[{"issue":"1","abstract":[{"text":"Given a continuous function f:X-R on a topological space, we consider the preimages of intervals and their homology groups and show how to read the ranks of these groups from the extended persistence diagram of f. In addition, we quantify the robustness of the homology classes under perturbations of f using well groups, and we show how to read the ranks of these groups from the same extended persistence diagram. The special case X=R3 has ramifications in the fields of medical imaging and scientific visualization.","lang":"eng"}],"type":"journal_article","oa_version":"Preprint","intvolume":" 15","status":"public","title":"Homology and robustness of level and interlevel sets","_id":"2859","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","day":"01","scopus_import":1,"date_published":"2013-05-01T00:00:00Z","page":"51 - 72","citation":{"chicago":"Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “Homology and Robustness of Level and Interlevel Sets.” Homology, Homotopy and Applications. International Press, 2013. https://doi.org/10.4310/HHA.2013.v15.n1.a3.","short":"P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, Homology, Homotopy and Applications 15 (2013) 51–72.","mla":"Bendich, Paul, et al. “Homology and Robustness of Level and Interlevel Sets.” Homology, Homotopy and Applications, vol. 15, no. 1, International Press, 2013, pp. 51–72, doi:10.4310/HHA.2013.v15.n1.a3.","ieee":"P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, “Homology and robustness of level and interlevel sets,” Homology, Homotopy and Applications, vol. 15, no. 1. International Press, pp. 51–72, 2013.","apa":"Bendich, P., Edelsbrunner, H., Morozov, D., & Patel, A. (2013). Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications. International Press. https://doi.org/10.4310/HHA.2013.v15.n1.a3","ista":"Bendich P, Edelsbrunner H, Morozov D, Patel A. 2013. Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications. 15(1), 51–72.","ama":"Bendich P, Edelsbrunner H, Morozov D, Patel A. Homology and robustness of level and interlevel sets. Homology, Homotopy and Applications. 2013;15(1):51-72. doi:10.4310/HHA.2013.v15.n1.a3"},"publication":"Homology, Homotopy and Applications","publist_id":"3930","volume":15,"date_updated":"2021-01-12T07:00:18Z","date_created":"2018-12-11T11:59:58Z","author":[{"full_name":"Bendich, Paul","last_name":"Bendich","first_name":"Paul","id":"43F6EC54-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"last_name":"Morozov","first_name":"Dmitriy","full_name":"Morozov, Dmitriy"},{"full_name":"Patel, Amit","last_name":"Patel","first_name":"Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87"}],"department":[{"_id":"HeEd"}],"publisher":"International Press","publication_status":"published","year":"2013","month":"05","language":[{"iso":"eng"}],"doi":"10.4310/HHA.2013.v15.n1.a3","quality_controlled":"1","external_id":{"arxiv":["1102.3389"]},"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1102.3389v1"}],"oa":1},{"oa_version":"Submitted Version","status":"public","title":"Quantifying transversality by measuring the robustness of intersections","intvolume":" 11","_id":"3377","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","abstract":[{"text":"By definition, transverse intersections are stable under in- finitesimal perturbations. Using persistent homology, we ex- tend this notion to sizeable perturbations. Specifically, we assign to each homology class of the intersection its robust- ness, the magnitude of a perturbation necessary to kill it, and prove that robustness is stable. Among the applications of this result is a stable notion of robustness for fixed points of continuous mappings and a statement of stability for con- tours of smooth mappings.","lang":"eng"}],"issue":"3","type":"journal_article","date_published":"2011-06-01T00:00:00Z","page":"345 - 361","publication":"Foundations of Computational Mathematics","citation":{"chicago":"Edelsbrunner, Herbert, Dmitriy Morozov, and Amit Patel. “Quantifying Transversality by Measuring the Robustness of Intersections.” Foundations of Computational Mathematics. Springer, 2011. https://doi.org/10.1007/s10208-011-9090-8.","mla":"Edelsbrunner, Herbert, et al. “Quantifying Transversality by Measuring the Robustness of Intersections.” Foundations of Computational Mathematics, vol. 11, no. 3, Springer, 2011, pp. 345–61, doi:10.1007/s10208-011-9090-8.","short":"H. Edelsbrunner, D. Morozov, A. Patel, Foundations of Computational Mathematics 11 (2011) 345–361.","ista":"Edelsbrunner H, Morozov D, Patel A. 2011. Quantifying transversality by measuring the robustness of intersections. Foundations of Computational Mathematics. 11(3), 345–361.","apa":"Edelsbrunner, H., Morozov, D., & Patel, A. (2011). Quantifying transversality by measuring the robustness of intersections. Foundations of Computational Mathematics. Springer. https://doi.org/10.1007/s10208-011-9090-8","ieee":"H. Edelsbrunner, D. Morozov, and A. Patel, “Quantifying transversality by measuring the robustness of intersections,” Foundations of Computational Mathematics, vol. 11, no. 3. Springer, pp. 345–361, 2011.","ama":"Edelsbrunner H, Morozov D, Patel A. Quantifying transversality by measuring the robustness of intersections. Foundations of Computational Mathematics. 2011;11(3):345-361. doi:10.1007/s10208-011-9090-8"},"day":"01","scopus_import":1,"date_updated":"2021-01-12T07:43:04Z","date_created":"2018-12-11T12:02:59Z","volume":11,"author":[{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"last_name":"Morozov","first_name":"Dmitriy","full_name":"Morozov, Dmitriy"},{"id":"34A254A0-F248-11E8-B48F-1D18A9856A87","first_name":"Amit","last_name":"Patel","full_name":"Patel, Amit"}],"publication_status":"published","department":[{"_id":"HeEd"}],"publisher":"Springer","acknowledgement":"This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.","year":"2011","publist_id":"3230","language":[{"iso":"eng"}],"doi":"10.1007/s10208-011-9090-8","quality_controlled":"1","oa":1,"main_file_link":[{"url":"http://arxiv.org/abs/0911.2142","open_access":"1"}],"month":"06"},{"publisher":"Springer","department":[{"_id":"HeEd"}],"publication_status":"published","acknowledgement":"This research is partially supported by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.","year":"2010","date_updated":"2021-01-12T07:52:15Z","date_created":"2018-12-11T12:05:13Z","author":[{"full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Morozov, Dmitriy","last_name":"Morozov","first_name":"Dmitriy"},{"full_name":"Patel, Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87","last_name":"Patel","first_name":"Amit"}],"publist_id":"2428","file_date_updated":"2020-07-14T12:46:16Z","quality_controlled":"1","oa":1,"language":[{"iso":"eng"}],"doi":"10.1007/978-3-642-15014-2_3","month":"12","status":"public","ddc":["000"],"title":"The stability of the apparent contour of an orientable 2-manifold","_id":"3795","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa_version":"Submitted Version","file":[{"file_id":"4896","relation":"main_file","checksum":"f03a44c3d1c3e2d4fedb3b94404f3fd5","date_updated":"2020-07-14T12:46:16Z","date_created":"2018-12-12T10:11:40Z","access_level":"open_access","file_name":"IST-2016-538-v1+1_2011-B-02-ApparentContour.pdf","creator":"system","content_type":"application/pdf","file_size":210710}],"pubrep_id":"538","alternative_title":["Mathematics and Visualization"],"type":"book_chapter","abstract":[{"text":"The (apparent) contour of a smooth mapping from a 2-manifold to the plane, f: M → R2 , is the set of critical values, that is, the image of the points at which the gradients of the two component functions are linearly dependent. Assuming M is compact and orientable and measuring difference with the erosion distance, we prove that the contour is stable.","lang":"eng"}],"page":"27 - 42","citation":{"mla":"Edelsbrunner, Herbert, et al. “The Stability of the Apparent Contour of an Orientable 2-Manifold.” Topological Data Analysis and Visualization: Theory, Algorithms and Applications, Springer, 2010, pp. 27–42, doi:10.1007/978-3-642-15014-2_3.","short":"H. Edelsbrunner, D. Morozov, A. Patel, in:, Topological Data Analysis and Visualization: Theory, Algorithms and Applications, Springer, 2010, pp. 27–42.","chicago":"Edelsbrunner, Herbert, Dmitriy Morozov, and Amit Patel. “The Stability of the Apparent Contour of an Orientable 2-Manifold.” In Topological Data Analysis and Visualization: Theory, Algorithms and Applications, 27–42. Springer, 2010. https://doi.org/10.1007/978-3-642-15014-2_3.","ama":"Edelsbrunner H, Morozov D, Patel A. The stability of the apparent contour of an orientable 2-manifold. In: Topological Data Analysis and Visualization: Theory, Algorithms and Applications. Springer; 2010:27-42. doi:10.1007/978-3-642-15014-2_3","ista":"Edelsbrunner H, Morozov D, Patel A. 2010.The stability of the apparent contour of an orientable 2-manifold. In: Topological Data Analysis and Visualization: Theory, Algorithms and Applications. Mathematics and Visualization, , 27–42.","ieee":"H. Edelsbrunner, D. Morozov, and A. Patel, “The stability of the apparent contour of an orientable 2-manifold,” in Topological Data Analysis and Visualization: Theory, Algorithms and Applications, Springer, 2010, pp. 27–42.","apa":"Edelsbrunner, H., Morozov, D., & Patel, A. (2010). The stability of the apparent contour of an orientable 2-manifold. In Topological Data Analysis and Visualization: Theory, Algorithms and Applications (pp. 27–42). Springer. https://doi.org/10.1007/978-3-642-15014-2_3"},"publication":"Topological Data Analysis and Visualization: Theory, Algorithms and Applications","date_published":"2010-12-22T00:00:00Z","scopus_import":1,"has_accepted_license":"1","day":"22"},{"_id":"3848","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","year":"2010","status":"public","title":"The robustness of level sets","publication_status":"published","intvolume":" 6346","department":[{"_id":"HeEd"}],"publisher":"Springer","author":[{"full_name":"Bendich, Paul","first_name":"Paul","last_name":"Bendich","id":"43F6EC54-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert"},{"full_name":"Morozov, Dmitriy","last_name":"Morozov","first_name":"Dmitriy"},{"id":"34A254A0-F248-11E8-B48F-1D18A9856A87","first_name":"Amit","last_name":"Patel","full_name":"Patel, Amit"}],"date_updated":"2021-01-12T07:52:38Z","date_created":"2018-12-11T12:05:30Z","oa_version":"None","volume":6346,"type":"conference","alternative_title":["LNCS"],"abstract":[{"text":"We define the robustness of a level set homology class of a function f:XR as the magnitude of a perturbation necessary to kill the class. Casting this notion into a group theoretic framework, we compute the robustness for each class, using a connection to extended persistent homology. The special case X=R3 has ramifications in medical imaging and scientific visualization.","lang":"eng"}],"publist_id":"2336","citation":{"chicago":"Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “The Robustness of Level Sets,” 6346:1–10. Springer, 2010. https://doi.org/10.1007/978-3-642-15775-2_1.","mla":"Bendich, Paul, et al. The Robustness of Level Sets. Vol. 6346, Springer, 2010, pp. 1–10, doi:10.1007/978-3-642-15775-2_1.","short":"P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, in:, Springer, 2010, pp. 1–10.","ista":"Bendich P, Edelsbrunner H, Morozov D, Patel A. 2010. The robustness of level sets. ESA: European Symposium on Algorithms, LNCS, vol. 6346, 1–10.","ieee":"P. Bendich, H. Edelsbrunner, D. Morozov, and A. Patel, “The robustness of level sets,” presented at the ESA: European Symposium on Algorithms, Liverpool, UK, 2010, vol. 6346, pp. 1–10.","apa":"Bendich, P., Edelsbrunner, H., Morozov, D., & Patel, A. (2010). The robustness of level sets (Vol. 6346, pp. 1–10). Presented at the ESA: European Symposium on Algorithms, Liverpool, UK: Springer. https://doi.org/10.1007/978-3-642-15775-2_1","ama":"Bendich P, Edelsbrunner H, Morozov D, Patel A. The robustness of level sets. In: Vol 6346. Springer; 2010:1-10. doi:10.1007/978-3-642-15775-2_1"},"quality_controlled":"1","page":"1 - 10","conference":{"name":"ESA: European Symposium on Algorithms","end_date":"2010-09-08","location":"Liverpool, UK","start_date":"2010-09-06"},"doi":"10.1007/978-3-642-15775-2_1","date_published":"2010-09-01T00:00:00Z","language":[{"iso":"eng"}],"scopus_import":1,"day":"01","month":"09"},{"date_published":"2010-08-10T00:00:00Z","citation":{"ista":"Bendich P, Edelsbrunner H, Kerber M, Patel A. 2010. Persistent homology under non-uniform error. MFCS: Mathematical Foundations of Computer Science, LNCS, vol. 6281, 12–23.","apa":"Bendich, P., Edelsbrunner, H., Kerber, M., & Patel, A. (2010). Persistent homology under non-uniform error (Vol. 6281, pp. 12–23). Presented at the MFCS: Mathematical Foundations of Computer Science, Brno, Czech Republic: Springer. https://doi.org/10.1007/978-3-642-15155-2_2","ieee":"P. Bendich, H. Edelsbrunner, M. Kerber, and A. Patel, “Persistent homology under non-uniform error,” presented at the MFCS: Mathematical Foundations of Computer Science, Brno, Czech Republic, 2010, vol. 6281, pp. 12–23.","ama":"Bendich P, Edelsbrunner H, Kerber M, Patel A. Persistent homology under non-uniform error. In: Vol 6281. Springer; 2010:12-23. doi:10.1007/978-3-642-15155-2_2","chicago":"Bendich, Paul, Herbert Edelsbrunner, Michael Kerber, and Amit Patel. “Persistent Homology under Non-Uniform Error,” 6281:12–23. Springer, 2010. https://doi.org/10.1007/978-3-642-15155-2_2.","mla":"Bendich, Paul, et al. Persistent Homology under Non-Uniform Error. Vol. 6281, Springer, 2010, pp. 12–23, doi:10.1007/978-3-642-15155-2_2.","short":"P. Bendich, H. Edelsbrunner, M. Kerber, A. Patel, in:, Springer, 2010, pp. 12–23."},"page":"12 - 23","day":"10","has_accepted_license":"1","scopus_import":1,"pubrep_id":"537","file":[{"file_name":"IST-2016-537-v1+1_2010-P-05-NonuniformError.pdf","access_level":"open_access","content_type":"application/pdf","file_size":142357,"creator":"system","relation":"main_file","file_id":"4994","date_created":"2018-12-12T10:13:13Z","date_updated":"2020-07-14T12:46:17Z","checksum":"af61e1c2bb42f3d556179d4692caeb1b"}],"oa_version":"Submitted Version","_id":"3849","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","ddc":["000"],"status":"public","title":"Persistent homology under non-uniform error","intvolume":" 6281","abstract":[{"text":"Using ideas from persistent homology, the robustness of a level set of a real-valued function is defined in terms of the magnitude of the perturbation necessary to kill the classes. Prior work has shown that the homology and robustness information can be read off the extended persistence diagram of the function. This paper extends these results to a non-uniform error model in which perturbations vary in their magnitude across the domain.","lang":"eng"}],"type":"conference","alternative_title":["LNCS"],"conference":{"name":"MFCS: Mathematical Foundations of Computer Science","end_date":"2010-08-27","start_date":"2010-08-23","location":"Brno, Czech Republic"},"doi":"10.1007/978-3-642-15155-2_2","language":[{"iso":"eng"}],"oa":1,"quality_controlled":"1","month":"08","author":[{"full_name":"Bendich, Paul","last_name":"Bendich","first_name":"Paul","id":"43F6EC54-F248-11E8-B48F-1D18A9856A87"},{"orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert"},{"last_name":"Kerber","first_name":"Michael","orcid":"0000-0002-8030-9299","id":"36E4574A-F248-11E8-B48F-1D18A9856A87","full_name":"Kerber, Michael"},{"full_name":"Patel, Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87","first_name":"Amit","last_name":"Patel"}],"date_created":"2018-12-11T12:05:30Z","date_updated":"2021-01-12T07:52:38Z","volume":6281,"year":"2010","publication_status":"published","publisher":"Springer","department":[{"_id":"HeEd"}],"file_date_updated":"2020-07-14T12:46:17Z","publist_id":"2333"},{"extern":1,"publist_id":"2155","abstract":[{"text":"Generalizing the concept of a Reeb graph, the Reeb space of a multivariate continuous mapping identifies points of the domain that belong to a common component of the preimage of a point in the range. We study the local and global structure of this space for generic, piecewise linear mappings on a combinatorial manifold.","lang":"eng"}],"type":"conference","date_updated":"2021-01-12T07:53:35Z","date_created":"2018-12-11T12:06:13Z","author":[{"full_name":"Herbert Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner"},{"full_name":"Harer, John","last_name":"Harer","first_name":"John"},{"full_name":"Amit Patel","last_name":"Patel","first_name":"Amit","id":"34A254A0-F248-11E8-B48F-1D18A9856A87"}],"publisher":"ACM","title":"Reeb spaces of piecewise linear mappings","publication_status":"published","status":"public","_id":"3974","year":"2008","day":"01","month":"01","date_published":"2008-01-01T00:00:00Z","doi":"10.1145/1377676.1377720","conference":{"name":"SCG: Symposium on Computational Geometry"},"page":"242 - 250","quality_controlled":0,"citation":{"short":"H. Edelsbrunner, J. Harer, A. Patel, in:, ACM, 2008, pp. 242–250.","mla":"Edelsbrunner, Herbert, et al. Reeb Spaces of Piecewise Linear Mappings. ACM, 2008, pp. 242–50, doi:10.1145/1377676.1377720.","chicago":"Edelsbrunner, Herbert, John Harer, and Amit Patel. “Reeb Spaces of Piecewise Linear Mappings,” 242–50. ACM, 2008. https://doi.org/10.1145/1377676.1377720.","ama":"Edelsbrunner H, Harer J, Patel A. Reeb spaces of piecewise linear mappings. In: ACM; 2008:242-250. doi:10.1145/1377676.1377720","apa":"Edelsbrunner, H., Harer, J., & Patel, A. (2008). Reeb spaces of piecewise linear mappings (pp. 242–250). Presented at the SCG: Symposium on Computational Geometry, ACM. https://doi.org/10.1145/1377676.1377720","ieee":"H. Edelsbrunner, J. Harer, and A. Patel, “Reeb spaces of piecewise linear mappings,” presented at the SCG: Symposium on Computational Geometry, 2008, pp. 242–250.","ista":"Edelsbrunner H, Harer J, Patel A. 2008. Reeb spaces of piecewise linear mappings. SCG: Symposium on Computational Geometry, 242–250."}}]