":"Ethier"},{"id":"4483EF78-F248-11E8-B48F-1D18A9856A87","first_name":"Grzegorz","orcid":"0000-0002-3536-9866","full_name":"Jablonski, Grzegorz","last_name":"Jablonski"},{"full_name":"Mrozek, Marian","last_name":"Mrozek","first_name":"Marian"}],"abstract":[{"lang":"eng","text":"Recent research has examined how to study the topological features of a continuous self-map by means of the persistence of the eigenspaces, for given eigenvalues, of the endomorphism induced in homology over a field. This raised the question of how to select dynamically significant eigenvalues. The present paper aims to answer this question, giving an algorithm that computes the persistence of eigenspaces for every eigenvalue simultaneously, also expressing said eigenspaces as direct sums of “finite” and “singular” subspaces."}],"publication_status":"published","citation":{"mla":"Ethier, Marc, et al. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical Form.” <i>Special Sessions in Applications of Computer Algebra</i>, vol. 198, Springer, 2017, pp. 119–36, doi:<a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">10.1007/978-3-319-56932-1_8</a>.","ama":"Ethier M, Jablonski G, Mrozek M. Finding eigenvalues of self-maps with the Kronecker canonical form. In: <i>Special Sessions in Applications of Computer Algebra</i>. Vol 198. Springer; 2017:119-136. doi:<a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">10.1007/978-3-319-56932-1_8</a>","ista":"Ethier M, Jablonski G, Mrozek M. 2017. Finding eigenvalues of self-maps with the Kronecker canonical form. Special Sessions in Applications of Computer Algebra. ACA: Applications of Computer Algebra, PROMS, vol. 198, 119–136.","short":"M. Ethier, G. Jablonski, M. Mrozek, in:, Special Sessions in Applications of Computer Algebra, Springer, 2017, pp. 119–136.","apa":"Ethier, M., Jablonski, G., &#38; Mrozek, M. (2017). Finding eigenvalues of self-maps with the Kronecker canonical form. In <i>Special Sessions in Applications of Computer Algebra</i> (Vol. 198, pp. 119–136). Kalamata, Greece: Springer. <a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">https://doi.org/10.1007/978-3-319-56932-1_8</a>","ieee":"M. Ethier, G. Jablonski, and M. Mrozek, “Finding eigenvalues of self-maps with the Kronecker canonical form,” in <i>Special Sessions in Applications of Computer Algebra</i>, Kalamata, Greece, 2017, vol. 198, pp. 119–136.","chicago":"Ethier, Marc, Grzegorz Jablonski, and Marian Mrozek. “Finding Eigenvalues of Self-Maps with the Kronecker Canonical Form.” In <i>Special Sessions in Applications of Computer Algebra</i>, 198:119–36. Springer, 2017. <a href=\"https://doi.org/10.1007/978-3-319-56932-1_8\">https://doi.org/10.1007/978-3-319-56932-1_8</a>."},"date_created":"2018-12-11T11:48:46Z","conference":{"start_date":"2015-07-20","location":"Kalamata, Greece","end_date":"2015-07-23","name":"ACA: Applications of Computer Algebra"},"department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publisher":"Springer","scopus_import":"1","date_published":"2017-07-27T00:00:00Z","month":"07","page":"119 - 136","publication":"Special Sessions in Applications of Computer Algebra","status":"public","intvolume":"       198","type":"conference","day":"27"},{"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert"},{"full_name":"Nikitenko, Anton","last_name":"Nikitenko","orcid":"0000-0002-0659-3201","first_name":"Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Reitzner","full_name":"Reitzner, Matthias","first_name":"Matthias"}],"abstract":[{"lang":"eng","text":"Mapping every simplex in the Delaunay mosaic of a discrete point set to the radius of the smallest empty circumsphere gives a generalized discrete Morse function. Choosing the points from a Poisson point process in ℝ n , we study the expected number of simplices in the Delaunay mosaic as well as the expected number of critical simplices and nonsingular intervals in the corresponding generalized discrete gradient. Observing connections with other probabilistic models, we obtain precise expressions for the expected numbers in low dimensions. In particular, we obtain the expected numbers of simplices in the Poisson–Delaunay mosaic in dimensions n ≤ 4."}],"publication_status":"published","citation":{"chicago":"Edelsbrunner, Herbert, Anton Nikitenko, and Matthias Reitzner. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>. Cambridge University Press, 2017. <a href=\"https://doi.org/10.1017/apr.2017.20\">https://doi.org/10.1017/apr.2017.20</a>.","apa":"Edelsbrunner, H., Nikitenko, A., &#38; Reitzner, M. (2017). Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>. Cambridge University Press. <a href=\"https://doi.org/10.1017/apr.2017.20\">https://doi.org/10.1017/apr.2017.20</a>","ieee":"H. Edelsbrunner, A. Nikitenko, and M. Reitzner, “Expected sizes of poisson Delaunay mosaics and their discrete Morse functions,” <i>Advances in Applied Probability</i>, vol. 49, no. 3. Cambridge University Press, pp. 745–767, 2017.","short":"H. Edelsbrunner, A. Nikitenko, M. Reitzner, Advances in Applied Probability 49 (2017) 745–767.","ista":"Edelsbrunner H, Nikitenko A, Reitzner M. 2017. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. Advances in Applied Probability. 49(3), 745–767.","ama":"Edelsbrunner H, Nikitenko A, Reitzner M. Expected sizes of poisson Delaunay mosaics and their discrete Morse functions. <i>Advances in Applied Probability</i>. 2017;49(3):745-767. doi:<a href=\"https://doi.org/10.1017/apr.2017.20\">10.1017/apr.2017.20</a>","mla":"Edelsbrunner, Herbert, et al. “Expected Sizes of Poisson Delaunay Mosaics and Their Discrete Morse Functions.” <i>Advances in Applied Probability</i>, vol. 49, no. 3, Cambridge University Press, 2017, pp. 745–67, doi:<a href=\"https://doi.org/10.1017/apr.2017.20\">10.1017/apr.2017.20</a>."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Preprint","project":[{"grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"},{"grant_number":"I02979-N35","call_identifier":"FWF","name":"Persistence and stability of geometric complexes","_id":"2561EBF4-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","_id":"718","publication_identifier":{"issn":["00018678"]},"volume":49,"publist_id":"6962","date_updated":"2023-09-07T12:07:12Z","oa":1,"arxiv":1,"title":"Expected sizes of poisson Delaunay mosaics and their discrete Morse functions","external_id":{"arxiv":["1607.05915"]},"ec_funded":1,"doi":"10.1017/apr.2017.20","year":"2017","related_material":{"record":[{"status":"public","id":"6287","relation":"dissertation_contains"}]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1607.05915"}],"status":"public","intvolume":"        49","type":"journal_article","day":"01","page":"745 - 767","issue":"3","publication":"Advances in Applied Probability","language":[{"iso":"eng"}],"publisher":"Cambridge University Press","scopus_import":1,"date_published":"2017-09-01T00:00:00Z","month":"09","date_created":"2018-12-11T11:48:07Z","department":[{"_id":"HeEd"}]},{"date_created":"2018-12-11T11:51:59Z","department":[{"_id":"HeEd"}],"scopus_import":"1","publisher":"Academic Press","language":[{"iso":"eng"}],"month":"01","article_type":"original","date_published":"2017-01-01T00:00:00Z","publication":"Journal of Symbolic Computation","page":"76 - 90","intvolume":"        78","status":"public","day":"01","type":"journal_article","related_material":{"record":[{"status":"public","id":"10894","relation":"earlier_version"}]},"isi":1,"main_file_link":[{"open_access":"1","url":"https://doi.org/10.1016/j.jsc.2016.03.008"}],"external_id":{"isi":["000384396000005"]},"title":"Phat - Persistent homology algorithms toolbox","doi":"10.1016/j.jsc.2016.03.008","year":"2017","ec_funded":1,"publication_identifier":{"issn":[" 07477171"]},"_id":"1433","oa_version":"Published Version","quality_controlled":"1","project":[{"grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","call_identifier":"FP7"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","article_processing_charge":"No","publist_id":"5765","date_updated":"2023-09-20T09:42:40Z","oa":1,"volume":78,"abstract":[{"lang":"eng","text":"Phat is an open-source C. ++ library for the computation of persistent homology by matrix reduction, targeted towards developers of software for topological data analysis. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. We provide numerous different reduction strategies as well as data types to store and manipulate the boundary matrix. We compare the different combinations through extensive experimental evaluation and identify optimization techniques that work well in practical situations. We also compare our software with various other publicly available libraries for persistent homology."}],"author":[{"last_name":"Bauer","full_name":"Bauer, Ulrich","first_name":"Ulrich"},{"last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"},{"last_name":"Reininghaus","full_name":"Reininghaus, Jan","first_name":"Jan"},{"first_name":"Hubert","last_name":"Wagner","full_name":"Wagner, Hubert","id":"379CA8B8-F248-11E8-B48F-1D18A9856A87"}],"citation":{"apa":"Bauer, U., Kerber, M., Reininghaus, J., &#38; Wagner, H. (2017). Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>","ieee":"U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “Phat - Persistent homology algorithms toolbox,” <i>Journal of Symbolic Computation</i>, vol. 78. Academic Press, pp. 76–90, 2017.","chicago":"Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>. Academic Press, 2017. <a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">https://doi.org/10.1016/j.jsc.2016.03.008</a>.","mla":"Bauer, Ulrich, et al. “Phat - Persistent Homology Algorithms Toolbox.” <i>Journal of Symbolic Computation</i>, vol. 78, Academic Press, 2017, pp. 76–90, doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>.","ama":"Bauer U, Kerber M, Reininghaus J, Wagner H. Phat - Persistent homology algorithms toolbox. <i>Journal of Symbolic Computation</i>. 2017;78:76-90. doi:<a href=\"https://doi.org/10.1016/j.jsc.2016.03.008\">10.1016/j.jsc.2016.03.008</a>","short":"U. Bauer, M. Kerber, J. Reininghaus, H. Wagner, Journal of Symbolic Computation 78 (2017) 76–90.","ista":"Bauer U, Kerber M, Reininghaus J, Wagner H. 2017. Phat - Persistent homology algorithms toolbox. Journal of Symbolic Computation. 78, 76–90."},"publication_status":"published"},{"tmp":{"legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","image":"/images/cc_by_nc_nd.png","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","short":"CC BY-NC-ND (4.0)"},"ddc":["004"],"related_material":{"record":[{"status":"public","relation":"dissertation_contains","id":"1399"}]},"ec_funded":1,"year":"2016","doi":"10.1016/j.aim.2015.10.004","pubrep_id":"774","title":"Approximation and convergence of the intrinsic volume","volume":287,"publist_id":"5488","date_updated":"2023-09-07T11:41:25Z","oa":1,"acknowledgement":"This research is partially supported by the Toposys project FP7-ICT-318493-STREP, and by ESF under the ACAT Research Network Programme.\r\nBoth authors thank Anne Marie Svane for her comments on an early version of this paper. The second author wishes to thank Eva B. Vedel Jensen and Markus Kiderlen from Aarhus University for enlightening discussions and their kind hospitality during a visit of their department in 2014.","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","project":[{"grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","oa_version":"Published Version","_id":"1662","publication_status":"published","citation":{"short":"H. Edelsbrunner, F. Pausinger, Advances in Mathematics 287 (2016) 674–703.","ista":"Edelsbrunner H, Pausinger F. 2016. Approximation and convergence of the intrinsic volume. Advances in Mathematics. 287, 674–703.","mla":"Edelsbrunner, Herbert, and Florian Pausinger. “Approximation and Convergence of the Intrinsic Volume.” <i>Advances in Mathematics</i>, vol. 287, Academic Press, 2016, pp. 674–703, doi:<a href=\"https://doi.org/10.1016/j.aim.2015.10.004\">10.1016/j.aim.2015.10.004</a>.","ama":"Edelsbrunner H, Pausinger F. Approximation and convergence of the intrinsic volume. <i>Advances in Mathematics</i>. 2016;287:674-703. doi:<a href=\"https://doi.org/10.1016/j.aim.2015.10.004\">10.1016/j.aim.2015.10.004</a>","chicago":"Edelsbrunner, Herbert, and Florian Pausinger. “Approximation and Convergence of the Intrinsic Volume.” <i>Advances in Mathematics</i>. Academic Press, 2016. <a href=\"https://doi.org/10.1016/j.aim.2015.10.004\">https://doi.org/10.1016/j.aim.2015.10.004</a>.","apa":"Edelsbrunner, H., &#38; Pausinger, F. (2016). Approximation and convergence of the intrinsic volume. <i>Advances in Mathematics</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.aim.2015.10.004\">https://doi.org/10.1016/j.aim.2015.10.004</a>","ieee":"H. Edelsbrunner and F. Pausinger, “Approximation and convergence of the intrinsic volume,” <i>Advances in Mathematics</i>, vol. 287. Academic Press, pp. 674–703, 2016."},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833"},{"id":"2A77D7A2-F248-11E8-B48F-1D18A9856A87","first_name":"Florian","last_name":"Pausinger","full_name":"Pausinger, Florian","orcid":"0000-0002-8379-3768"}],"abstract":[{"lang":"eng","text":"We introduce a modification of the classic notion of intrinsic volume using persistence moments of height functions. Evaluating the modified first intrinsic volume on digital approximations of a compact body with smoothly embedded boundary in Rn, we prove convergence to the first intrinsic volume of the body as the resolution of the approximation improves. We have weaker results for the other modified intrinsic volumes, proving they converge to the corresponding intrinsic volumes of the n-dimensional unit ball."}],"department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"file_id":"4928","creator":"system","relation":"main_file","content_type":"application/pdf","access_level":"open_access","date_updated":"2020-07-14T12:45:10Z","checksum":"f8869ec110c35c852ef6a37425374af7","date_created":"2018-12-12T10:12:10Z","file_size":248985,"file_name":"IST-2017-774-v1+1_2016-J-03-FirstIntVolume.pdf"}],"date_created":"2018-12-11T11:53:20Z","date_published":"2016-01-10T00:00:00Z","month":"01","language":[{"iso":"eng"}],"publisher":"Academic Press","scopus_import":1,"page":"674 - 703","file_date_updated":"2020-07-14T12:45:10Z","publication":"Advances in Mathematics","type":"journal_article","day":"10","status":"public","intvolume":"       287"},{"department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:51:12Z","month":"10","doi":"10.1016/j.endm.2016.09.030","year":"2016","date_published":"2016-10-01T00:00:00Z","ec_funded":1,"publisher":"Elsevier","title":"Multiple covers with balls II: Weighted averages","scopus_import":1,"language":[{"iso":"eng"}],"publication":"Electronic Notes in Discrete Mathematics","date_updated":"2021-01-12T06:49:41Z","volume":54,"publist_id":"5976","page":"169 - 174","_id":"1295","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","acknowledgement":"This work is partially supported by the Toposys project FP7-ICT-318493-STREP, and by ESF under the ACAT Research Network Programme.","project":[{"grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","call_identifier":"FP7"}],"quality_controlled":"1","oa_version":"None","publication_status":"published","citation":{"ama":"Edelsbrunner H, Iglesias Ham M. Multiple covers with balls II: Weighted averages. <i>Electronic Notes in Discrete Mathematics</i>. 2016;54:169-174. doi:<a href=\"https://doi.org/10.1016/j.endm.2016.09.030\">10.1016/j.endm.2016.09.030</a>","mla":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls II: Weighted Averages.” <i>Electronic Notes in Discrete Mathematics</i>, vol. 54, Elsevier, 2016, pp. 169–74, doi:<a href=\"https://doi.org/10.1016/j.endm.2016.09.030\">10.1016/j.endm.2016.09.030</a>.","short":"H. Edelsbrunner, M. Iglesias Ham, Electronic Notes in Discrete Mathematics 54 (2016) 169–174.","ista":"Edelsbrunner H, Iglesias Ham M. 2016. Multiple covers with balls II: Weighted averages. Electronic Notes in Discrete Mathematics. 54, 169–174.","apa":"Edelsbrunner, H., &#38; Iglesias Ham, M. (2016). Multiple covers with balls II: Weighted averages. <i>Electronic Notes in Discrete Mathematics</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.endm.2016.09.030\">https://doi.org/10.1016/j.endm.2016.09.030</a>","ieee":"H. Edelsbrunner and M. Iglesias Ham, “Multiple covers with balls II: Weighted averages,” <i>Electronic Notes in Discrete Mathematics</i>, vol. 54. Elsevier, pp. 169–174, 2016.","chicago":"Edelsbrunner, Herbert, and Mabel Iglesias Ham. “Multiple Covers with Balls II: Weighted Averages.” <i>Electronic Notes in Discrete Mathematics</i>. Elsevier, 2016. <a href=\"https://doi.org/10.1016/j.endm.2016.09.030\">https://doi.org/10.1016/j.endm.2016.09.030</a>."},"day":"01","type":"journal_article","intvolume":"        54","abstract":[{"text":"Voronoi diagrams and Delaunay triangulations have been extensively used to represent and compute geometric features of point configurations. We introduce a generalization to poset diagrams and poset complexes, which contain order-k and degree-k Voronoi diagrams and their duals as special cases. Extending a result of Aurenhammer from 1990, we show how to construct poset diagrams as weighted Voronoi diagrams of average balls.","lang":"eng"}],"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert"},{"last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel","first_name":"Mabel","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87"}],"status":"public"},{"issue":"8","publication":"Computational Geometry: Theory and Applications","page":"606 - 621","day":"03","type":"journal_article","intvolume":"        48","status":"public","department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:54:06Z","month":"06","date_published":"2015-06-03T00:00:00Z","publisher":"Elsevier","scopus_import":1,"language":[{"iso":"eng"}],"volume":48,"publist_id":"5305","date_updated":"2023-02-23T10:59:19Z","_id":"1805","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"None","project":[{"name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7","grant_number":"318493"}],"quality_controlled":"1","publication_status":"published","citation":{"ieee":"D. Attali, U. Bauer, O. Devillers, M. Glisse, and A. Lieutier, “Homological reconstruction and simplification in R3,” <i>Computational Geometry: Theory and Applications</i>, vol. 48, no. 8. Elsevier, pp. 606–621, 2015.","apa":"Attali, D., Bauer, U., Devillers, O., Glisse, M., &#38; Lieutier, A. (2015). Homological reconstruction and simplification in R3. <i>Computational Geometry: Theory and Applications</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.comgeo.2014.08.010\">https://doi.org/10.1016/j.comgeo.2014.08.010</a>","chicago":"Attali, Dominique, Ulrich Bauer, Olivier Devillers, Marc Glisse, and André Lieutier. “Homological Reconstruction and Simplification in R3.” <i>Computational Geometry: Theory and Applications</i>. Elsevier, 2015. <a href=\"https://doi.org/10.1016/j.comgeo.2014.08.010\">https://doi.org/10.1016/j.comgeo.2014.08.010</a>.","ama":"Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. Homological reconstruction and simplification in R3. <i>Computational Geometry: Theory and Applications</i>. 2015;48(8):606-621. doi:<a href=\"https://doi.org/10.1016/j.comgeo.2014.08.010\">10.1016/j.comgeo.2014.08.010</a>","mla":"Attali, Dominique, et al. “Homological Reconstruction and Simplification in R3.” <i>Computational Geometry: Theory and Applications</i>, vol. 48, no. 8, Elsevier, 2015, pp. 606–21, doi:<a href=\"https://doi.org/10.1016/j.comgeo.2014.08.010\">10.1016/j.comgeo.2014.08.010</a>.","ista":"Attali D, Bauer U, Devillers O, Glisse M, Lieutier A. 2015. Homological reconstruction and simplification in R3. Computational Geometry: Theory and Applications. 48(8), 606–621.","short":"D. Attali, U. Bauer, O. Devillers, M. Glisse, A. Lieutier, Computational Geometry: Theory and Applications 48 (2015) 606–621."},"abstract":[{"lang":"eng","text":"We consider the problem of deciding whether the persistent homology group of a simplicial pair (K,L) can be realized as the homology H∗(X) of some complex X with L ⊂ X ⊂ K. We show that this problem is NP-complete even if K is embedded in double-struck R3. As a consequence, we show that it is NP-hard to simplify level and sublevel sets of scalar functions on double-struck S3 within a given tolerance constraint. This problem has relevance to the visualization of medical images by isosurfaces. We also show an implication to the theory of well groups of scalar functions: not every well group can be realized by some level set, and deciding whether a well group can be realized is NP-hard."}],"author":[{"last_name":"Attali","full_name":"Attali, Dominique","first_name":"Dominique"},{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer","first_name":"Ulrich"},{"first_name":"Olivier","last_name":"Devillers","full_name":"Devillers, Olivier"},{"first_name":"Marc","full_name":"Glisse, Marc","last_name":"Glisse"},{"first_name":"André","full_name":"Lieutier, André","last_name":"Lieutier"}],"related_material":{"record":[{"relation":"earlier_version","id":"2812","status":"public"}]},"doi":"10.1016/j.comgeo.2014.08.010","year":"2015","ec_funded":1,"title":"Homological reconstruction and simplification in R3"},{"volume":15,"oa":1,"publist_id":"5022","date_updated":"2021-01-12T06:54:53Z","_id":"2035","acknowledgement":"This research is partially supported by the Toposys project FP7-ICT-318493-STREP, by ESF under the ACAT Research Network Programme, by the Russian Government under mega project 11.G34.31.0053, and by the Polish National Science Center under Grant No. N201 419639.","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"quality_controlled":"1","oa_version":"Published Version","publication_status":"published","citation":{"chicago":"Edelsbrunner, Herbert, Grzegorz Jablonski, and Marian Mrozek. “The Persistent Homology of a Self-Map.” <i>Foundations of Computational Mathematics</i>. Springer, 2015. <a href=\"https://doi.org/10.1007/s10208-014-9223-y\">https://doi.org/10.1007/s10208-014-9223-y</a>.","apa":"Edelsbrunner, H., Jablonski, G., &#38; Mrozek, M. (2015). The persistent homology of a self-map. <i>Foundations of Computational Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s10208-014-9223-y\">https://doi.org/10.1007/s10208-014-9223-y</a>","ieee":"H. Edelsbrunner, G. Jablonski, and M. Mrozek, “The persistent homology of a self-map,” <i>Foundations of Computational Mathematics</i>, vol. 15, no. 5. Springer, pp. 1213–1244, 2015.","ista":"Edelsbrunner H, Jablonski G, Mrozek M. 2015. The persistent homology of a self-map. Foundations of Computational Mathematics. 15(5), 1213–1244.","short":"H. Edelsbrunner, G. Jablonski, M. Mrozek, Foundations of Computational Mathematics 15 (2015) 1213–1244.","ama":"Edelsbrunner H, Jablonski G, Mrozek M. The persistent homology of a self-map. <i>Foundations of Computational Mathematics</i>. 2015;15(5):1213-1244. doi:<a href=\"https://doi.org/10.1007/s10208-014-9223-y\">10.1007/s10208-014-9223-y</a>","mla":"Edelsbrunner, Herbert, et al. “The Persistent Homology of a Self-Map.” <i>Foundations of Computational Mathematics</i>, vol. 15, no. 5, Springer, 2015, pp. 1213–44, doi:<a href=\"https://doi.org/10.1007/s10208-014-9223-y\">10.1007/s10208-014-9223-y</a>."},"abstract":[{"lang":"eng","text":"Considering a continuous self-map and the induced endomorphism on homology, we study the eigenvalues and eigenspaces of the latter. Taking a filtration of representations, we define the persistence of the eigenspaces, effectively introducing a hierarchical organization of the map. The algorithm that computes this information for a finite sample is proved to be stable, and to give the correct answer for a sufficiently dense sample. Results computed with an implementation of the algorithm provide evidence of its practical utility.\r\n"}],"author":[{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"4483EF78-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-3536-9866","full_name":"Jablonski, Grzegorz","last_name":"Jablonski","first_name":"Grzegorz"},{"full_name":"Mrozek, Marian","last_name":"Mrozek","first_name":"Marian"}],"tmp":{"image":"/images/cc_by.png","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","short":"CC BY (4.0)"},"ddc":["000"],"year":"2015","doi":"10.1007/s10208-014-9223-y","ec_funded":1,"pubrep_id":"486","title":"The persistent homology of a self-map","issue":"5","publication":"Foundations of Computational Mathematics","page":"1213 - 1244","file_date_updated":"2020-07-14T12:45:26Z","day":"01","type":"journal_article","intvolume":"        15","status":"public","has_accepted_license":"1","department":[{"_id":"HeEd"}],"file":[{"file_size":1317546,"file_name":"IST-2016-486-v1+1_s10208-014-9223-y.pdf","checksum":"3566f3a8b0c1bc550e62914a88c584ff","date_created":"2018-12-12T10:08:10Z","access_level":"open_access","date_updated":"2020-07-14T12:45:26Z","relation":"main_file","content_type":"application/pdf","creator":"system","file_id":"4670"}],"date_created":"2018-12-11T11:55:20Z","month":"10","date_published":"2015-10-01T00:00:00Z","publisher":"Springer","scopus_import":1,"language":[{"iso":"eng"}]},{"publisher":"Queen's University","title":"Relaxed disk packing","scopus_import":1,"language":[{"iso":"eng"}],"month":"08","year":"2015","ec_funded":1,"date_published":"2015-08-01T00:00:00Z","conference":{"location":"Ontario, Canada","end_date":"2015-08-12","name":"CCCG: Canadian Conference on Computational Geometry","start_date":"2015-08-10"},"date_created":"2018-12-11T11:52:21Z","department":[{"_id":"HeEd"}],"main_file_link":[{"url":"https://arxiv.org/abs/1505.03402","open_access":"1"}],"abstract":[{"text":"Motivated by biological questions, we study configurations of equal-sized disks in the Euclidean plane that neither pack nor cover. Measuring the quality by the probability that a random point lies in exactly one disk, we show that the regular hexagonal grid gives the maximum among lattice configurations. ","lang":"eng"}],"author":[{"first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Mabel","last_name":"Iglesias Ham","full_name":"Iglesias Ham, Mabel","id":"41B58C0C-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Vitaliy","last_name":"Kurlin","full_name":"Kurlin, Vitaliy"}],"status":"public","publication_status":"published","day":"01","citation":{"ista":"Edelsbrunner H, Iglesias Ham M, Kurlin V. 2015. Relaxed disk packing. Proceedings of the 27th Canadian Conference on Computational Geometry. CCCG: Canadian Conference on Computational Geometry vol. 2015–August, 128–135.","short":"H. Edelsbrunner, M. Iglesias Ham, V. Kurlin, in:, Proceedings of the 27th Canadian Conference on Computational Geometry, Queen’s University, 2015, pp. 128–135.","ama":"Edelsbrunner H, Iglesias Ham M, Kurlin V. Relaxed disk packing. In: <i>Proceedings of the 27th Canadian Conference on Computational Geometry</i>. Vol 2015-August. Queen’s University; 2015:128-135.","mla":"Edelsbrunner, Herbert, et al. “Relaxed Disk Packing.” <i>Proceedings of the 27th Canadian Conference on Computational Geometry</i>, vol. 2015–August, Queen’s University, 2015, pp. 128–35.","chicago":"Edelsbrunner, Herbert, Mabel Iglesias Ham, and Vitaliy Kurlin. “Relaxed Disk Packing.” In <i>Proceedings of the 27th Canadian Conference on Computational Geometry</i>, 2015–August:128–35. Queen’s University, 2015.","ieee":"H. Edelsbrunner, M. Iglesias Ham, and V. Kurlin, “Relaxed disk packing,” in <i>Proceedings of the 27th Canadian Conference on Computational Geometry</i>, Ontario, Canada, 2015, vol. 2015–August, pp. 128–135.","apa":"Edelsbrunner, H., Iglesias Ham, M., &#38; Kurlin, V. (2015). Relaxed disk packing. In <i>Proceedings of the 27th Canadian Conference on Computational Geometry</i> (Vol. 2015–August, pp. 128–135). Ontario, Canada: Queen’s University."},"type":"conference","_id":"1495","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","project":[{"call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493"}],"oa_version":"Submitted Version","publication":"Proceedings of the 27th Canadian Conference on Computational Geometry","volume":"2015-August","date_updated":"2021-01-12T06:51:09Z","oa":1,"publist_id":"5684","page":"128-135"},{"date_created":"2022-03-04T08:33:57Z","department":[{"_id":"HeEd"}],"publisher":"Springer Nature","scopus_import":"1","language":[{"iso":"eng"}],"month":"03","date_published":"2014-03-19T00:00:00Z","publication":"Topological Methods in Data Analysis and Visualization III.","page":"135-150","status":"public","day":"19","type":"book_chapter","series_title":"Mathematics and Visualization","title":"Notes on the simplification of the Morse-Smale complex","year":"2014","doi":"10.1007/978-3-319-04099-8_9","place":"Cham","ec_funded":1,"_id":"10817","publication_identifier":{"eissn":["2197-666X"],"isbn":["9783319040981"],"issn":["1612-3786"],"eisbn":["9783319040998"]},"acknowledgement":"This research is supported and funded by the Digiteo unTopoVis project, the TOPOSYS project FP7-ICT-318493-STREP, and MPC-VCC.","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"None","project":[{"grant_number":"318493","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"quality_controlled":"1","editor":[{"first_name":"Peer-Timo","full_name":"Bremer, Peer-Timo","last_name":"Bremer"},{"first_name":"Ingrid","last_name":"Hotz","full_name":"Hotz, Ingrid"},{"full_name":"Pascucci, Valerio","last_name":"Pascucci","first_name":"Valerio"},{"last_name":"Peikert","full_name":"Peikert, Ronald","first_name":"Ronald"}],"date_updated":"2023-09-05T15:33:45Z","article_processing_charge":"No","abstract":[{"text":"The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption.","lang":"eng"}],"author":[{"first_name":"David","last_name":"Günther","full_name":"Günther, David"},{"first_name":"Jan","last_name":"Reininghaus","full_name":"Reininghaus, Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Hans-Peter","full_name":"Seidel, Hans-Peter","last_name":"Seidel"},{"first_name":"Tino","full_name":"Weinkauf, Tino","last_name":"Weinkauf"}],"publication_status":"published","citation":{"chicago":"Günther, David, Jan Reininghaus, Hans-Peter Seidel, and Tino Weinkauf. “Notes on the Simplification of the Morse-Smale Complex.” In <i>Topological Methods in Data Analysis and Visualization III.</i>, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 135–50. Mathematics and Visualization. Cham: Springer Nature, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">https://doi.org/10.1007/978-3-319-04099-8_9</a>.","ieee":"D. Günther, J. Reininghaus, H.-P. Seidel, and T. Weinkauf, “Notes on the simplification of the Morse-Smale complex,” in <i>Topological Methods in Data Analysis and Visualization III.</i>, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer Nature, 2014, pp. 135–150.","apa":"Günther, D., Reininghaus, J., Seidel, H.-P., &#38; Weinkauf, T. (2014). Notes on the simplification of the Morse-Smale complex. In P.-T. Bremer, I. Hotz, V. Pascucci, &#38; R. Peikert (Eds.), <i>Topological Methods in Data Analysis and Visualization III.</i> (pp. 135–150). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">https://doi.org/10.1007/978-3-319-04099-8_9</a>","short":"D. Günther, J. Reininghaus, H.-P. Seidel, T. Weinkauf, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III., Springer Nature, Cham, 2014, pp. 135–150.","ista":"Günther D, Reininghaus J, Seidel H-P, Weinkauf T. 2014.Notes on the simplification of the Morse-Smale complex. In: Topological Methods in Data Analysis and Visualization III. , 135–150.","ama":"Günther D, Reininghaus J, Seidel H-P, Weinkauf T. Notes on the simplification of the Morse-Smale complex. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. <i>Topological Methods in Data Analysis and Visualization III.</i> Mathematics and Visualization. Cham: Springer Nature; 2014:135-150. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">10.1007/978-3-319-04099-8_9</a>","mla":"Günther, David, et al. “Notes on the Simplification of the Morse-Smale Complex.” <i>Topological Methods in Data Analysis and Visualization III.</i>, edited by Peer-Timo Bremer et al., Springer Nature, 2014, pp. 135–50, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">10.1007/978-3-319-04099-8_9</a>."}},{"scopus_import":"1","publisher":"Springer","language":[{"iso":"eng"}],"month":"03","date_published":"2014-03-19T00:00:00Z","date_created":"2022-03-21T07:11:23Z","department":[{"_id":"HeEd"}],"intvolume":"         1","status":"public","day":"19","series_title":"Mathematics and Visualization","type":"book_chapter","publication":"Topological Methods in Data Analysis and Visualization III ","page":"55-69","title":"Toward the extraction of saddle periodic orbits","doi":"10.1007/978-3-319-04099-8_4","year":"2014","place":"Cham","ec_funded":1,"abstract":[{"lang":"eng","text":"Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data."}],"author":[{"full_name":"Kasten, Jens","last_name":"Kasten","first_name":"Jens"},{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","full_name":"Reininghaus, Jan","last_name":"Reininghaus"},{"first_name":"Wieland","full_name":"Reich, Wieland","last_name":"Reich"},{"full_name":"Scheuermann, Gerik","last_name":"Scheuermann","first_name":"Gerik"}],"citation":{"mla":"Kasten, Jens, et al. “Toward the Extraction of Saddle Periodic Orbits.” <i>Topological Methods in Data Analysis and Visualization III </i>, edited by Peer-Timo Bremer et al., vol. 1, Springer, 2014, pp. 55–69, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">10.1007/978-3-319-04099-8_4</a>.","ama":"Kasten J, Reininghaus J, Reich W, Scheuermann G. Toward the extraction of saddle periodic orbits. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. <i>Topological Methods in Data Analysis and Visualization III </i>. Vol 1. Mathematics and Visualization. Cham: Springer; 2014:55-69. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">10.1007/978-3-319-04099-8_4</a>","ista":"Kasten J, Reininghaus J, Reich W, Scheuermann G. 2014.Toward the extraction of saddle periodic orbits. In: Topological Methods in Data Analysis and Visualization III . vol. 1, 55–69.","short":"J. Kasten, J. Reininghaus, W. Reich, G. Scheuermann, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III , Springer, Cham, 2014, pp. 55–69.","apa":"Kasten, J., Reininghaus, J., Reich, W., &#38; Scheuermann, G. (2014). Toward the extraction of saddle periodic orbits. In P.-T. Bremer, I. Hotz, V. Pascucci, &#38; R. Peikert (Eds.), <i>Topological Methods in Data Analysis and Visualization III </i> (Vol. 1, pp. 55–69). Cham: Springer. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">https://doi.org/10.1007/978-3-319-04099-8_4</a>","ieee":"J. Kasten, J. Reininghaus, W. Reich, and G. Scheuermann, “Toward the extraction of saddle periodic orbits,” in <i>Topological Methods in Data Analysis and Visualization III </i>, vol. 1, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer, 2014, pp. 55–69.","chicago":"Kasten, Jens, Jan Reininghaus, Wieland Reich, and Gerik Scheuermann. “Toward the Extraction of Saddle Periodic Orbits.” In <i>Topological Methods in Data Analysis and Visualization III </i>, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 1:55–69. Mathematics and Visualization. Cham: Springer, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">https://doi.org/10.1007/978-3-319-04099-8_4</a>."},"publication_status":"published","publication_identifier":{"eissn":["2197-666X"],"isbn":["9783319040981"],"issn":["1612-3786"],"eisbn":["9783319040998"]},"_id":"10893","project":[{"grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"oa_version":"None","quality_controlled":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","acknowledgement":"First, we thank the reviewers of this paper for their ideas and critical comments. In addition, we thank Ronny Peikert and Filip Sadlo for a fruitful discussions. This research is supported by the European Commission under the TOPOSYS project FP7-ICT-318493-STREP, the European Social Fund (ESF App. No. 100098251), and the European Science Foundation under the ACAT Research Network Program.","editor":[{"full_name":"Bremer, Peer-Timo","last_name":"Bremer","first_name":"Peer-Timo"},{"full_name":"Hotz, Ingrid","last_name":"Hotz","first_name":"Ingrid"},{"last_name":"Pascucci","full_name":"Pascucci, Valerio","first_name":"Valerio"},{"first_name":"Ronald","full_name":"Peikert, Ronald","last_name":"Peikert"}],"article_processing_charge":"No","volume":1,"date_updated":"2022-06-21T12:01:47Z"},{"month":"01","date_published":"2014-01-01T00:00:00Z","publisher":"Society of Industrial and Applied Mathematics","scopus_import":1,"language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"conference":{"name":"ALENEX: Algorithm Engineering and Experiments","location":"Portland, USA","end_date":"2014-01-05","start_date":"2014-01-05"},"date_created":"2018-12-11T11:55:23Z","day":"01","type":"conference","status":"public","publication":"Proceedings of the Workshop on Algorithm Engineering and Experiments","page":"31 - 38","doi":"10.1137/1.9781611973198.4","year":"2014","ec_funded":1,"title":"Distributed computation of persistent homology","main_file_link":[{"url":"http://arxiv.org/abs/1310.0710","open_access":"1"}],"publication_status":"published","citation":{"ista":"Bauer U, Kerber M, Reininghaus J. 2014. Distributed computation of persistent homology. Proceedings of the Workshop on Algorithm Engineering and Experiments. ALENEX: Algorithm Engineering and Experiments, 31–38.","short":"U. Bauer, M. Kerber, J. Reininghaus, in:, C.  McGeoch, U. Meyer (Eds.), Proceedings of the Workshop on Algorithm Engineering and Experiments, Society of Industrial and Applied Mathematics, 2014, pp. 31–38.","mla":"Bauer, Ulrich, et al. “Distributed Computation of Persistent Homology.” <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>, edited by Catherine  McGeoch and Ulrich Meyer, Society of Industrial and Applied Mathematics, 2014, pp. 31–38, doi:<a href=\"https://doi.org/10.1137/1.9781611973198.4\">10.1137/1.9781611973198.4</a>.","ama":"Bauer U, Kerber M, Reininghaus J. Distributed computation of persistent homology. In:  McGeoch C, Meyer U, eds. <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>. Society of Industrial and Applied Mathematics; 2014:31-38. doi:<a href=\"https://doi.org/10.1137/1.9781611973198.4\">10.1137/1.9781611973198.4</a>","chicago":"Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Distributed Computation of Persistent Homology.” In <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>, edited by Catherine  McGeoch and Ulrich Meyer, 31–38. Society of Industrial and Applied Mathematics, 2014. <a href=\"https://doi.org/10.1137/1.9781611973198.4\">https://doi.org/10.1137/1.9781611973198.4</a>.","apa":"Bauer, U., Kerber, M., &#38; Reininghaus, J. (2014). Distributed computation of persistent homology. In C.  McGeoch &#38; U. Meyer (Eds.), <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i> (pp. 31–38). Portland, USA: Society of Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611973198.4\">https://doi.org/10.1137/1.9781611973198.4</a>","ieee":"U. Bauer, M. Kerber, and J. Reininghaus, “Distributed computation of persistent homology,” in <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>, Portland, USA, 2014, pp. 31–38."},"abstract":[{"lang":"eng","text":"Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node."}],"author":[{"orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-8030-9299","last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"},{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","full_name":"Reininghaus, Jan","last_name":"Reininghaus","first_name":"Jan"}],"editor":[{"last_name":" McGeoch","full_name":" McGeoch, Catherine","first_name":"Catherine"},{"full_name":"Meyer, Ulrich","last_name":"Meyer","first_name":"Ulrich"}],"date_updated":"2021-01-12T06:54:56Z","publist_id":"5008","oa":1,"_id":"2043","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","quality_controlled":"1","project":[{"grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"oa_version":"Submitted Version"},{"page":"103 - 117","publication":"Topological Methods in Data Analysis and Visualization III","series_title":"Mathematics and Visualization","type":"book_chapter","day":"19","status":"public","department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:55:23Z","date_published":"2014-03-19T00:00:00Z","month":"03","language":[{"iso":"eng"}],"scopus_import":1,"publisher":"Springer","oa":1,"publist_id":"5007","date_updated":"2021-01-12T06:54:56Z","editor":[{"last_name":"Bremer","full_name":"Bremer, Peer-Timo","first_name":"Peer-Timo"},{"full_name":"Hotz, Ingrid","last_name":"Hotz","first_name":"Ingrid"},{"first_name":"Valerio","last_name":"Pascucci","full_name":"Pascucci, Valerio"},{"full_name":"Peikert, Ronald","last_name":"Peikert","first_name":"Ronald"}],"oa_version":"Submitted Version","project":[{"grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"quality_controlled":"1","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","_id":"2044","citation":{"short":"U. Bauer, M. Kerber, J. Reininghaus, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III, Springer, 2014, pp. 103–117.","ista":"Bauer U, Kerber M, Reininghaus J. 2014.Clear and Compress: Computing Persistent Homology in Chunks. In: Topological Methods in Data Analysis and Visualization III. , 103–117.","mla":"Bauer, Ulrich, et al. “Clear and Compress: Computing Persistent Homology in Chunks.” <i>Topological Methods in Data Analysis and Visualization III</i>, edited by Peer-Timo Bremer et al., Springer, 2014, pp. 103–17, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">10.1007/978-3-319-04099-8_7</a>.","ama":"Bauer U, Kerber M, Reininghaus J. Clear and Compress: Computing Persistent Homology in Chunks. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. <i>Topological Methods in Data Analysis and Visualization III</i>. Mathematics and Visualization. Springer; 2014:103-117. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">10.1007/978-3-319-04099-8_7</a>","chicago":"Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Clear and Compress: Computing Persistent Homology in Chunks.” In <i>Topological Methods in Data Analysis and Visualization III</i>, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 103–17. Mathematics and Visualization. Springer, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">https://doi.org/10.1007/978-3-319-04099-8_7</a>.","apa":"Bauer, U., Kerber, M., &#38; Reininghaus, J. (2014). Clear and Compress: Computing Persistent Homology in Chunks. In P.-T. Bremer, I. Hotz, V. Pascucci, &#38; R. Peikert (Eds.), <i>Topological Methods in Data Analysis and Visualization III</i> (pp. 103–117). Springer. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">https://doi.org/10.1007/978-3-319-04099-8_7</a>","ieee":"U. Bauer, M. Kerber, and J. Reininghaus, “Clear and Compress: Computing Persistent Homology in Chunks,” in <i>Topological Methods in Data Analysis and Visualization III</i>, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Springer, 2014, pp. 103–117."},"publication_status":"published","author":[{"first_name":"Ulrich","last_name":"Bauer","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Kerber, Michael","last_name":"Kerber","orcid":"0000-0002-8030-9299","first_name":"Michael"},{"first_name":"Jan","full_name":"Reininghaus, Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1303.0477"}],"ec_funded":1,"year":"2014","doi":"10.1007/978-3-319-04099-8_7","title":"Clear and Compress: Computing Persistent Homology in Chunks"},{"publication":"Proceedings of the Annual Symposium on Computational Geometry","page":"355 - 364","oa":1,"date_updated":"2021-01-12T06:55:38Z","publist_id":"4853","_id":"2153","project":[{"grant_number":"318493","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"quality_controlled":"1","oa_version":"Submitted Version","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"ieee":"U. Bauer and M. Lesnick, “Induced matchings of barcodes and the algebraic stability of persistence,” in <i>Proceedings of the Annual Symposium on Computational Geometry</i>, Kyoto, Japan, 2014, pp. 355–364.","apa":"Bauer, U., &#38; Lesnick, M. (2014). Induced matchings of barcodes and the algebraic stability of persistence. In <i>Proceedings of the Annual Symposium on Computational Geometry</i> (pp. 355–364). Kyoto, Japan: ACM. <a href=\"https://doi.org/10.1145/2582112.2582168\">https://doi.org/10.1145/2582112.2582168</a>","chicago":"Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the Algebraic Stability of Persistence.” In <i>Proceedings of the Annual Symposium on Computational Geometry</i>, 355–64. ACM, 2014. <a href=\"https://doi.org/10.1145/2582112.2582168\">https://doi.org/10.1145/2582112.2582168</a>.","mla":"Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the Algebraic Stability of Persistence.” <i>Proceedings of the Annual Symposium on Computational Geometry</i>, ACM, 2014, pp. 355–64, doi:<a href=\"https://doi.org/10.1145/2582112.2582168\">10.1145/2582112.2582168</a>.","ama":"Bauer U, Lesnick M. Induced matchings of barcodes and the algebraic stability of persistence. In: <i>Proceedings of the Annual Symposium on Computational Geometry</i>. ACM; 2014:355-364. doi:<a href=\"https://doi.org/10.1145/2582112.2582168\">10.1145/2582112.2582168</a>","ista":"Bauer U, Lesnick M. 2014. Induced matchings of barcodes and the algebraic stability of persistence. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 355–364.","short":"U. Bauer, M. Lesnick, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–364."},"day":"01","publication_status":"published","type":"conference","abstract":[{"lang":"eng","text":"We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s)."}],"status":"public","author":[{"first_name":"Ulrich","full_name":"Bauer, Ulrich","last_name":"Bauer","orcid":"0000-0002-9683-0724","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Michael","full_name":"Lesnick, Michael","last_name":"Lesnick"}],"main_file_link":[{"url":"http://arxiv.org/abs/1311.3681","open_access":"1"}],"department":[{"_id":"HeEd"}],"conference":{"start_date":"2014-06-08","location":"Kyoto, Japan","name":"SoCG: Symposium on Computational Geometry","end_date":"2014-06-11"},"date_created":"2018-12-11T11:56:01Z","year":"2014","month":"06","doi":"10.1145/2582112.2582168","ec_funded":1,"date_published":"2014-06-01T00:00:00Z","title":"Induced matchings of barcodes and the algebraic stability of persistence","scopus_import":1,"publisher":"ACM","language":[{"iso":"eng"}]},{"conference":{"end_date":"2014-06-11","name":"SoCG: Symposium on Computational Geometry","location":"Kyoto, Japan","start_date":"2014-06-08"},"date_created":"2018-12-11T11:56:01Z","department":[{"_id":"HeEd"}],"scopus_import":1,"publisher":"ACM","language":[{"iso":"eng"}],"month":"06","date_published":"2014-06-01T00:00:00Z","publication":"Proceedings of the Annual Symposium on Computational Geometry","page":"484 - 490","status":"public","day":"01","type":"conference","main_file_link":[{"url":"http://arxiv.org/abs/1312.1231","open_access":"1"}],"title":"The morse theory of Čech and Delaunay filtrations","year":"2014","doi":"10.1145/2582112.2582167","ec_funded":1,"_id":"2155","project":[{"grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"quality_controlled":"1","oa_version":"Submitted Version","acknowledgement":"This research is partially supported by ESF under the ACAT Research Network Programme, and by the Russian Government under mega project 11.G34.31.0053","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","publist_id":"4851","date_updated":"2021-01-12T06:55:38Z","oa":1,"abstract":[{"text":"Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).","lang":"eng"}],"author":[{"first_name":"Ulrich","full_name":"Bauer, Ulrich","last_name":"Bauer","orcid":"0000-0002-9683-0724","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"citation":{"ieee":"U. Bauer and H. Edelsbrunner, “The morse theory of Čech and Delaunay filtrations,” in <i>Proceedings of the Annual Symposium on Computational Geometry</i>, Kyoto, Japan, 2014, pp. 484–490.","apa":"Bauer, U., &#38; Edelsbrunner, H. (2014). The morse theory of Čech and Delaunay filtrations. In <i>Proceedings of the Annual Symposium on Computational Geometry</i> (pp. 484–490). Kyoto, Japan: ACM. <a href=\"https://doi.org/10.1145/2582112.2582167\">https://doi.org/10.1145/2582112.2582167</a>","chicago":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” In <i>Proceedings of the Annual Symposium on Computational Geometry</i>, 484–90. ACM, 2014. <a href=\"https://doi.org/10.1145/2582112.2582167\">https://doi.org/10.1145/2582112.2582167</a>.","mla":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” <i>Proceedings of the Annual Symposium on Computational Geometry</i>, ACM, 2014, pp. 484–90, doi:<a href=\"https://doi.org/10.1145/2582112.2582167\">10.1145/2582112.2582167</a>.","ama":"Bauer U, Edelsbrunner H. The morse theory of Čech and Delaunay filtrations. In: <i>Proceedings of the Annual Symposium on Computational Geometry</i>. ACM; 2014:484-490. doi:<a href=\"https://doi.org/10.1145/2582112.2582167\">10.1145/2582112.2582167</a>","short":"U. Bauer, H. Edelsbrunner, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–490.","ista":"Bauer U, Edelsbrunner H. 2014. The morse theory of Čech and Delaunay filtrations. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 484–490."},"publication_status":"published"},{"conference":{"location":"Kyoto, Japan","name":"SoCG: Symposium on Computational Geometry","end_date":"2014-06-11","start_date":"2014-06-08"},"date_created":"2018-12-11T11:56:02Z","department":[{"_id":"HeEd"}],"scopus_import":1,"publisher":"ACM","language":[{"iso":"eng"}],"month":"06","date_published":"2014-06-01T00:00:00Z","publication":"Proceedings of the Annual Symposium on Computational Geometry","page":"464 - 473","status":"public","day":"01","type":"conference","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1307.2839"}],"title":"Measuring distance between Reeb graphs","year":"2014","doi":"10.1145/2582112.2582169","ec_funded":1,"_id":"2156","quality_controlled":"1","oa_version":"Submitted Version","project":[{"call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493"}],"acknowledgement":"National Science Foundation under grants CCF-1319406, CCF-1116258.","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","date_updated":"2021-01-12T06:55:39Z","oa":1,"publist_id":"4850","abstract":[{"lang":"eng","text":"We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s)."}],"author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer","first_name":"Ulrich"},{"last_name":"Ge","full_name":"Ge, Xiaoyin","first_name":"Xiaoyin"},{"last_name":"Wang","full_name":"Wang, Yusu","first_name":"Yusu"}],"citation":{"mla":"Bauer, Ulrich, et al. “Measuring Distance between Reeb Graphs.” <i>Proceedings of the Annual Symposium on Computational Geometry</i>, ACM, 2014, pp. 464–73, doi:<a href=\"https://doi.org/10.1145/2582112.2582169\">10.1145/2582112.2582169</a>.","ama":"Bauer U, Ge X, Wang Y. Measuring distance between Reeb graphs. In: <i>Proceedings of the Annual Symposium on Computational Geometry</i>. ACM; 2014:464-473. doi:<a href=\"https://doi.org/10.1145/2582112.2582169\">10.1145/2582112.2582169</a>","short":"U. Bauer, X. Ge, Y. Wang, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–473.","ista":"Bauer U, Ge X, Wang Y. 2014. Measuring distance between Reeb graphs. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 464–473.","ieee":"U. Bauer, X. Ge, and Y. Wang, “Measuring distance between Reeb graphs,” in <i>Proceedings of the Annual Symposium on Computational Geometry</i>, Kyoto, Japan, 2014, pp. 464–473.","apa":"Bauer, U., Ge, X., &#38; Wang, Y. (2014). Measuring distance between Reeb graphs. In <i>Proceedings of the Annual Symposium on Computational Geometry</i> (pp. 464–473). Kyoto, Japan: ACM. <a href=\"https://doi.org/10.1145/2582112.2582169\">https://doi.org/10.1145/2582112.2582169</a>","chicago":"Bauer, Ulrich, Xiaoyin Ge, and Yusu Wang. “Measuring Distance between Reeb Graphs.” In <i>Proceedings of the Annual Symposium on Computational Geometry</i>, 464–73. ACM, 2014. <a href=\"https://doi.org/10.1145/2582112.2582169\">https://doi.org/10.1145/2582112.2582169</a>."},"publication_status":"published"},{"date_published":"2014-09-01T00:00:00Z","month":"09","language":[{"iso":"eng"}],"publisher":"Springer","scopus_import":1,"department":[{"_id":"HeEd"}],"has_accepted_license":"1","file":[{"creator":"system","file_id":"5204","relation":"main_file","content_type":"application/pdf","checksum":"2f93f3e63a38a85cd4404d7953913b14","date_created":"2018-12-12T10:16:18Z","file_size":3941391,"file_name":"IST-2016-549-v1+1_2014-J-06-LengthEstimate.pdf","access_level":"open_access","date_updated":"2020-07-14T12:45:35Z"}],"date_created":"2018-12-11T11:56:36Z","type":"journal_article","day":"01","status":"public","intvolume":"        50","file_date_updated":"2020-07-14T12:45:35Z","page":"164 - 177","issue":"1","publication":"Journal of Mathematical Imaging and Vision","ec_funded":1,"doi":"10.1007/s10851-013-0468-x","year":"2014","pubrep_id":"549","title":"Stable length estimates of tube-like shapes","ddc":["000"],"related_material":{"record":[{"status":"public","relation":"earlier_version","id":"2843"},{"relation":"dissertation_contains","id":"1399","status":"public"}]},"publication_status":"published","citation":{"chicago":"Edelsbrunner, Herbert, and Florian Pausinger. “Stable Length Estimates of Tube-like Shapes.” <i>Journal of Mathematical Imaging and Vision</i>. Springer, 2014. <a href=\"https://doi.org/10.1007/s10851-013-0468-x\">https://doi.org/10.1007/s10851-013-0468-x</a>.","apa":"Edelsbrunner, H., &#38; Pausinger, F. (2014). Stable length estimates of tube-like shapes. <i>Journal of Mathematical Imaging and Vision</i>. Springer. <a href=\"https://doi.org/10.1007/s10851-013-0468-x\">https://doi.org/10.1007/s10851-013-0468-x</a>","ieee":"H. Edelsbrunner and F. Pausinger, “Stable length estimates of tube-like shapes,” <i>Journal of Mathematical Imaging and Vision</i>, vol. 50, no. 1. Springer, pp. 164–177, 2014.","ista":"Edelsbrunner H, Pausinger F. 2014. Stable length estimates of tube-like shapes. Journal of Mathematical Imaging and Vision. 50(1), 164–177.","short":"H. Edelsbrunner, F. Pausinger, Journal of Mathematical Imaging and Vision 50 (2014) 164–177.","ama":"Edelsbrunner H, Pausinger F. Stable length estimates of tube-like shapes. <i>Journal of Mathematical Imaging and Vision</i>. 2014;50(1):164-177. doi:<a href=\"https://doi.org/10.1007/s10851-013-0468-x\">10.1007/s10851-013-0468-x</a>","mla":"Edelsbrunner, Herbert, and Florian Pausinger. “Stable Length Estimates of Tube-like Shapes.” <i>Journal of Mathematical Imaging and Vision</i>, vol. 50, no. 1, Springer, 2014, pp. 164–77, doi:<a href=\"https://doi.org/10.1007/s10851-013-0468-x\">10.1007/s10851-013-0468-x</a>."},"author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert"},{"first_name":"Florian","orcid":"0000-0002-8379-3768","last_name":"Pausinger","full_name":"Pausinger, Florian","id":"2A77D7A2-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"lang":"eng","text":"Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm."}],"publist_id":"4691","volume":50,"oa":1,"date_updated":"2023-09-07T11:41:25Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","project":[{"grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems"}],"oa_version":"Submitted Version","quality_controlled":"1","_id":"2255","publication_identifier":{"issn":["09249907"]}},{"page":"182-183","publication":"Graph-Based Representations in Pattern Recognition","series_title":"LNCS","type":"conference","day":"01","status":"public","intvolume":"      7877","department":[{"_id":"HeEd"}],"date_created":"2022-03-21T07:30:33Z","conference":{"start_date":"2013-05-15","end_date":"2013-05-17","name":"GbRPR: Graph-based Representations in Pattern Recognition","location":"Vienna, Austria"},"date_published":"2013-06-01T00:00:00Z","month":"06","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"Springer Nature","article_processing_charge":"No","volume":7877,"date_updated":"2023-09-05T15:10:20Z","project":[{"grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","call_identifier":"FP7"}],"quality_controlled":"1","oa_version":"None","acknowledgement":"This research is partially supported by the European Science Foundation (ESF) under the Research Network Programme, the European Union under the Toposys Project FP7-ICT-318493-STREP, the Russian Government under the Mega Project 11.G34.31.0053.","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publication_identifier":{"isbn":["9783642382208"],"eisbn":["9783642382215"],"issn":["0302-9743"],"eissn":["1611-3349"]},"_id":"10897","citation":{"ista":"Edelsbrunner H. 2013. Persistent homology in image processing. Graph-Based Representations in Pattern Recognition. GbRPR: Graph-based Representations in Pattern RecognitionLNCS vol. 7877, 182–183.","short":"H. Edelsbrunner, in:, Graph-Based Representations in Pattern Recognition, Springer Nature, Berlin, Heidelberg, 2013, pp. 182–183.","ama":"Edelsbrunner H. Persistent homology in image processing. In: <i>Graph-Based Representations in Pattern Recognition</i>. Vol 7877. LNCS. Berlin, Heidelberg: Springer Nature; 2013:182-183. doi:<a href=\"https://doi.org/10.1007/978-3-642-38221-5_19\">10.1007/978-3-642-38221-5_19</a>","mla":"Edelsbrunner, Herbert. “Persistent Homology in Image Processing.” <i>Graph-Based Representations in Pattern Recognition</i>, vol. 7877, Springer Nature, 2013, pp. 182–83, doi:<a href=\"https://doi.org/10.1007/978-3-642-38221-5_19\">10.1007/978-3-642-38221-5_19</a>.","chicago":"Edelsbrunner, Herbert. “Persistent Homology in Image Processing.” In <i>Graph-Based Representations in Pattern Recognition</i>, 7877:182–83. LNCS. Berlin, Heidelberg: Springer Nature, 2013. <a href=\"https://doi.org/10.1007/978-3-642-38221-5_19\">https://doi.org/10.1007/978-3-642-38221-5_19</a>.","apa":"Edelsbrunner, H. (2013). Persistent homology in image processing. In <i>Graph-Based Representations in Pattern Recognition</i> (Vol. 7877, pp. 182–183). Berlin, Heidelberg: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-642-38221-5_19\">https://doi.org/10.1007/978-3-642-38221-5_19</a>","ieee":"H. Edelsbrunner, “Persistent homology in image processing,” in <i>Graph-Based Representations in Pattern Recognition</i>, Vienna, Austria, 2013, vol. 7877, pp. 182–183."},"publication_status":"published","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"}],"abstract":[{"text":"Taking images is an efficient way to collect data about the physical world. It can be done fast and in exquisite detail. By definition, image processing is the field that concerns itself with the computation aimed at harnessing the information contained in images [10]. This talk is concerned with topological information. Our main thesis is that persistent homology [5] is a useful method to quantify and summarize topological information, building a bridge that connects algebraic topology with applications. We provide supporting evidence for this thesis by touching upon four technical developments in the overlap between persistent homology and image processing.","lang":"eng"}],"ec_funded":1,"place":"Berlin, Heidelberg","year":"2013","doi":"10.1007/978-3-642-38221-5_19","title":"Persistent homology in image processing"}]