[{"author":[{"last_name":"Gonzalez-Diaz","first_name":"Rocio","full_name":"Gonzalez-Diaz, Rocio"},{"first_name":"Adrian","full_name":"Ion, Adrian","last_name":"Ion","id":"29F89302-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Maria Jose","full_name":"Jimenez, Maria Jose","last_name":"Jimenez"},{"last_name":"Poyatos","full_name":"Poyatos, Regina","first_name":"Regina"}],"quality_controlled":"1","date_updated":"2026-06-18T19:56:17Z","month":"08","publication":"Computer Analysis of Images and Patterns","abstract":[{"lang":"eng","text":"In this paper, we establish a correspondence between the incremental algorithm for computing AT-models [8,9] and the one for computing persistent homology [6,14,15]. We also present a decremental algorithm for computing AT-models that allows to extend the persistence computation to a wider setting. Finally, we show how to combine incremental and decremental techniques for persistent homology computation."}],"department":[{"_id":"HeEd"}],"_id":"9648","year":"2011","main_file_link":[{"url":"http://hdl.handle.net/11441/30766","open_access":"1"}],"scopus_import":"1","publication_identifier":{"isbn":["9783642236716"],"issn":["0302-9743"],"eissn":["1611-3349"]},"title":"Incremental-decremental algorithm for computing AT-models and persistent homology","day":"01","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","status":"public","date_published":"2011-08-01T00:00:00Z","doi":"10.1007/978-3-642-23672-3_35","oa_version":"Published Version","ddc":["500"],"intvolume":"      6854","publisher":"Springer Nature","volume":6854,"article_processing_charge":"No","type":"conference","date_created":"2021-07-11T22:01:19Z","citation":{"short":"R. Gonzalez-Diaz, A. Ion, M.J. Jimenez, R. Poyatos, in:, Computer Analysis of Images and Patterns, Springer Nature, 2011, pp. 286–293.","ama":"Gonzalez-Diaz R, Ion A, Jimenez MJ, Poyatos R. Incremental-decremental algorithm for computing AT-models and persistent homology. In: <i>Computer Analysis of Images and Patterns</i>. Vol 6854. Springer Nature; 2011:286-293. doi:<a href=\"https://doi.org/10.1007/978-3-642-23672-3_35\">10.1007/978-3-642-23672-3_35</a>","ieee":"R. Gonzalez-Diaz, A. Ion, M. J. Jimenez, and R. Poyatos, “Incremental-decremental algorithm for computing AT-models and persistent homology,” in <i>Computer Analysis of Images and Patterns</i>, Seville, Spain, 2011, vol. 6854, pp. 286–293.","ista":"Gonzalez-Diaz R, Ion A, Jimenez MJ, Poyatos R. 2011. Incremental-decremental algorithm for computing AT-models and persistent homology. Computer Analysis of Images and Patterns. CAIP: International Conference on Computer Analysis of Images and Patterns, LNCS, vol. 6854, 286–293.","mla":"Gonzalez-Diaz, Rocio, et al. “Incremental-Decremental Algorithm for Computing AT-Models and Persistent Homology.” <i>Computer Analysis of Images and Patterns</i>, vol. 6854, Springer Nature, 2011, pp. 286–93, doi:<a href=\"https://doi.org/10.1007/978-3-642-23672-3_35\">10.1007/978-3-642-23672-3_35</a>.","apa":"Gonzalez-Diaz, R., Ion, A., Jimenez, M. J., &#38; Poyatos, R. (2011). Incremental-decremental algorithm for computing AT-models and persistent homology. In <i>Computer Analysis of Images and Patterns</i> (Vol. 6854, pp. 286–293). Seville, Spain: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-642-23672-3_35\">https://doi.org/10.1007/978-3-642-23672-3_35</a>","chicago":"Gonzalez-Diaz, Rocio, Adrian Ion, Maria Jose Jimenez, and Regina Poyatos. “Incremental-Decremental Algorithm for Computing AT-Models and Persistent Homology.” In <i>Computer Analysis of Images and Patterns</i>, 6854:286–93. Springer Nature, 2011. <a href=\"https://doi.org/10.1007/978-3-642-23672-3_35\">https://doi.org/10.1007/978-3-642-23672-3_35</a>."},"publication_status":"published","language":[{"iso":"eng"}],"page":"286-293","alternative_title":["LNCS"],"conference":{"end_date":"2011-08-31","location":"Seville, Spain","start_date":"2011-08-29","name":"CAIP: International Conference on Computer Analysis of Images and Patterns"},"oa":1},{"_id":"10909","year":"2010","date_created":"2022-03-21T08:24:07Z","scopus_import":"1","month":"02","type":"conference","publication":"Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms","publisher":"Society for Industrial and Applied Mathematics","article_processing_charge":"No","abstract":[{"text":"We address the problem of localizing homology classes, namely, finding the cycle representing a given class with the most concise geometric measure. We focus on the volume measure, that is, the 1-norm of a cycle. Two main results are presented. First, we prove the problem is NP-hard to approximate within any constant factor. Second, we prove that for homology of dimension two or higher, the problem is NP-hard to approximate even when the Betti number is O(1). A side effect is the inapproximability of the problem of computing the nonbounding cycle with the smallest volume, and computing cycles representing a homology basis with the minimal total volume. We also discuss other geometric measures (diameter and radius) and show their disadvantages in homology localization. Our work is restricted to homology over the ℤ2 field.","lang":"eng"}],"acknowledgement":"Partially supported by the Austrian Science Fund under grantFSP-S9103-N04 and P20134-N13.","department":[{"_id":"HeEd"}],"related_material":{"record":[{"relation":"later_version","id":"3267","status":"public"}]},"corr_author":"1","author":[{"last_name":"Chen","full_name":"Chen, Chao","first_name":"Chao","id":"3E92416E-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Daniel","full_name":"Freedman, Daniel","last_name":"Freedman"}],"quality_controlled":"1","date_updated":"2025-09-30T09:22:32Z","date_published":"2010-02-01T00:00:00Z","doi":"10.1137/1.9781611973075.129","conference":{"start_date":"2010-01-17","name":"SODA: Symposium on Discrete Algorithms","end_date":"2010-01-19","location":"Austin, TX, United States"},"oa_version":"None","page":"1594-1604","day":"01","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publication_status":"published","language":[{"iso":"eng"}],"publication_identifier":{"eisbn":["9781611973075"]},"citation":{"ista":"Chen C, Freedman D. 2010. Hardness results for homology localization. Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 1594–1604.","chicago":"Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.” In <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, 1594–1604. Society for Industrial and Applied Mathematics, 2010. <a href=\"https://doi.org/10.1137/1.9781611973075.129\">https://doi.org/10.1137/1.9781611973075.129</a>.","apa":"Chen, C., &#38; Freedman, D. (2010). Hardness results for homology localization. In <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i> (pp. 1594–1604). Austin, TX, United States: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611973075.129\">https://doi.org/10.1137/1.9781611973075.129</a>","mla":"Chen, Chao, and Daniel Freedman. “Hardness Results for Homology Localization.” <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Society for Industrial and Applied Mathematics, 2010, pp. 1594–604, doi:<a href=\"https://doi.org/10.1137/1.9781611973075.129\">10.1137/1.9781611973075.129</a>.","short":"C. Chen, D. Freedman, in:, Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, 2010, pp. 1594–1604.","ama":"Chen C, Freedman D. Hardness results for homology localization. In: <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>. Society for Industrial and Applied Mathematics; 2010:1594-1604. doi:<a href=\"https://doi.org/10.1137/1.9781611973075.129\">10.1137/1.9781611973075.129</a>","ieee":"C. Chen and D. Freedman, “Hardness results for homology localization,” in <i>Proceedings of the 2010 Annual ACM-SIAM Symposium on Discrete Algorithms</i>, Austin, TX, United States, 2010, pp. 1594–1604."},"title":"Hardness results for homology localization"},{"doi":"10.1007/978-3-642-18421-5_4","conference":{"location":"Beijing, China","end_date":"2010-09-20","name":"MCV: Medical Computer Vision","start_date":"2010-09-20"},"alternative_title":["LNCS"],"date_published":"2010-12-31T00:00:00Z","oa_version":"None","page":"31 - 42","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","status":"public","day":"31","publication_status":"published","language":[{"iso":"eng"}],"publist_id":"2445","title":"Topology noise removal for curve  and surface evolution","citation":{"mla":"Chen, Chao, and Daniel Freedman. “Topology Noise Removal for Curve  and Surface Evolution.” <i> Conference Proceedings MCV 2010</i>, vol. 6533, Springer, 2010, pp. 31–42, doi:<a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">10.1007/978-3-642-18421-5_4</a>.","chicago":"Chen, Chao, and Daniel Freedman. “Topology Noise Removal for Curve  and Surface Evolution.” In <i> Conference Proceedings MCV 2010</i>, 6533:31–42. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">https://doi.org/10.1007/978-3-642-18421-5_4</a>.","apa":"Chen, C., &#38; Freedman, D. (2010). Topology noise removal for curve  and surface evolution. In <i> Conference proceedings MCV 2010</i> (Vol. 6533, pp. 31–42). Beijing, China: Springer. <a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">https://doi.org/10.1007/978-3-642-18421-5_4</a>","ista":"Chen C, Freedman D. 2010. Topology noise removal for curve  and surface evolution.  Conference proceedings MCV 2010. MCV: Medical Computer Vision, LNCS, vol. 6533, 31–42.","ama":"Chen C, Freedman D. Topology noise removal for curve  and surface evolution. In: <i> Conference Proceedings MCV 2010</i>. Vol 6533. Springer; 2010:31-42. doi:<a href=\"https://doi.org/10.1007/978-3-642-18421-5_4\">10.1007/978-3-642-18421-5_4</a>","short":"C. Chen, D. Freedman, in:,  Conference Proceedings MCV 2010, Springer, 2010, pp. 31–42.","ieee":"C. Chen and D. Freedman, “Topology noise removal for curve  and surface evolution,” in <i> Conference proceedings MCV 2010</i>, Beijing, China, 2010, vol. 6533, pp. 31–42."},"_id":"3782","date_created":"2018-12-11T12:05:08Z","scopus_import":1,"year":"2010","abstract":[{"text":"In cortex surface segmentation, the extracted surface is required to have a particular topology, namely, a two-sphere. We present a new method for removing topology noise of a curve or surface within the level set framework, and thus produce a cortical surface with correct topology. We define a new energy term which quantifies topology noise. We then show how to minimize this term by computing its functional derivative with respect to the level set function. This method differs from existing methods in that it is inherently continuous and not digital; and in the way that our energy directly relates to the topology of the underlying curve or surface, versus existing knot-based measures which are related in a more indirect fashion. The proposed flow is validated empirically.","lang":"eng"}],"acknowledgement":"Partially supported by the Austri an Science Fund unde r grant P20134-N13.\r\nWe thank Helena Molina-Abril for very helpful discussion. We thank anonymous reviewers for helpful comments.","publication":" Conference proceedings MCV 2010","publisher":"Springer","month":"12","volume":6533,"type":"conference","department":[{"_id":"HeEd"}],"quality_controlled":"1","intvolume":"      6533","author":[{"id":"3E92416E-F248-11E8-B48F-1D18A9856A87","first_name":"Chao","full_name":"Chen, Chao","last_name":"Chen"},{"last_name":"Freedman","first_name":"Daniel","full_name":"Freedman, Daniel"}],"date_updated":"2021-01-12T07:52:10Z"},{"title":"The stability of the apparent contour of an orientable 2-manifold","publist_id":"2428","day":"22","status":"public","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","oa_version":"Submitted Version","date_published":"2010-12-22T00:00:00Z","doi":"10.1007/978-3-642-15014-2_3","date_updated":"2024-10-09T20:54:11Z","pubrep_id":"538","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"},{"last_name":"Morozov","first_name":"Dmitriy","full_name":"Morozov, Dmitriy"},{"id":"34A254A0-F248-11E8-B48F-1D18A9856A87","full_name":"Patel, Amit","first_name":"Amit","last_name":"Patel"}],"quality_controlled":"1","file":[{"relation":"main_file","date_created":"2018-12-12T10:11:40Z","checksum":"f03a44c3d1c3e2d4fedb3b94404f3fd5","file_size":210710,"creator":"system","content_type":"application/pdf","file_id":"4896","file_name":"IST-2016-538-v1+1_2011-B-02-ApparentContour.pdf","date_updated":"2020-07-14T12:46:16Z","access_level":"open_access"}],"corr_author":"1","department":[{"_id":"HeEd"}],"publication":"Topological Data Analysis and Visualization: Theory, Algorithms and Applications","month":"12","abstract":[{"lang":"eng","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."}],"year":"2010","file_date_updated":"2020-07-14T12:46:16Z","scopus_import":1,"_id":"3795","citation":{"ieee":"H. Edelsbrunner, D. Morozov, and A. Patel, “The stability of the apparent contour of an orientable 2-manifold,” in <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>, Springer, 2010, pp. 27–42.","short":"H. Edelsbrunner, D. Morozov, A. Patel, in:, Topological Data Analysis and Visualization: Theory, Algorithms and Applications, Springer, 2010, pp. 27–42.","ama":"Edelsbrunner H, Morozov D, Patel A. The stability of the apparent contour of an orientable 2-manifold. In: <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>. Springer; 2010:27-42. doi:<a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">10.1007/978-3-642-15014-2_3</a>","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.","mla":"Edelsbrunner, Herbert, et al. “The Stability of the Apparent Contour of an Orientable 2-Manifold.” <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>, Springer, 2010, pp. 27–42, doi:<a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">10.1007/978-3-642-15014-2_3</a>.","apa":"Edelsbrunner, H., Morozov, D., &#38; Patel, A. (2010). The stability of the apparent contour of an orientable 2-manifold. In <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i> (pp. 27–42). Springer. <a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">https://doi.org/10.1007/978-3-642-15014-2_3</a>","chicago":"Edelsbrunner, Herbert, Dmitriy Morozov, and Amit Patel. “The Stability of the Apparent Contour of an Orientable 2-Manifold.” In <i>Topological Data Analysis and Visualization: Theory, Algorithms and Applications</i>, 27–42. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15014-2_3\">https://doi.org/10.1007/978-3-642-15014-2_3</a>."},"language":[{"iso":"eng"}],"publication_status":"published","page":"27 - 42","oa":1,"alternative_title":["Mathematics and Visualization"],"ddc":["000"],"publisher":"Springer","type":"book_chapter","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.","has_accepted_license":"1","date_created":"2018-12-11T12:05:13Z"},{"conference":{"start_date":"2010-09-06","name":"ESA: European Symposium on Algorithms","end_date":"2010-09-08","location":"Liverpool, UK"},"doi":"10.1007/978-3-642-15775-2_1","alternative_title":["LNCS"],"date_published":"2010-09-01T00:00:00Z","oa_version":"None","page":"1 - 10","status":"public","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"01","publication_status":"published","language":[{"iso":"eng"}],"title":"The robustness of level sets","publist_id":"2336","citation":{"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.","ama":"Bendich P, Edelsbrunner H, Morozov D, Patel A. The robustness of level sets. In: Vol 6346. Springer; 2010:1-10. doi:<a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">10.1007/978-3-642-15775-2_1</a>","short":"P. Bendich, H. Edelsbrunner, D. Morozov, A. Patel, in:, Springer, 2010, pp. 1–10.","apa":"Bendich, P., Edelsbrunner, H., Morozov, D., &#38; Patel, A. (2010). The robustness of level sets (Vol. 6346, pp. 1–10). Presented at the ESA: European Symposium on Algorithms, Liverpool, UK: Springer. <a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">https://doi.org/10.1007/978-3-642-15775-2_1</a>","mla":"Bendich, Paul, et al. <i>The Robustness of Level Sets</i>. Vol. 6346, Springer, 2010, pp. 1–10, doi:<a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">10.1007/978-3-642-15775-2_1</a>.","chicago":"Bendich, Paul, Herbert Edelsbrunner, Dmitriy Morozov, and Amit Patel. “The Robustness of Level Sets,” 6346:1–10. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15775-2_1\">https://doi.org/10.1007/978-3-642-15775-2_1</a>.","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."},"_id":"3848","scopus_import":1,"date_created":"2018-12-11T12:05:30Z","year":"2010","abstract":[{"lang":"eng","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."}],"publisher":"Springer","month":"09","volume":6346,"type":"conference","department":[{"_id":"HeEd"}],"corr_author":"1","intvolume":"      6346","quality_controlled":"1","author":[{"id":"43F6EC54-F248-11E8-B48F-1D18A9856A87","full_name":"Bendich, Paul","first_name":"Paul","last_name":"Bendich"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Morozov","first_name":"Dmitriy","full_name":"Morozov, Dmitriy"},{"id":"34A254A0-F248-11E8-B48F-1D18A9856A87","first_name":"Amit","full_name":"Patel, Amit","last_name":"Patel"}],"date_updated":"2024-10-09T20:54:09Z"},{"has_accepted_license":"1","date_created":"2018-12-11T12:05:30Z","volume":6281,"type":"conference","publisher":"Springer","intvolume":"      6281","ddc":["000"],"conference":{"location":"Brno, Czech Republic","end_date":"2010-08-27","name":"MFCS: Mathematical Foundations of Computer Science","start_date":"2010-08-23"},"alternative_title":["LNCS"],"oa":1,"page":"12 - 23","language":[{"iso":"eng"}],"publication_status":"published","citation":{"ama":"Bendich P, Edelsbrunner H, Kerber M, Patel A. Persistent homology under non-uniform error. In: Vol 6281. Springer; 2010:12-23. doi:<a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">10.1007/978-3-642-15155-2_2</a>","short":"P. Bendich, H. Edelsbrunner, M. Kerber, A. Patel, in:, Springer, 2010, pp. 12–23.","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.","chicago":"Bendich, Paul, Herbert Edelsbrunner, Michael Kerber, and Amit Patel. “Persistent Homology under Non-Uniform Error,” 6281:12–23. Springer, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">https://doi.org/10.1007/978-3-642-15155-2_2</a>.","apa":"Bendich, P., Edelsbrunner, H., Kerber, M., &#38; 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. <a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">https://doi.org/10.1007/978-3-642-15155-2_2</a>","mla":"Bendich, Paul, et al. <i>Persistent Homology under Non-Uniform Error</i>. Vol. 6281, Springer, 2010, pp. 12–23, doi:<a href=\"https://doi.org/10.1007/978-3-642-15155-2_2\">10.1007/978-3-642-15155-2_2</a>.","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."},"_id":"3849","file_date_updated":"2020-07-14T12:46:17Z","scopus_import":1,"year":"2010","abstract":[{"lang":"eng","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."}],"month":"08","department":[{"_id":"HeEd"}],"file":[{"checksum":"af61e1c2bb42f3d556179d4692caeb1b","date_created":"2018-12-12T10:13:13Z","relation":"main_file","content_type":"application/pdf","file_id":"4994","creator":"system","file_size":142357,"file_name":"IST-2016-537-v1+1_2010-P-05-NonuniformError.pdf","date_updated":"2020-07-14T12:46:17Z","access_level":"open_access"}],"quality_controlled":"1","pubrep_id":"537","author":[{"id":"43F6EC54-F248-11E8-B48F-1D18A9856A87","full_name":"Bendich, Paul","first_name":"Paul","last_name":"Bendich"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber","orcid":"0000-0002-8030-9299","id":"36E4574A-F248-11E8-B48F-1D18A9856A87"},{"id":"34A254A0-F248-11E8-B48F-1D18A9856A87","full_name":"Patel, Amit","first_name":"Amit","last_name":"Patel"}],"date_updated":"2021-01-12T07:52:38Z","doi":"10.1007/978-3-642-15155-2_2","date_published":"2010-08-10T00:00:00Z","oa_version":"Submitted Version","status":"public","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"10","title":"Persistent homology under non-uniform error","publist_id":"2333"},{"date_published":"2010-01-01T00:00:00Z","conference":{"start_date":"2010-03-22","name":"EuroCG: European Workshop on Computational Geometry","end_date":"2010-03-24","location":"Dortmund, Germany"},"oa_version":"None","page":"12 - 23","day":"01","status":"public","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","language":[{"iso":"eng"}],"publication_status":"published","citation":{"chicago":"Berberich, Eric, Dan Halperin, Michael Kerber, and Roza Pogalnikova. “Polygonal Reconstruction from Approximate Offsets,” 12–23. TU Dortmund, 2010.","mla":"Berberich, Eric, et al. <i>Polygonal Reconstruction from Approximate Offsets</i>. TU Dortmund, 2010, pp. 12–23.","apa":"Berberich, E., Halperin, D., Kerber, M., &#38; Pogalnikova, R. (2010). Polygonal reconstruction from approximate offsets (pp. 12–23). Presented at the EuroCG: European Workshop on Computational Geometry, Dortmund, Germany: TU Dortmund.","ista":"Berberich E, Halperin D, Kerber M, Pogalnikova R. 2010. Polygonal reconstruction from approximate offsets. EuroCG: European Workshop on Computational Geometry, 12–23.","ama":"Berberich E, Halperin D, Kerber M, Pogalnikova R. Polygonal reconstruction from approximate offsets. In: TU Dortmund; 2010:12-23.","short":"E. Berberich, D. Halperin, M. Kerber, R. Pogalnikova, in:, TU Dortmund, 2010, pp. 12–23.","ieee":"E. Berberich, D. Halperin, M. Kerber, and R. Pogalnikova, “Polygonal reconstruction from approximate offsets,” presented at the EuroCG: European Workshop on Computational Geometry, Dortmund, Germany, 2010, pp. 12–23."},"title":"Polygonal reconstruction from approximate offsets","publist_id":"2334","_id":"3850","year":"2010","date_created":"2018-12-11T12:05:30Z","month":"01","publisher":"TU Dortmund","type":"conference","abstract":[{"lang":"eng","text":"Given a polygonal shape Q with n vertices, can it be expressed, up to a tolerance ε in Hausdorff distance, as the Minkowski sum of another polygonal shape with a disk of fixed radius? If it does, we also seek a preferably simple solution shape P;P’s offset constitutes an accurate, vertex-reduced, and smoothened approximation of Q. We give a decision algorithm for fixed radius in O(nlogn) time that handles any polygonal shape. For convex shapes, the complexity drops to O(n), which is also the time required to compute a solution shape P with at most one more vertex than a vertex-minimal one."}],"department":[{"_id":"HeEd"}],"author":[{"first_name":"Eric","full_name":"Berberich, Eric","last_name":"Berberich"},{"first_name":"Dan","full_name":"Halperin, Dan","last_name":"Halperin"},{"id":"36E4574A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8030-9299","last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"},{"last_name":"Pogalnikova","full_name":"Pogalnikova, Roza","first_name":"Roza"}],"quality_controlled":"1","date_updated":"2021-01-12T07:52:39Z"},{"publist_id":"2328","title":"Mean-payoff automaton expressions","oa_version":"Submitted Version","doi":"10.1007/978-3-642-15375-4_19","date_published":"2010-11-18T00:00:00Z","status":"public","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","day":"18","corr_author":"1","date_updated":"2024-10-09T20:54:08Z","quality_controlled":"1","file":[{"file_id":"5163","content_type":"application/pdf","creator":"system","file_size":233260,"checksum":"4f753ae99d076553fb8733e2c8b390e2","relation":"main_file","date_created":"2018-12-12T10:15:41Z","access_level":"open_access","date_updated":"2020-07-14T12:46:17Z","file_name":"IST-2012-62-v1+1_Mean-payoff_automaton_expressions.pdf"}],"pubrep_id":"62","author":[{"id":"2E5DCA20-F248-11E8-B48F-1D18A9856A87","last_name":"Chatterjee","full_name":"Chatterjee, Krishnendu","first_name":"Krishnendu","orcid":"0000-0002-4561-241X"},{"first_name":"Laurent","full_name":"Doyen, Laurent","last_name":"Doyen"},{"orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"40876CD8-F248-11E8-B48F-1D18A9856A87","first_name":"Thomas A","full_name":"Henzinger, Thomas A","last_name":"Henzinger","orcid":"0000−0002−2985−7724"},{"last_name":"Rannou","first_name":"Philippe","full_name":"Rannou, Philippe"}],"file_date_updated":"2020-07-14T12:46:17Z","scopus_import":1,"year":"2010","_id":"3853","department":[{"_id":"KrCh"},{"_id":"HeEd"},{"_id":"ToHe"}],"abstract":[{"text":"Quantitative languages are an extension of boolean languages that assign to each word a real number. Mean-payoff automata are finite automata with numerical weights on transitions that assign to each infinite path the long-run average of the transition weights. When the mode of branching of the automaton is deterministic, nondeterministic, or alternating, the corresponding class of quantitative languages is not robust as it is not closed under the pointwise operations of max, min, sum, and numerical complement. Nondeterministic and alternating mean-payoff automata are not decidable either, as the quantitative generalization of the problems of universality and language inclusion is undecidable. We introduce a new class of quantitative languages, defined by mean-payoff automaton expressions, which is robust and decidable: it is closed under the four pointwise operations, and we show that all decision problems are decidable for this class. Mean-payoff automaton expressions subsume deterministic meanpayoff automata, and we show that they have expressive power incomparable to nondeterministic and alternating mean-payoff automata. We also present for the first time an algorithm to compute distance between two quantitative languages, and in our case the quantitative languages are given as mean-payoff automaton expressions.","lang":"eng"}],"month":"11","language":[{"iso":"eng"}],"publication_status":"published","citation":{"ama":"Chatterjee K, Doyen L, Edelsbrunner H, Henzinger TA, Rannou P. Mean-payoff automaton expressions. In: Vol 6269. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2010:269-283. doi:<a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">10.1007/978-3-642-15375-4_19</a>","short":"K. Chatterjee, L. Doyen, H. Edelsbrunner, T.A. Henzinger, P. Rannou, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010, pp. 269–283.","ieee":"K. Chatterjee, L. Doyen, H. Edelsbrunner, T. A. Henzinger, and P. Rannou, “Mean-payoff automaton expressions,” presented at the CONCUR: Concurrency Theory, Paris, France, 2010, vol. 6269, pp. 269–283.","apa":"Chatterjee, K., Doyen, L., Edelsbrunner, H., Henzinger, T. A., &#38; Rannou, P. (2010). Mean-payoff automaton expressions (Vol. 6269, pp. 269–283). Presented at the CONCUR: Concurrency Theory, Paris, France: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">https://doi.org/10.1007/978-3-642-15375-4_19</a>","chicago":"Chatterjee, Krishnendu, Laurent Doyen, Herbert Edelsbrunner, Thomas A Henzinger, and Philippe Rannou. “Mean-Payoff Automaton Expressions,” 6269:269–83. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010. <a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">https://doi.org/10.1007/978-3-642-15375-4_19</a>.","mla":"Chatterjee, Krishnendu, et al. <i>Mean-Payoff Automaton Expressions</i>. Vol. 6269, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010, pp. 269–83, doi:<a href=\"https://doi.org/10.1007/978-3-642-15375-4_19\">10.1007/978-3-642-15375-4_19</a>.","ista":"Chatterjee K, Doyen L, Edelsbrunner H, Henzinger TA, Rannou P. 2010. Mean-payoff automaton expressions. CONCUR: Concurrency Theory, LNCS, vol. 6269, 269–283."},"oa":1,"conference":{"name":"CONCUR: Concurrency Theory","start_date":"2010-08-31","end_date":"2010-09-03","location":"Paris, France"},"alternative_title":["LNCS"],"page":"269 - 283","intvolume":"      6269","project":[{"name":"COMponent-Based Embedded Systems design Techniques","call_identifier":"FP7","grant_number":"215543","_id":"25EFB36C-B435-11E9-9278-68D0E5697425"},{"_id":"25F1337C-B435-11E9-9278-68D0E5697425","name":"Design for Embedded Systems","call_identifier":"FP7","grant_number":"214373"}],"ddc":["000","005"],"date_created":"2018-12-11T12:05:31Z","has_accepted_license":"1","publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","volume":6269,"type":"conference","ec_funded":1},{"publist_id":"2253","title":"Computing robustness and persistence for images","doi":"10.1109/TVCG.2010.139","date_published":"2010-10-28T00:00:00Z","oa_version":"Submitted Version","issue":"6","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","status":"public","day":"28","corr_author":"1","file":[{"date_updated":"2020-07-14T12:46:21Z","access_level":"open_access","file_name":"IST-2016-536-v1+1_2010-J-02-PersistenceforImages.pdf","file_size":721994,"file_id":"5262","content_type":"application/pdf","creator":"system","relation":"main_file","date_created":"2018-12-12T10:17:10Z","checksum":"f6d813c04f4b46023cec6b9a17f15472"}],"quality_controlled":"1","author":[{"id":"43F6EC54-F248-11E8-B48F-1D18A9856A87","last_name":"Bendich","full_name":"Bendich, Paul","first_name":"Paul"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"id":"36E4574A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8030-9299","full_name":"Kerber, Michael","first_name":"Michael","last_name":"Kerber"}],"pubrep_id":"536","date_updated":"2025-09-30T09:30:22Z","_id":"3901","file_date_updated":"2020-07-14T12:46:21Z","scopus_import":"1","year":"2010","abstract":[{"text":"We are interested in 3-dimensional images given as arrays of voxels with intensity values. Extending these values to acontinuous function, we study the robustness of homology classes in its level and interlevel sets, that is, the amount of perturbationneeded to destroy these classes. The structure of the homology classes and their robustness, over all level and interlevel sets, can bevisualized by a triangular diagram of dots obtained by computing the extended persistence of the function. We give a fast hierarchicalalgorithm using the dual complexes of oct-tree approximations of the function. In addition, we show that for balanced oct-trees, thedual complexes are geometrically realized in $R^3$ and can thus be used to construct level and interlevel sets. We apply these tools tostudy 3-dimensional images of plant root systems.","lang":"eng"}],"publication":"IEEE Transactions of Visualization and Computer Graphics","month":"10","department":[{"_id":"HeEd"}],"publication_status":"published","language":[{"iso":"eng"}],"citation":{"ama":"Bendich P, Edelsbrunner H, Kerber M. Computing robustness and persistence for images. <i>IEEE Transactions of Visualization and Computer Graphics</i>. 2010;16(6):1251-1260. doi:<a href=\"https://doi.org/10.1109/TVCG.2010.139\">10.1109/TVCG.2010.139</a>","short":"P. Bendich, H. Edelsbrunner, M. Kerber, IEEE Transactions of Visualization and Computer Graphics 16 (2010) 1251–1260.","ieee":"P. Bendich, H. Edelsbrunner, and M. Kerber, “Computing robustness and persistence for images,” <i>IEEE Transactions of Visualization and Computer Graphics</i>, vol. 16, no. 6. IEEE, pp. 1251–1260, 2010.","mla":"Bendich, Paul, et al. “Computing Robustness and Persistence for Images.” <i>IEEE Transactions of Visualization and Computer Graphics</i>, vol. 16, no. 6, IEEE, 2010, pp. 1251–60, doi:<a href=\"https://doi.org/10.1109/TVCG.2010.139\">10.1109/TVCG.2010.139</a>.","chicago":"Bendich, Paul, Herbert Edelsbrunner, and Michael Kerber. “Computing Robustness and Persistence for Images.” <i>IEEE Transactions of Visualization and Computer Graphics</i>. IEEE, 2010. <a href=\"https://doi.org/10.1109/TVCG.2010.139\">https://doi.org/10.1109/TVCG.2010.139</a>.","apa":"Bendich, P., Edelsbrunner, H., &#38; Kerber, M. (2010). Computing robustness and persistence for images. <i>IEEE Transactions of Visualization and Computer Graphics</i>. IEEE. <a href=\"https://doi.org/10.1109/TVCG.2010.139\">https://doi.org/10.1109/TVCG.2010.139</a>","ista":"Bendich P, Edelsbrunner H, Kerber M. 2010. Computing robustness and persistence for images. IEEE Transactions of Visualization and Computer Graphics. 16(6), 1251–1260."},"isi":1,"oa":1,"external_id":{"isi":["000283758600045"]},"page":"1251 - 1260","intvolume":"        16","ddc":["000"],"has_accepted_license":"1","date_created":"2018-12-11T12:05:47Z","type":"journal_article","volume":16,"publisher":"IEEE","article_processing_charge":"No"},{"year":"2009","file_date_updated":"2020-07-14T12:46:21Z","scopus_import":1,"_id":"3968","department":[{"_id":"HeEd"}],"month":"11","abstract":[{"text":"We describe an algorithm for segmenting three-dimensional medical imaging data modeled as a continuous function on a 3-manifold. It is related to watershed algorithms developed in image processing but is closer to its mathematical roots, which are Morse theory and homological algebra. It allows for the implicit treatment of an underlying mesh, thus combining the structural integrity of its mathematical foundations with the computational efficiency of image processing.","lang":"eng"}],"corr_author":"1","date_updated":"2024-10-09T20:53:56Z","author":[{"full_name":"Edelsbrunner, Herbert","first_name":"Herbert","last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Harer, John","first_name":"John","last_name":"Harer"}],"pubrep_id":"535","quality_controlled":"1","file":[{"content_type":"application/pdf","creator":"system","file_id":"4694","file_size":165090,"checksum":"11fc85bcc19bab1f020e706a4b8a4660","date_created":"2018-12-12T10:08:33Z","relation":"main_file","access_level":"open_access","date_updated":"2020-07-14T12:46:21Z","file_name":"IST-2016-535-v1+1_2009-P-04-3ManifoldSegmentation.pdf"}],"oa_version":"Submitted Version","date_published":"2009-11-17T00:00:00Z","doi":"10.1007/978-3-642-10470-1_4","day":"17","status":"public","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","publist_id":"2160","title":"The persistent Morse complex segmentation of a 3-manifold","has_accepted_license":"1","date_created":"2018-12-11T12:06:10Z","publisher":"Springer","type":"conference","volume":5903,"acknowledgement":"This research was partially supported by Geomagic, Inc., and by the Defense Advanced Research Projects Agency (DARPA) under grants HR0011-05-1-0007 and HR0011-05-1-0057.","ddc":["000"],"intvolume":"      5903","oa":1,"alternative_title":["LNCS"],"conference":{"name":"3DPH: Modelling the Physiological Human","start_date":"2009-11-29","end_date":"2009-12-02","location":"Zermatt, Switzerland"},"page":"36 - 50","language":[{"iso":"eng"}],"publication_status":"published","citation":{"ama":"Edelsbrunner H, Harer J. The persistent Morse complex segmentation of a 3-manifold. In: Vol 5903. Springer; 2009:36-50. doi:<a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">10.1007/978-3-642-10470-1_4</a>","short":"H. Edelsbrunner, J. Harer, in:, Springer, 2009, pp. 36–50.","ieee":"H. Edelsbrunner and J. Harer, “The persistent Morse complex segmentation of a 3-manifold,” presented at the 3DPH: Modelling the Physiological Human, Zermatt, Switzerland, 2009, vol. 5903, pp. 36–50.","chicago":"Edelsbrunner, Herbert, and John Harer. “The Persistent Morse Complex Segmentation of a 3-Manifold,” 5903:36–50. Springer, 2009. <a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">https://doi.org/10.1007/978-3-642-10470-1_4</a>.","apa":"Edelsbrunner, H., &#38; Harer, J. (2009). The persistent Morse complex segmentation of a 3-manifold (Vol. 5903, pp. 36–50). Presented at the 3DPH: Modelling the Physiological Human, Zermatt, Switzerland: Springer. <a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">https://doi.org/10.1007/978-3-642-10470-1_4</a>","mla":"Edelsbrunner, Herbert, and John Harer. <i>The Persistent Morse Complex Segmentation of a 3-Manifold</i>. Vol. 5903, Springer, 2009, pp. 36–50, doi:<a href=\"https://doi.org/10.1007/978-3-642-10470-1_4\">10.1007/978-3-642-10470-1_4</a>.","ista":"Edelsbrunner H, Harer J. 2009. The persistent Morse complex segmentation of a 3-manifold. 3DPH: Modelling the Physiological Human, LNCS, vol. 5903, 36–50."}}]