[{"title":"Optimal packings of congruent circles on a square flat torus","date_published":"2016-01-01T00:00:00Z","abstract":[{"lang":"eng","text":"We consider packings of congruent circles on a square flat torus, i.e., periodic (w.r.t. a square lattice) planar circle packings, with the maximal circle radius. This problem is interesting due to a practical reason—the problem of “super resolution of images.” We have found optimal arrangements for N=6, 7 and 8 circles. Surprisingly, for the case N=7 there are three different optimal arrangements. Our proof is based on a computer enumeration of toroidal irreducible contact graphs."}],"doi":"10.1007/s00454-015-9742-6","date_updated":"2025-09-22T09:32:23Z","oa":1,"publication":"Discrete & Computational Geometry","publication_status":"published","language":[{"iso":"eng"}],"acknowledgement":"We wish to thank Alexey Tarasov, Vladislav Volkov and Brittany Fasy for some useful comments and remarks, and especially Thom Sulanke for modifying surftri to suit our purposes. Oleg R. Musin was partially supported by the NSF Grant DMS-1400876 and by the RFBR Grant 15-01-99563. Anton V. Nikitenko was supported by the Chebyshev Laboratory (Department of Mathematics and Mechanics, St. Petersburg State University) under RF Government Grant 11.G34.31.0026.","main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1212.0649"}],"article_processing_charge":"No","arxiv":1,"_id":"1222","page":"1 - 20","date_created":"2018-12-11T11:50:48Z","type":"journal_article","issue":"1","intvolume":"        55","day":"01","scopus_import":"1","status":"public","publisher":"Springer","citation":{"ista":"Musin O, Nikitenko A. 2016. Optimal packings of congruent circles on a square flat torus. Discrete &#38; Computational Geometry. 55(1), 1–20.","ama":"Musin O, Nikitenko A. Optimal packings of congruent circles on a square flat torus. <i>Discrete &#38; Computational Geometry</i>. 2016;55(1):1-20. doi:<a href=\"https://doi.org/10.1007/s00454-015-9742-6\">10.1007/s00454-015-9742-6</a>","ieee":"O. Musin and A. Nikitenko, “Optimal packings of congruent circles on a square flat torus,” <i>Discrete &#38; Computational Geometry</i>, vol. 55, no. 1. Springer, pp. 1–20, 2016.","chicago":"Musin, Oleg, and Anton Nikitenko. “Optimal Packings of Congruent Circles on a Square Flat Torus.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s00454-015-9742-6\">https://doi.org/10.1007/s00454-015-9742-6</a>.","apa":"Musin, O., &#38; Nikitenko, A. (2016). Optimal packings of congruent circles on a square flat torus. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-015-9742-6\">https://doi.org/10.1007/s00454-015-9742-6</a>","short":"O. Musin, A. Nikitenko, Discrete &#38; Computational Geometry 55 (2016) 1–20.","mla":"Musin, Oleg, and Anton Nikitenko. “Optimal Packings of Congruent Circles on a Square Flat Torus.” <i>Discrete &#38; Computational Geometry</i>, vol. 55, no. 1, Springer, 2016, pp. 1–20, doi:<a href=\"https://doi.org/10.1007/s00454-015-9742-6\">10.1007/s00454-015-9742-6</a>."},"publist_id":"6111","month":"01","department":[{"_id":"HeEd"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"arxiv":["1212.0649"],"isi":["000367625500001"]},"volume":55,"author":[{"full_name":"Musin, Oleg","first_name":"Oleg","last_name":"Musin"},{"last_name":"Nikitenko","orcid":"0000-0002-0659-3201","first_name":"Anton","full_name":"Nikitenko, Anton","id":"3E4FF1BA-F248-11E8-B48F-1D18A9856A87"}],"isi":1,"year":"2016","oa_version":"Preprint","quality_controlled":"1"},{"scopus_import":"1","publisher":"Springer","ec_funded":1,"status":"public","day":"02","department":[{"_id":"UlWa"},{"_id":"HeEd"}],"month":"06","volume":9667,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["000389805800013"]},"conference":{"end_date":"2016-06-17","start_date":"2016-06-15","location":"Marseille, France","name":"CTIC: Computational Topology in Image Context"},"citation":{"apa":"Krcál, M., &#38; Pilarczyk, P. (2016). Computation of cubical Steenrod squares (Vol. 9667, pp. 140–151). Presented at the CTIC: Computational Topology in Image Context, Marseille, France: Springer. <a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">https://doi.org/10.1007/978-3-319-39441-1_13</a>","ieee":"M. Krcál and P. Pilarczyk, “Computation of cubical Steenrod squares,” presented at the CTIC: Computational Topology in Image Context, Marseille, France, 2016, vol. 9667, pp. 140–151.","chicago":"Krcál, Marek, and Pawel Pilarczyk. “Computation of Cubical Steenrod Squares,” 9667:140–51. Springer, 2016. <a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">https://doi.org/10.1007/978-3-319-39441-1_13</a>.","ama":"Krcál M, Pilarczyk P. Computation of cubical Steenrod squares. In: Vol 9667. Springer; 2016:140-151. doi:<a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">10.1007/978-3-319-39441-1_13</a>","ista":"Krcál M, Pilarczyk P. 2016. Computation of cubical Steenrod squares. CTIC: Computational Topology in Image Context, LNCS, vol. 9667, 140–151.","mla":"Krcál, Marek, and Pawel Pilarczyk. <i>Computation of Cubical Steenrod Squares</i>. Vol. 9667, Springer, 2016, pp. 140–51, doi:<a href=\"https://doi.org/10.1007/978-3-319-39441-1_13\">10.1007/978-3-319-39441-1_13</a>.","short":"M. Krcál, P. Pilarczyk, in:, Springer, 2016, pp. 140–151."},"publist_id":"6096","isi":1,"project":[{"call_identifier":"FP7","grant_number":"291734","name":"International IST Postdoc Fellowship Programme","_id":"25681D80-B435-11E9-9278-68D0E5697425"},{"_id":"255F06BE-B435-11E9-9278-68D0E5697425","grant_number":"622033","name":"Persistent Homology - Images, Data and Maps","call_identifier":"FP7"}],"author":[{"first_name":"Marek","last_name":"Krcál","id":"33E21118-F248-11E8-B48F-1D18A9856A87","full_name":"Krcál, Marek"},{"first_name":"Pawel","last_name":"Pilarczyk","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel"}],"quality_controlled":"1","year":"2016","oa_version":"None","doi":"10.1007/978-3-319-39441-1_13","date_updated":"2025-09-22T09:19:17Z","date_published":"2016-06-02T00:00:00Z","title":"Computation of cubical Steenrod squares","abstract":[{"text":"Bitmap images of arbitrary dimension may be formally perceived as unions of m-dimensional boxes aligned with respect to a rectangular grid in ℝm. Cohomology and homology groups are well known topological invariants of such sets. Cohomological operations, such as the cup product, provide higher-order algebraic topological invariants, especially important for digital images of dimension higher than 3. If such an operation is determined at the level of simplicial chains [see e.g. González-Díaz, Real, Homology, Homotopy Appl, 2003, 83-93], then it is effectively computable. However, decomposing a cubical complex into a simplicial one deleteriously affects the efficiency of such an approach. In order to avoid this overhead, a direct cubical approach was applied in [Pilarczyk, Real, Adv. Comput. Math., 2015, 253-275] for the cup product in cohomology, and implemented in the ChainCon software package [http://www.pawelpilarczyk.com/chaincon/]. We establish a formula for the Steenrod square operations [see Steenrod, Annals of Mathematics. Second Series, 1947, 290-320] directly at the level of cubical chains, and we prove the correctness of this formula. An implementation of this formula is programmed in C++ within the ChainCon software framework. We provide a few examples and discuss the effectiveness of this approach. One specific application follows from the fact that Steenrod squares yield tests for the topological extension problem: Can a given map A → Sd to a sphere Sd be extended to a given super-complex X of A? In particular, the ROB-SAT problem, which is to decide for a given function f: X → ℝm and a value r &gt; 0 whether every g: X → ℝm with ∥g - f ∥∞ ≤ r has a root, reduces to the extension problem.","lang":"eng"}],"acknowledgement":"The research conducted by both authors has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreements no. 291734 (for M. K.) and no. 622033 (for P. P.).","publication_status":"published","alternative_title":["LNCS"],"language":[{"iso":"eng"}],"date_created":"2018-12-11T11:50:52Z","page":"140 - 151","type":"conference","article_processing_charge":"No","_id":"1237","intvolume":"      9667"},{"intvolume":"       144","corr_author":"1","issue":"4","type":"journal_article","date_created":"2018-12-11T11:50:57Z","page":"1787 - 1801","_id":"1252","arxiv":1,"article_processing_charge":"No","main_file_link":[{"url":"https://arxiv.org/abs/1411.7563","open_access":"1"}],"acknowledgement":"The authors gratefully acknowledge the support of the Lorenz Center which\r\nprovided an opportunity for us to discuss in depth the work of this paper. Research leading to these results has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Funda¸c˜ao para a Ciˆencia e a Tecnologia (FCT) in the framework of the research\r\nproject FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008),\r\nas well as from the People Programme (Marie Curie Actions) of the European\r\nUnion’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033 (supporting PP). The work of the first and third author has\r\nbeen partially supported by NSF grants NSF-DMS-0835621, 0915019, 1125174,\r\n1248071, and contracts from AFOSR and DARPA. The work of the second author\r\nwas supported by Grant-in-Aid for Scientific Research (No. 25287029), Ministry of\r\nEducation, Science, Technology, Culture and Sports, Japan.","language":[{"iso":"eng"}],"publication_status":"published","publication":"Proceedings of the American Mathematical Society","oa":1,"date_updated":"2025-09-22T09:09:31Z","doi":"10.1090/proc/12812","publication_identifier":{"issn":["1088-6826"]},"abstract":[{"lang":"eng","text":"We study the homomorphism induced in homology by a closed correspondence between topological spaces, using projections from the graph of the correspondence to its domain and codomain. We provide assumptions under which the homomorphism induced by an outer approximation of a continuous map coincides with the homomorphism induced in homology by the map. In contrast to more classical results we do not require that the projection to the domain have acyclic preimages. Moreover, we show that it is possible to retrieve correct homological information from a correspondence even if some data is missing or perturbed. Finally, we describe an application to combinatorial maps that are either outer approximations of continuous maps or reconstructions of such maps from a finite set of data points."}],"date_published":"2016-04-01T00:00:00Z","title":"Inducing a map on homology from a correspondence","quality_controlled":"1","oa_version":"Preprint","year":"2016","isi":1,"author":[{"first_name":"Shaun","last_name":"Harker","full_name":"Harker, Shaun"},{"first_name":"Hiroshi","last_name":"Kokubu","full_name":"Kokubu, Hiroshi"},{"full_name":"Mischaikow, Konstantin","last_name":"Mischaikow","first_name":"Konstantin"},{"full_name":"Pilarczyk, Pawel","id":"3768D56A-F248-11E8-B48F-1D18A9856A87","last_name":"Pilarczyk","first_name":"Pawel"}],"project":[{"call_identifier":"FP7","grant_number":"622033","name":"Persistent Homology - Images, Data and Maps","_id":"255F06BE-B435-11E9-9278-68D0E5697425"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","volume":144,"external_id":{"arxiv":["1411.7563"],"isi":["000369298400038"]},"department":[{"_id":"HeEd"}],"month":"04","publist_id":"6075","citation":{"mla":"Harker, Shaun, et al. “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4, American Mathematical Society, 2016, pp. 1787–801, doi:<a href=\"https://doi.org/10.1090/proc/12812\">10.1090/proc/12812</a>.","short":"S. Harker, H. Kokubu, K. Mischaikow, P. Pilarczyk, Proceedings of the American Mathematical Society 144 (2016) 1787–1801.","ieee":"S. Harker, H. Kokubu, K. Mischaikow, and P. Pilarczyk, “Inducing a map on homology from a correspondence,” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 4. American Mathematical Society, pp. 1787–1801, 2016.","chicago":"Harker, Shaun, Hiroshi Kokubu, Konstantin Mischaikow, and Pawel Pilarczyk. “Inducing a Map on Homology from a Correspondence.” <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society, 2016. <a href=\"https://doi.org/10.1090/proc/12812\">https://doi.org/10.1090/proc/12812</a>.","ama":"Harker S, Kokubu H, Mischaikow K, Pilarczyk P. Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>. 2016;144(4):1787-1801. doi:<a href=\"https://doi.org/10.1090/proc/12812\">10.1090/proc/12812</a>","apa":"Harker, S., Kokubu, H., Mischaikow, K., &#38; Pilarczyk, P. (2016). Inducing a map on homology from a correspondence. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/proc/12812\">https://doi.org/10.1090/proc/12812</a>","ista":"Harker S, Kokubu H, Mischaikow K, Pilarczyk P. 2016. Inducing a map on homology from a correspondence. Proceedings of the American Mathematical Society. 144(4), 1787–1801."},"scopus_import":"1","publisher":"American Mathematical Society","status":"public","ec_funded":1,"article_type":"original","day":"01"},{"isi":1,"author":[{"first_name":"Olga","last_name":"Dunaeva","full_name":"Dunaeva, Olga"},{"full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","last_name":"Edelsbrunner","first_name":"Herbert","orcid":"0000-0002-9823-6833"},{"first_name":"Anton","last_name":"Lukyanov","full_name":"Lukyanov, Anton"},{"last_name":"Machin","first_name":"Michael","full_name":"Machin, Michael"},{"last_name":"Malkova","first_name":"Daria","full_name":"Malkova, Daria"},{"first_name":"Roman","last_name":"Kuvaev","full_name":"Kuvaev, Roman"},{"full_name":"Kashin, Sergey","last_name":"Kashin","first_name":"Sergey"}],"quality_controlled":"1","oa_version":"Submitted Version","file_date_updated":"2020-07-14T12:44:42Z","year":"2016","has_accepted_license":"1","file":[{"relation":"main_file","creator":"dernst","file_name":"2016-Edelsbrunner_The_classification.pdf","file_id":"6334","checksum":"33458bbb8c32a339e1adeca6d5a1112d","file_size":1921113,"access_level":"open_access","content_type":"application/pdf","date_created":"2019-04-17T07:55:51Z","date_updated":"2020-07-14T12:44:42Z"}],"publisher":"Elsevier","status":"public","scopus_import":"1","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"1568"}]},"tmp":{"image":"/images/cc_by_nc_nd.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","short":"CC BY-NC-ND (4.0)"},"day":"01","volume":83,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["000386874700003"]},"month":"11","department":[{"_id":"HeEd"}],"publist_id":"6027","citation":{"short":"O. Dunaeva, H. Edelsbrunner, A. Lukyanov, M. Machin, D. Malkova, R. Kuvaev, S. Kashin, Pattern Recognition Letters 83 (2016) 13–22.","mla":"Dunaeva, Olga, et al. “The Classification of Endoscopy Images with Persistent Homology.” <i>Pattern Recognition Letters</i>, vol. 83, no. 1, Elsevier, 2016, pp. 13–22, doi:<a href=\"https://doi.org/10.1016/j.patrec.2015.12.012\">10.1016/j.patrec.2015.12.012</a>.","ista":"Dunaeva O, Edelsbrunner H, Lukyanov A, Machin M, Malkova D, Kuvaev R, Kashin S. 2016. The classification of endoscopy images with persistent homology. Pattern Recognition Letters. 83(1), 13–22.","apa":"Dunaeva, O., Edelsbrunner, H., Lukyanov, A., Machin, M., Malkova, D., Kuvaev, R., &#38; Kashin, S. (2016). The classification of endoscopy images with persistent homology. <i>Pattern Recognition Letters</i>. Elsevier. <a href=\"https://doi.org/10.1016/j.patrec.2015.12.012\">https://doi.org/10.1016/j.patrec.2015.12.012</a>","ama":"Dunaeva O, Edelsbrunner H, Lukyanov A, et al. The classification of endoscopy images with persistent homology. <i>Pattern Recognition Letters</i>. 2016;83(1):13-22. doi:<a href=\"https://doi.org/10.1016/j.patrec.2015.12.012\">10.1016/j.patrec.2015.12.012</a>","chicago":"Dunaeva, Olga, Herbert Edelsbrunner, Anton Lukyanov, Michael Machin, Daria Malkova, Roman Kuvaev, and Sergey Kashin. “The Classification of Endoscopy Images with Persistent Homology.” <i>Pattern Recognition Letters</i>. Elsevier, 2016. <a href=\"https://doi.org/10.1016/j.patrec.2015.12.012\">https://doi.org/10.1016/j.patrec.2015.12.012</a>.","ieee":"O. Dunaeva <i>et al.</i>, “The classification of endoscopy images with persistent homology,” <i>Pattern Recognition Letters</i>, vol. 83, no. 1. Elsevier, pp. 13–22, 2016."},"type":"journal_article","page":"13 - 22","date_created":"2018-12-11T11:51:10Z","_id":"1289","ddc":["004","514"],"article_processing_charge":"No","intvolume":"        83","pubrep_id":"975","issue":"1","date_updated":"2025-09-23T13:44:16Z","doi":"10.1016/j.patrec.2015.12.012","license":"https://creativecommons.org/licenses/by-nc-nd/4.0/","abstract":[{"text":"Aiming at the automatic diagnosis of tumors using narrow band imaging (NBI) magnifying endoscopic (ME) images of the stomach, we combine methods from image processing, topology, geometry, and machine learning to classify patterns into three classes: oval, tubular and irregular. Training the algorithm on a small number of images of each type, we achieve a high rate of correct classifications. The analysis of the learning algorithm reveals that a handful of geometric and topological features are responsible for the overwhelming majority of decisions.","lang":"eng"}],"title":"The classification of endoscopy images with persistent homology","date_published":"2016-11-01T00:00:00Z","language":[{"iso":"eng"}],"oa":1,"publication_status":"published","publication":"Pattern Recognition Letters"},{"citation":{"apa":"Durst, S., Kegel, M., &#38; Klukas, M. D. (2016). Computing the Thurston–Bennequin invariant in open books. <i>Acta Mathematica Hungarica</i>. Springer. <a href=\"https://doi.org/10.1007/s10474-016-0648-4\">https://doi.org/10.1007/s10474-016-0648-4</a>","ieee":"S. Durst, M. Kegel, and M. D. Klukas, “Computing the Thurston–Bennequin invariant in open books,” <i>Acta Mathematica Hungarica</i>, vol. 150, no. 2. Springer, pp. 441–455, 2016.","ama":"Durst S, Kegel M, Klukas MD. Computing the Thurston–Bennequin invariant in open books. <i>Acta Mathematica Hungarica</i>. 2016;150(2):441-455. doi:<a href=\"https://doi.org/10.1007/s10474-016-0648-4\">10.1007/s10474-016-0648-4</a>","chicago":"Durst, Sebastian, Marc Kegel, and Mirko D Klukas. “Computing the Thurston–Bennequin Invariant in Open Books.” <i>Acta Mathematica Hungarica</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s10474-016-0648-4\">https://doi.org/10.1007/s10474-016-0648-4</a>.","ista":"Durst S, Kegel M, Klukas MD. 2016. Computing the Thurston–Bennequin invariant in open books. Acta Mathematica Hungarica. 150(2), 441–455.","mla":"Durst, Sebastian, et al. “Computing the Thurston–Bennequin Invariant in Open Books.” <i>Acta Mathematica Hungarica</i>, vol. 150, no. 2, Springer, 2016, pp. 441–55, doi:<a href=\"https://doi.org/10.1007/s10474-016-0648-4\">10.1007/s10474-016-0648-4</a>.","short":"S. Durst, M. Kegel, M.D. Klukas, Acta Mathematica Hungarica 150 (2016) 441–455."},"publist_id":"6023","month":"12","department":[{"_id":"HeEd"}],"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["000387471100014"],"arxiv":["1605.00794"]},"volume":150,"day":"01","scopus_import":"1","status":"public","publisher":"Springer","year":"2016","oa_version":"Preprint","quality_controlled":"1","author":[{"first_name":"Sebastian","last_name":"Durst","full_name":"Durst, Sebastian"},{"full_name":"Kegel, Marc","last_name":"Kegel","first_name":"Marc"},{"last_name":"Klukas","first_name":"Mirko D","full_name":"Klukas, Mirko D","id":"34927512-F248-11E8-B48F-1D18A9856A87"}],"isi":1,"oa":1,"publication":"Acta Mathematica Hungarica","publication_status":"published","language":[{"iso":"eng"}],"acknowledgement":"The authors are veryg rateful to Hansj ̈org Geiges \r\nfor fruitful discussions and advice and Christian Evers for helpful remarks on a draft version.","main_file_link":[{"url":"https://arxiv.org/abs/1605.00794","open_access":"1"}],"title":"Computing the Thurston–Bennequin invariant in open books","date_published":"2016-12-01T00:00:00Z","abstract":[{"text":"We give explicit formulas and algorithms for the computation of the Thurston–Bennequin invariant of a nullhomologous Legendrian knot on a page of a contact open book and on Heegaard surfaces in convex position. Furthermore, we extend the results to rationally nullhomologous knots in arbitrary 3-manifolds.","lang":"eng"}],"doi":"10.1007/s10474-016-0648-4","date_updated":"2025-09-22T08:29:12Z","issue":"2","corr_author":"1","intvolume":"       150","article_processing_charge":"No","arxiv":1,"_id":"1292","page":"441 - 455","date_created":"2018-12-11T11:51:11Z","type":"journal_article"},{"author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","full_name":"Iglesias Ham, Mabel","first_name":"Mabel","last_name":"Iglesias Ham"}],"project":[{"grant_number":"318493","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"quality_controlled":"1","oa_version":"None","year":"2016","ec_funded":1,"scopus_import":1,"status":"public","publisher":"Elsevier","day":"01","user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","volume":54,"month":"10","department":[{"_id":"HeEd"}],"publist_id":"5976","citation":{"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>.","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>","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.","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>","ista":"Edelsbrunner H, Iglesias Ham M. 2016. Multiple covers with balls II: Weighted averages. Electronic Notes in Discrete Mathematics. 54, 169–174.","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."},"type":"journal_article","page":"169 - 174","date_created":"2018-12-11T11:51:12Z","_id":"1295","intvolume":"        54","corr_author":"1","date_updated":"2024-10-09T20:57:05Z","doi":"10.1016/j.endm.2016.09.030","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"}],"title":"Multiple covers with balls II: Weighted averages","date_published":"2016-10-01T00:00:00Z","acknowledgement":"This work is partially supported by the Toposys project FP7-ICT-318493-STREP, and by ESF under the ACAT Research Network Programme.","language":[{"iso":"eng"}],"publication":"Electronic Notes in Discrete Mathematics","publication_status":"published"},{"oa_version":"Preprint","year":"2016","quality_controlled":"1","author":[{"full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87","last_name":"Akopyan","orcid":"0000-0002-2548-617X","first_name":"Arseniy"},{"last_name":"Balitskiy","first_name":"Alexey","full_name":"Balitskiy, Alexey"}],"project":[{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"isi":1,"publist_id":"5938","citation":{"ista":"Akopyan A, Balitskiy A. 2016. Billiards in convex bodies with acute angles. Israel Journal of Mathematics. 216(2), 833–845.","chicago":"Akopyan, Arseniy, and Alexey Balitskiy. “Billiards in Convex Bodies with Acute Angles.” <i>Israel Journal of Mathematics</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s11856-016-1429-z\">https://doi.org/10.1007/s11856-016-1429-z</a>.","ieee":"A. Akopyan and A. Balitskiy, “Billiards in convex bodies with acute angles,” <i>Israel Journal of Mathematics</i>, vol. 216, no. 2. Springer, pp. 833–845, 2016.","ama":"Akopyan A, Balitskiy A. Billiards in convex bodies with acute angles. <i>Israel Journal of Mathematics</i>. 2016;216(2):833-845. doi:<a href=\"https://doi.org/10.1007/s11856-016-1429-z\">10.1007/s11856-016-1429-z</a>","apa":"Akopyan, A., &#38; Balitskiy, A. (2016). Billiards in convex bodies with acute angles. <i>Israel Journal of Mathematics</i>. Springer. <a href=\"https://doi.org/10.1007/s11856-016-1429-z\">https://doi.org/10.1007/s11856-016-1429-z</a>","short":"A. Akopyan, A. Balitskiy, Israel Journal of Mathematics 216 (2016) 833–845.","mla":"Akopyan, Arseniy, and Alexey Balitskiy. “Billiards in Convex Bodies with Acute Angles.” <i>Israel Journal of Mathematics</i>, vol. 216, no. 2, Springer, 2016, pp. 833–45, doi:<a href=\"https://doi.org/10.1007/s11856-016-1429-z\">10.1007/s11856-016-1429-z</a>."},"external_id":{"arxiv":["1506.06014"],"isi":["000386356400012"]},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","volume":216,"department":[{"_id":"HeEd"}],"month":"10","day":"15","scopus_import":"1","publisher":"Springer","status":"public","ec_funded":1,"issue":"2","intvolume":"       216","corr_author":"1","_id":"1330","arxiv":1,"article_processing_charge":"No","type":"journal_article","date_created":"2018-12-11T11:51:24Z","page":"833 - 845","language":[{"iso":"eng"}],"publication":"Israel Journal of Mathematics","publication_status":"published","oa":1,"main_file_link":[{"url":"https://arxiv.org/abs/1506.06014","open_access":"1"}],"acknowledgement":"Supported by People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement n°[291734]. Supported by the Russian Foundation for Basic Research grant 15-31-20403 (mol a ved), by the Russian Foundation for Basic Research grant 15-01-99563 A, in part by the Moebius Contest Foundation for Young Scientists, and in part by the Simons Foundation.","abstract":[{"text":"In this paper we investigate the existence of closed billiard trajectories in not necessarily smooth convex bodies. In particular, we show that if a body K ⊂ Rd has the property that the tangent cone of every non-smooth point q ∉ ∂K is acute (in a certain sense), then there is a closed billiard trajectory in K.","lang":"eng"}],"date_published":"2016-10-15T00:00:00Z","title":"Billiards in convex bodies with acute angles","date_updated":"2025-09-22T08:23:27Z","doi":"10.1007/s11856-016-1429-z"},{"publisher":"American Mathematical Society","scopus_import":"1","ec_funded":1,"status":"public","day":"01","department":[{"_id":"HeEd"}],"month":"10","user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","volume":144,"external_id":{"arxiv":["1401.0442"],"isi":["000383054200034"]},"citation":{"short":"A. Akopyan, A. Balitskiy, R. Karasev, A. Sharipova, Proceedings of the American Mathematical Society 144 (2016) 4501–4513.","mla":"Akopyan, Arseniy, et al. “Elementary Approach to Closed Billiard Trajectories in Asymmetric Normed Spaces.” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 10, American Mathematical Society, 2016, pp. 4501–13, doi:<a href=\"https://doi.org/10.1090/proc/13062\">10.1090/proc/13062</a>.","ista":"Akopyan A, Balitskiy A, Karasev R, Sharipova A. 2016. Elementary approach to closed billiard trajectories in asymmetric normed spaces. Proceedings of the American Mathematical Society. 144(10), 4501–4513.","apa":"Akopyan, A., Balitskiy, A., Karasev, R., &#38; Sharipova, A. (2016). Elementary approach to closed billiard trajectories in asymmetric normed spaces. <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society. <a href=\"https://doi.org/10.1090/proc/13062\">https://doi.org/10.1090/proc/13062</a>","ieee":"A. Akopyan, A. Balitskiy, R. Karasev, and A. Sharipova, “Elementary approach to closed billiard trajectories in asymmetric normed spaces,” <i>Proceedings of the American Mathematical Society</i>, vol. 144, no. 10. American Mathematical Society, pp. 4501–4513, 2016.","ama":"Akopyan A, Balitskiy A, Karasev R, Sharipova A. Elementary approach to closed billiard trajectories in asymmetric normed spaces. <i>Proceedings of the American Mathematical Society</i>. 2016;144(10):4501-4513. doi:<a href=\"https://doi.org/10.1090/proc/13062\">10.1090/proc/13062</a>","chicago":"Akopyan, Arseniy, Alexey Balitskiy, Roman Karasev, and Anastasia Sharipova. “Elementary Approach to Closed Billiard Trajectories in Asymmetric Normed Spaces.” <i>Proceedings of the American Mathematical Society</i>. American Mathematical Society, 2016. <a href=\"https://doi.org/10.1090/proc/13062\">https://doi.org/10.1090/proc/13062</a>."},"publist_id":"5885","isi":1,"project":[{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"author":[{"last_name":"Akopyan","orcid":"0000-0002-2548-617X","first_name":"Arseniy","full_name":"Akopyan, Arseniy","id":"430D2C90-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Balitskiy, Alexey","first_name":"Alexey","last_name":"Balitskiy"},{"first_name":"Roman","last_name":"Karasev","full_name":"Karasev, Roman"},{"last_name":"Sharipova","first_name":"Anastasia","full_name":"Sharipova, Anastasia"}],"quality_controlled":"1","year":"2016","oa_version":"Preprint","doi":"10.1090/proc/13062","date_updated":"2025-09-22T07:44:59Z","date_published":"2016-10-01T00:00:00Z","title":"Elementary approach to closed billiard trajectories in asymmetric normed spaces","abstract":[{"lang":"eng","text":"We apply the technique of Károly Bezdek and Daniel Bezdek to study billiard trajectories in convex bodies, when the length is measured with a (possibly asymmetric) norm. We prove a lower bound for the length of the shortest closed billiard trajectory, related to the non-symmetric Mahler problem. With this technique we are able to give short and elementary proofs to some known results. "}],"acknowledgement":"The first and third authors were supported by the Dynasty Foundation. The first, second and third authors were supported by the Russian Foundation for Basic Re- search grant 15-31-20403 (mol a ved).","main_file_link":[{"url":"https://arxiv.org/abs/1401.0442","open_access":"1"}],"publication_status":"published","publication":"Proceedings of the American Mathematical Society","oa":1,"language":[{"iso":"eng"}],"date_created":"2018-12-11T11:51:34Z","page":"4501 - 4513","type":"journal_article","arxiv":1,"article_processing_charge":"No","_id":"1360","intvolume":"       144","issue":"10"},{"license":"https://creativecommons.org/licenses/by/4.0/","abstract":[{"lang":"eng","text":"The concept of well group in a special but important case captures homological properties of the zero set of a continuous map (Formula presented.) on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within (Formula presented.) distance r from f for a given (Formula presented.). The main drawback of the approach is that the computability of well groups was shown only when (Formula presented.) or (Formula presented.). Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of (Formula presented.) by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and (Formula presented.), our approximation of the (Formula presented.)th well group is exact. For the second part, we find examples of maps (Formula presented.) with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status."}],"title":"On computability and triviality of well groups","date_published":"2016-07-01T00:00:00Z","date_updated":"2025-09-18T14:30:52Z","doi":"10.1007/s00454-016-9794-2","language":[{"iso":"eng"}],"oa":1,"publication_status":"published","publication":"Discrete & Computational Geometry","acknowledgement":"Open access funding provided by Institute of Science and Technology (IST Austria). ","_id":"1408","ddc":["510"],"article_processing_charge":"Yes (via OA deal)","type":"journal_article","page":"126 - 164","date_created":"2018-12-11T11:51:51Z","pubrep_id":"614","issue":"1","intvolume":"        56","corr_author":"1","day":"01","has_accepted_license":"1","file":[{"relation":"main_file","file_name":"IST-2016-614-v1+1_s00454-016-9794-2.pdf","creator":"system","file_id":"4846","checksum":"e0da023abf6b72abd8c6a8c76740d53c","file_size":905303,"access_level":"open_access","content_type":"application/pdf","date_created":"2018-12-12T10:10:55Z","date_updated":"2020-07-14T12:44:53Z"}],"status":"public","ec_funded":1,"publisher":"Springer","scopus_import":"1","related_material":{"record":[{"status":"public","relation":"earlier_version","id":"1510"}]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"publist_id":"5799","citation":{"ista":"Franek P, Krcál M. 2016. On computability and triviality of well groups. Discrete &#38; Computational Geometry. 56(1), 126–164.","ama":"Franek P, Krcál M. On computability and triviality of well groups. <i>Discrete &#38; Computational Geometry</i>. 2016;56(1):126-164. doi:<a href=\"https://doi.org/10.1007/s00454-016-9794-2\">10.1007/s00454-016-9794-2</a>","chicago":"Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well Groups.” <i>Discrete &#38; Computational Geometry</i>. Springer, 2016. <a href=\"https://doi.org/10.1007/s00454-016-9794-2\">https://doi.org/10.1007/s00454-016-9794-2</a>.","ieee":"P. Franek and M. Krcál, “On computability and triviality of well groups,” <i>Discrete &#38; Computational Geometry</i>, vol. 56, no. 1. Springer, pp. 126–164, 2016.","apa":"Franek, P., &#38; Krcál, M. (2016). On computability and triviality of well groups. <i>Discrete &#38; Computational Geometry</i>. Springer. <a href=\"https://doi.org/10.1007/s00454-016-9794-2\">https://doi.org/10.1007/s00454-016-9794-2</a>","short":"P. Franek, M. Krcál, Discrete &#38; Computational Geometry 56 (2016) 126–164.","mla":"Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well Groups.” <i>Discrete &#38; Computational Geometry</i>, vol. 56, no. 1, Springer, 2016, pp. 126–64, doi:<a href=\"https://doi.org/10.1007/s00454-016-9794-2\">10.1007/s00454-016-9794-2</a>."},"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["000377722100005"]},"volume":56,"month":"07","department":[{"_id":"UlWa"},{"_id":"HeEd"}],"project":[{"call_identifier":"FWF","_id":"25F8B9BC-B435-11E9-9278-68D0E5697425","name":"Robust Invariants of Nonlinear Systems","grant_number":"M01980"},{"call_identifier":"FP7","name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425"},{"_id":"B67AFEDC-15C9-11EA-A837-991A96BB2854","name":"IST Austria Open Access Fund"}],"author":[{"id":"473294AE-F248-11E8-B48F-1D18A9856A87","full_name":"Franek, Peter","first_name":"Peter","orcid":"0000-0001-8878-8397","last_name":"Franek"},{"full_name":"Krcál, Marek","id":"33E21118-F248-11E8-B48F-1D18A9856A87","last_name":"Krcál","first_name":"Marek"}],"isi":1,"oa_version":"Published Version","file_date_updated":"2020-07-14T12:44:53Z","year":"2016","quality_controlled":"1"},{"page":"199 - 216","date_created":"2018-12-11T11:53:03Z","type":"journal_article","article_processing_charge":"No","arxiv":1,"_id":"1617","intvolume":"        33","doi":"10.1016/j.jco.2015.11.003","date_updated":"2025-09-18T10:57:52Z","title":"On the discrepancy of jittered sampling","date_published":"2016-04-01T00:00:00Z","abstract":[{"lang":"eng","text":"We study the discrepancy of jittered sampling sets: such a set P⊂ [0,1]d is generated for fixed m∈ℕ by partitioning [0,1]d into md axis aligned cubes of equal measure and placing a random point inside each of the N=md cubes. We prove that, for N sufficiently large, 1/10 d/N1/2+1/2d ≤EDN∗(P)≤ √d(log N) 1/2/N1/2+1/2d, where the upper bound with an unspecified constant Cd was proven earlier by Beck. Our proof makes crucial use of the sharp Dvoretzky-Kiefer-Wolfowitz inequality and a suitably taylored Bernstein inequality; we have reasons to believe that the upper bound has the sharp scaling in N. Additional heuristics suggest that jittered sampling should be able to improve known bounds on the inverse of the star-discrepancy in the regime N≳dd. We also prove a partition principle showing that every partition of [0,1]d combined with a jittered sampling construction gives rise to a set whose expected squared L2-discrepancy is smaller than that of purely random points."}],"acknowledgement":"We are grateful to the referee whose suggestions greatly improved the quality and clarity of the exposition.","main_file_link":[{"url":"http://arxiv.org/abs/1510.00251","open_access":"1"}],"oa":1,"publication_status":"published","publication":"Journal of Complexity","language":[{"iso":"eng"}],"isi":1,"author":[{"id":"2A77D7A2-F248-11E8-B48F-1D18A9856A87","full_name":"Pausinger, Florian","orcid":"0000-0002-8379-3768","first_name":"Florian","last_name":"Pausinger"},{"last_name":"Steinerberger","first_name":"Stefan","full_name":"Steinerberger, Stefan"}],"quality_controlled":"1","year":"2016","oa_version":"Submitted Version","publisher":"Academic Press","scopus_import":"1","status":"public","day":"01","month":"04","department":[{"_id":"HeEd"}],"volume":33,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","external_id":{"isi":["000370090400011"],"arxiv":["1510.00251"]},"citation":{"mla":"Pausinger, Florian, and Stefan Steinerberger. “On the Discrepancy of Jittered Sampling.” <i>Journal of Complexity</i>, vol. 33, Academic Press, 2016, pp. 199–216, doi:<a href=\"https://doi.org/10.1016/j.jco.2015.11.003\">10.1016/j.jco.2015.11.003</a>.","short":"F. Pausinger, S. Steinerberger, Journal of Complexity 33 (2016) 199–216.","apa":"Pausinger, F., &#38; Steinerberger, S. (2016). On the discrepancy of jittered sampling. <i>Journal of Complexity</i>. Academic Press. <a href=\"https://doi.org/10.1016/j.jco.2015.11.003\">https://doi.org/10.1016/j.jco.2015.11.003</a>","ama":"Pausinger F, Steinerberger S. On the discrepancy of jittered sampling. <i>Journal of Complexity</i>. 2016;33:199-216. doi:<a href=\"https://doi.org/10.1016/j.jco.2015.11.003\">10.1016/j.jco.2015.11.003</a>","ieee":"F. Pausinger and S. Steinerberger, “On the discrepancy of jittered sampling,” <i>Journal of Complexity</i>, vol. 33. Academic Press, pp. 199–216, 2016.","chicago":"Pausinger, Florian, and Stefan Steinerberger. “On the Discrepancy of Jittered Sampling.” <i>Journal of Complexity</i>. Academic Press, 2016. <a href=\"https://doi.org/10.1016/j.jco.2015.11.003\">https://doi.org/10.1016/j.jco.2015.11.003</a>.","ista":"Pausinger F, Steinerberger S. 2016. On the discrepancy of jittered sampling. Journal of Complexity. 33, 199–216."},"publist_id":"5549"},{"quality_controlled":"1","oa_version":"None","year":"2016","author":[{"full_name":"Sen, Nabhasmita","first_name":"Nabhasmita","last_name":"Sen"},{"id":"3C2B033E-F248-11E8-B48F-1D18A9856A87","full_name":"Biswas, Ranita","orcid":"0000-0002-5372-7890","first_name":"Ranita","last_name":"Biswas"},{"first_name":"Partha","last_name":"Bhowmick","full_name":"Bhowmick, Partha"}],"conference":{"start_date":"2016-06-15","end_date":"2016-06-17","name":"CTIC: Computational Topology in Image Context","location":"Marseille, France"},"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","volume":9667,"month":"06","department":[{"_id":"HeEd"}],"citation":{"short":"N. Sen, R. Biswas, P. Bhowmick, in:, Computational Topology in Image Context, Springer Nature, Cham, 2016, pp. 253–264.","mla":"Sen, Nabhasmita, et al. “On Some Local Topological Properties of Naive Discrete Sphere.” <i>Computational Topology in Image Context</i>, vol. 9667, Springer Nature, 2016, pp. 253–64, doi:<a href=\"https://doi.org/10.1007/978-3-319-39441-1_23\">10.1007/978-3-319-39441-1_23</a>.","ista":"Sen N, Biswas R, Bhowmick P. 2016.On some local topological properties of naive discrete sphere. In: Computational Topology in Image Context. LNCS, vol. 9667, 253–264.","apa":"Sen, N., Biswas, R., &#38; Bhowmick, P. (2016). On some local topological properties of naive discrete sphere. In <i>Computational Topology in Image Context</i> (Vol. 9667, pp. 253–264). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-319-39441-1_23\">https://doi.org/10.1007/978-3-319-39441-1_23</a>","ieee":"N. Sen, R. Biswas, and P. Bhowmick, “On some local topological properties of naive discrete sphere,” in <i>Computational Topology in Image Context</i>, vol. 9667, Cham: Springer Nature, 2016, pp. 253–264.","ama":"Sen N, Biswas R, Bhowmick P. On some local topological properties of naive discrete sphere. In: <i>Computational Topology in Image Context</i>. Vol 9667. Cham: Springer Nature; 2016:253-264. doi:<a href=\"https://doi.org/10.1007/978-3-319-39441-1_23\">10.1007/978-3-319-39441-1_23</a>","chicago":"Sen, Nabhasmita, Ranita Biswas, and Partha Bhowmick. “On Some Local Topological Properties of Naive Discrete Sphere.” In <i>Computational Topology in Image Context</i>, 9667:253–64. Cham: Springer Nature, 2016. <a href=\"https://doi.org/10.1007/978-3-319-39441-1_23\">https://doi.org/10.1007/978-3-319-39441-1_23</a>."},"place":"Cham","publisher":"Springer Nature","status":"public","day":"02","intvolume":"      9667","type":"book_chapter","page":"253-264","date_created":"2019-01-08T20:44:24Z","_id":"5805","article_processing_charge":"No","language":[{"iso":"eng"}],"extern":"1","alternative_title":["LNCS"],"publication":"Computational Topology in Image Context","publication_status":"published","date_updated":"2022-01-28T08:01:22Z","publication_identifier":{"issn":["0302-9743"],"eisbn":["978-3-319-39441-1"],"isbn":["978-3-319-39440-4"],"eissn":["1611-3349"]},"doi":"10.1007/978-3-319-39441-1_23","abstract":[{"lang":"eng","text":"Discretization of sphere in the integer space follows a particular discretization scheme, which, in principle, conforms to some topological model. This eventually gives rise to interesting topological properties of a discrete spherical surface, which need to be investigated for its analytical characterization. This paper presents some novel results on the local topological properties of the naive model of discrete sphere. They follow from the bijection of each quadraginta octant of naive sphere with its projection map called f -map on the corresponding functional plane and from the characterization of certain jumps in the f-map. As an application, we have shown how these properties can be used in designing an efficient reconstruction algorithm for a naive spherical surface from an input voxel set when it is sparse or noisy."}],"title":"On some local topological properties of naive discrete sphere","date_published":"2016-06-02T00:00:00Z"},{"intvolume":"      9647","article_processing_charge":"No","_id":"5806","date_created":"2019-01-08T20:44:37Z","page":"256-267","type":"conference","publication_status":"published","publication":"Discrete Geometry for Computer Imagery","alternative_title":["LNCS"],"extern":"1","language":[{"iso":"eng"}],"date_published":"2016-04-09T00:00:00Z","title":"On functionality of quadraginta octants of naive sphere with application to circle drawing","abstract":[{"lang":"eng","text":"Although the concept of functional plane for naive plane is studied and reported in the literature in great detail, no similar study is yet found for naive sphere. This article exposes the first study in this line, opening up further prospects of analyzing the topological properties of sphere in the discrete space. We show that each quadraginta octant Q of a naive sphere forms a bijection with its projected pixel set on a unique coordinate plane, which thereby serves as the functional plane of Q, and hence gives rise to merely mono-jumps during back projection. The other two coordinate planes serve as para-functional and dia-functional planes for Q, as the former is ‘mono-jumping’ but not bijective, whereas the latter holds neither of the two. Owing to this, the quadraginta octants form symmetry groups and subgroups with equivalent jump conditions. We also show a potential application in generating a special class of discrete 3D circles based on back projection and jump bridging by Steiner voxels. A circle in this class possesses 4-symmetry, uniqueness, and bounded distance from the underlying real sphere and real plane."}],"doi":"10.1007/978-3-319-32360-2_20","publication_identifier":{"eisbn":["978-3-319-32360-2"],"isbn":["978-3-319-32359-6"],"issn":["0302-9743","1611-3349"]},"date_updated":"2022-01-28T08:10:11Z","year":"2016","oa_version":"None","quality_controlled":"1","author":[{"last_name":"Biswas","first_name":"Ranita","orcid":"0000-0002-5372-7890","full_name":"Biswas, Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Bhowmick","first_name":"Partha","full_name":"Bhowmick, Partha"}],"citation":{"short":"R. Biswas, P. Bhowmick, in:, Discrete Geometry for Computer Imagery, Springer Nature, Cham, 2016, pp. 256–267.","mla":"Biswas, Ranita, and Partha Bhowmick. “On Functionality of Quadraginta Octants of Naive Sphere with Application to Circle Drawing.” <i>Discrete Geometry for Computer Imagery</i>, vol. 9647, Springer Nature, 2016, pp. 256–67, doi:<a href=\"https://doi.org/10.1007/978-3-319-32360-2_20\">10.1007/978-3-319-32360-2_20</a>.","ista":"Biswas R, Bhowmick P. 2016. On functionality of quadraginta octants of naive sphere with application to circle drawing. Discrete Geometry for Computer Imagery. DGCI: International Conference on Discrete Geometry for Computer Imagery, LNCS, vol. 9647, 256–267.","ieee":"R. Biswas and P. Bhowmick, “On functionality of quadraginta octants of naive sphere with application to circle drawing,” in <i>Discrete Geometry for Computer Imagery</i>, Nantes, France, 2016, vol. 9647, pp. 256–267.","ama":"Biswas R, Bhowmick P. On functionality of quadraginta octants of naive sphere with application to circle drawing. In: <i>Discrete Geometry for Computer Imagery</i>. Vol 9647. Cham: Springer Nature; 2016:256-267. doi:<a href=\"https://doi.org/10.1007/978-3-319-32360-2_20\">10.1007/978-3-319-32360-2_20</a>","chicago":"Biswas, Ranita, and Partha Bhowmick. “On Functionality of Quadraginta Octants of Naive Sphere with Application to Circle Drawing.” In <i>Discrete Geometry for Computer Imagery</i>, 9647:256–67. Cham: Springer Nature, 2016. <a href=\"https://doi.org/10.1007/978-3-319-32360-2_20\">https://doi.org/10.1007/978-3-319-32360-2_20</a>.","apa":"Biswas, R., &#38; Bhowmick, P. (2016). On functionality of quadraginta octants of naive sphere with application to circle drawing. In <i>Discrete Geometry for Computer Imagery</i> (Vol. 9647, pp. 256–267). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-319-32360-2_20\">https://doi.org/10.1007/978-3-319-32360-2_20</a>"},"department":[{"_id":"HeEd"}],"month":"04","user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","volume":9647,"conference":{"name":"DGCI: International Conference on Discrete Geometry for Computer Imagery","location":"Nantes, France","start_date":"2016-04-18","end_date":"2016-04-20"},"day":"09","publisher":"Springer Nature","status":"public","place":"Cham"},{"type":"book_chapter","date_created":"2019-01-08T20:45:19Z","page":"86-100","_id":"5809","author":[{"last_name":"Biswas","orcid":"0000-0002-5372-7890","first_name":"Ranita","full_name":"Biswas, Ranita","id":"3C2B033E-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Bhowmick, Partha","last_name":"Bhowmick","first_name":"Partha"},{"full_name":"Brimkov, Valentin E.","first_name":"Valentin E.","last_name":"Brimkov"}],"article_processing_charge":"No","intvolume":"      9448","quality_controlled":"1","oa_version":"None","year":"2016","date_updated":"2022-01-28T08:13:03Z","place":"Cham","doi":"10.1007/978-3-319-26145-4_7","status":"public","publication_identifier":{"issn":["0302-9743"],"eisbn":["978-3-319-26145-4"],"isbn":["978-3-319-26144-7"],"eissn":["1611-3349"]},"publisher":"Springer Nature","abstract":[{"text":"A discrete spherical circle is a topologically well-connected 3D circle in the integer space, which belongs to a discrete sphere as well as a discrete plane. It is one of the most important 3D geometric primitives, but has not possibly yet been studied up to its merit. This paper is a maiden exposition of some of its elementary properties, which indicates a sense of its profound theoretical prospects in the framework of digital geometry. We have shown how different types of discretization can lead to forbidden and admissible classes, when one attempts to define the discretization of a spherical circle in terms of intersection between a discrete sphere and a discrete plane. Several fundamental theoretical results have been presented, the algorithm for construction of discrete spherical circles has been discussed, and some test results have been furnished to demonstrate its practicality and usefulness.","lang":"eng"}],"date_published":"2016-01-06T00:00:00Z","title":"On the connectivity and smoothness of discrete spherical circles","day":"06","volume":9448,"user_id":"8b945eb4-e2f2-11eb-945a-df72226e66a9","conference":{"end_date":"2015-11-27","start_date":"2015-11-24","location":"Kolkata, India","name":"IWCIA: International Workshop on Combinatorial Image Analysis"},"department":[{"_id":"HeEd"}],"month":"01","language":[{"iso":"eng"}],"publication":"Combinatorial image analysis","citation":{"short":"R. Biswas, P. Bhowmick, V.E. Brimkov, in:, Combinatorial Image Analysis, Springer Nature, Cham, 2016, pp. 86–100.","mla":"Biswas, Ranita, et al. “On the Connectivity and Smoothness of Discrete Spherical Circles.” <i>Combinatorial Image Analysis</i>, vol. 9448, Springer Nature, 2016, pp. 86–100, doi:<a href=\"https://doi.org/10.1007/978-3-319-26145-4_7\">10.1007/978-3-319-26145-4_7</a>.","ista":"Biswas R, Bhowmick P, Brimkov VE. 2016.On the connectivity and smoothness of discrete spherical circles. In: Combinatorial image analysis. vol. 9448, 86–100.","apa":"Biswas, R., Bhowmick, P., &#38; Brimkov, V. E. (2016). On the connectivity and smoothness of discrete spherical circles. In <i>Combinatorial image analysis</i> (Vol. 9448, pp. 86–100). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-319-26145-4_7\">https://doi.org/10.1007/978-3-319-26145-4_7</a>","ama":"Biswas R, Bhowmick P, Brimkov VE. On the connectivity and smoothness of discrete spherical circles. In: <i>Combinatorial Image Analysis</i>. Vol 9448. Cham: Springer Nature; 2016:86-100. doi:<a href=\"https://doi.org/10.1007/978-3-319-26145-4_7\">10.1007/978-3-319-26145-4_7</a>","ieee":"R. Biswas, P. Bhowmick, and V. E. Brimkov, “On the connectivity and smoothness of discrete spherical circles,” in <i>Combinatorial image analysis</i>, vol. 9448, Cham: Springer Nature, 2016, pp. 86–100.","chicago":"Biswas, Ranita, Partha Bhowmick, and Valentin E. Brimkov. “On the Connectivity and Smoothness of Discrete Spherical Circles.” In <i>Combinatorial Image Analysis</i>, 9448:86–100. Cham: Springer Nature, 2016. <a href=\"https://doi.org/10.1007/978-3-319-26145-4_7\">https://doi.org/10.1007/978-3-319-26145-4_7</a>."},"publication_status":"published","extern":"1"},{"volume":13,"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"09","department":[{"_id":"HeEd"}],"publist_id":"6048","citation":{"short":"M. Held, S. Huber, P. Palfrader, Computer-Aided Design and Applications 13 (2016) 712–721.","mla":"Held, Martin, et al. “Generalized Offsetting of Planar Structures Using Skeletons.” <i>Computer-Aided Design and Applications</i>, vol. 13, no. 5, Taylor &#38; Francis, 2016, pp. 712–21, doi:<a href=\"https://doi.org/10.1080/16864360.2016.1150718\">10.1080/16864360.2016.1150718</a>.","ista":"Held M, Huber S, Palfrader P. 2016. Generalized offsetting of planar structures using skeletons. Computer-Aided Design and Applications. 13(5), 712–721.","apa":"Held, M., Huber, S., &#38; Palfrader, P. (2016). Generalized offsetting of planar structures using skeletons. <i>Computer-Aided Design and Applications</i>. Taylor &#38; Francis. <a href=\"https://doi.org/10.1080/16864360.2016.1150718\">https://doi.org/10.1080/16864360.2016.1150718</a>","ama":"Held M, Huber S, Palfrader P. Generalized offsetting of planar structures using skeletons. <i>Computer-Aided Design and Applications</i>. 2016;13(5):712-721. doi:<a href=\"https://doi.org/10.1080/16864360.2016.1150718\">10.1080/16864360.2016.1150718</a>","chicago":"Held, Martin, Stefan Huber, and Peter Palfrader. “Generalized Offsetting of Planar Structures Using Skeletons.” <i>Computer-Aided Design and Applications</i>. Taylor &#38; Francis, 2016. <a href=\"https://doi.org/10.1080/16864360.2016.1150718\">https://doi.org/10.1080/16864360.2016.1150718</a>.","ieee":"M. Held, S. Huber, and P. Palfrader, “Generalized offsetting of planar structures using skeletons,” <i>Computer-Aided Design and Applications</i>, vol. 13, no. 5. Taylor &#38; Francis, pp. 712–721, 2016."},"has_accepted_license":"1","scopus_import":"1","status":"public","file":[{"content_type":"application/pdf","date_created":"2018-12-12T10:16:20Z","date_updated":"2020-07-14T12:44:42Z","creator":"system","file_name":"IST-2016-694-v1+1_Generalized_offsetting_of_planar_structures_using_skeletons.pdf","checksum":"c746f3a48edb62b588d92ea5d0fd2c0e","file_id":"5206","access_level":"open_access","file_size":1678369,"relation":"main_file"}],"publisher":"Taylor & Francis","tmp":{"image":"/images/cc_by_nc_nd.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","short":"CC BY-NC-ND (4.0)"},"day":"02","quality_controlled":"1","oa_version":"Published Version","file_date_updated":"2020-07-14T12:44:42Z","year":"2016","author":[{"last_name":"Held","first_name":"Martin","full_name":"Held, Martin"},{"first_name":"Stefan","orcid":"0000-0002-8871-5814","last_name":"Huber","id":"4700A070-F248-11E8-B48F-1D18A9856A87","full_name":"Huber, Stefan"},{"first_name":"Peter","last_name":"Palfrader","full_name":"Palfrader, Peter"}],"acknowledgement":"This work was supported by Austrian Science Fund (FWF): P25816-N15.","language":[{"iso":"eng"}],"oa":1,"publication":"Computer-Aided Design and Applications","publication_status":"published","date_updated":"2026-07-07T05:34:31Z","doi":"10.1080/16864360.2016.1150718","abstract":[{"lang":"eng","text":"We study different means to extend offsetting based on skeletal structures beyond the well-known constant-radius and mitered offsets supported by Voronoi diagrams and straight skeletons, for which the orthogonal distance of offset elements to their respective input elements is constant and uniform over all input elements. Our main contribution is a new geometric structure, called variable-radius Voronoi diagram, which supports the computation of variable-radius offsets, i.e., offsets whose distance to the input is allowed to vary along the input. We discuss properties of this structure and sketch a prototype implementation that supports the computation of variable-radius offsets based on this new variant of Voronoi diagrams."}],"title":"Generalized offsetting of planar structures using skeletons","date_published":"2016-09-02T00:00:00Z","intvolume":"        13","pubrep_id":"694","issue":"5","type":"journal_article","page":"712 - 721","date_created":"2018-12-11T11:51:04Z","ddc":["004","516"],"_id":"1272","das_tickbox":"1","article_processing_charge":"No"},{"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1504.00116"}],"acknowledgement":"AG and PP were partially supported by Abdus Salam International Centre for Theoretical Physics (ICTP). Additionally, AG was supported by BREUDS, and research conducted by PP has received funding from Fundo Europeu de Desenvolvimento Regional (FEDER) through COMPETE—Programa Operacional Factores de Competitividade (POFC) and from the Portuguese national funds through Fundação para a Ciência e a Tecnologia (FCT) in the framework of the research project FCOMP-01-0124-FEDER-010645 (ref. FCT PTDC/MAT/098871/2008); and from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement no. 622033. The  authors  gratefully  acknowledge  the  Department  of\r\nMathematics  of  Kyoto  University  for  providing  access\r\nto  their  server  for  conducting  computations  for  this\r\nproject.","language":[{"iso":"eng"}],"publication":"Experimental Mathematics","publication_status":"published","oa":1,"date_updated":"2026-07-07T05:34:16Z","doi":"10.1080/10586458.2015.1048011","abstract":[{"text":"We use rigorous numerical techniques to compute a lower bound for the exponent of expansivity outside a neighborhood of the critical point for thousands of intervals of parameter values in the quadratic family. We first compute a radius of the critical neighborhood outside which the map is uniformly expanding. This radius is taken as small as possible, yet large enough for our numerical procedure to succeed in proving that the expansivity exponent outside this neighborhood is positive. Then, for each of the intervals, we compute a lower bound for this expansivity exponent, valid for all the parameters in that interval. We illustrate and study the distribution of the radii and the expansivity exponents. The results of our computations are mathematically rigorous. The source code of the software and the results of the computations are made publicly available at http://www.pawelpilarczyk.com/quadratic/.","lang":"eng"}],"date_published":"2016-04-02T00:00:00Z","title":"Uniform expansivity outside a critical neighborhood in the quadratic family","intvolume":"        25","corr_author":"1","issue":"2","type":"journal_article","date_created":"2018-12-11T11:50:58Z","page":"116 - 124","das_tickbox":"1","_id":"1254","arxiv":1,"article_processing_charge":"No","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":25,"external_id":{"arxiv":["1504.00116"],"isi":["000372490500002"]},"department":[{"_id":"HeEd"}],"month":"04","publist_id":"6071","citation":{"apa":"Golmakani, A., Luzzatto, S., &#38; Pilarczyk, P. (2016). Uniform expansivity outside a critical neighborhood in the quadratic family. <i>Experimental Mathematics</i>. Taylor &#38; Francis. <a href=\"https://doi.org/10.1080/10586458.2015.1048011\">https://doi.org/10.1080/10586458.2015.1048011</a>","ama":"Golmakani A, Luzzatto S, Pilarczyk P. Uniform expansivity outside a critical neighborhood in the quadratic family. <i>Experimental Mathematics</i>. 2016;25(2):116-124. doi:<a href=\"https://doi.org/10.1080/10586458.2015.1048011\">10.1080/10586458.2015.1048011</a>","ieee":"A. Golmakani, S. Luzzatto, and P. Pilarczyk, “Uniform expansivity outside a critical neighborhood in the quadratic family,” <i>Experimental Mathematics</i>, vol. 25, no. 2. Taylor &#38; Francis, pp. 116–124, 2016.","chicago":"Golmakani, Ali, Stefano Luzzatto, and Pawel Pilarczyk. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” <i>Experimental Mathematics</i>. Taylor &#38; Francis, 2016. <a href=\"https://doi.org/10.1080/10586458.2015.1048011\">https://doi.org/10.1080/10586458.2015.1048011</a>.","ista":"Golmakani A, Luzzatto S, Pilarczyk P. 2016. Uniform expansivity outside a critical neighborhood in the quadratic family. Experimental Mathematics. 25(2), 116–124.","mla":"Golmakani, Ali, et al. “Uniform Expansivity Outside a Critical Neighborhood in the Quadratic Family.” <i>Experimental Mathematics</i>, vol. 25, no. 2, Taylor &#38; Francis, 2016, pp. 116–24, doi:<a href=\"https://doi.org/10.1080/10586458.2015.1048011\">10.1080/10586458.2015.1048011</a>.","short":"A. Golmakani, S. Luzzatto, P. Pilarczyk, Experimental Mathematics 25 (2016) 116–124."},"publisher":"Taylor & Francis","ec_funded":1,"scopus_import":"1","status":"public","day":"02","quality_controlled":"1","oa_version":"Preprint","year":"2016","isi":1,"project":[{"call_identifier":"FP7","name":"Persistent Homology - Images, Data and Maps","grant_number":"622033","_id":"255F06BE-B435-11E9-9278-68D0E5697425"}],"author":[{"last_name":"Golmakani","first_name":"Ali","full_name":"Golmakani, Ali"},{"full_name":"Luzzatto, Stefano","last_name":"Luzzatto","first_name":"Stefano"},{"id":"3768D56A-F248-11E8-B48F-1D18A9856A87","full_name":"Pilarczyk, Pawel","first_name":"Pawel","last_name":"Pilarczyk"}]},{"isi":1,"project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7"}],"author":[{"orcid":"0000-0002-9823-6833","first_name":"Herbert","last_name":"Edelsbrunner","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","full_name":"Edelsbrunner, Herbert"},{"first_name":"Florian","orcid":"0000-0002-8379-3768","last_name":"Pausinger","id":"2A77D7A2-F248-11E8-B48F-1D18A9856A87","full_name":"Pausinger, Florian"}],"quality_controlled":"1","year":"2016","oa_version":"Published Version","file_date_updated":"2020-07-14T12:45:10Z","file":[{"file_name":"IST-2017-774-v1+1_2016-J-03-FirstIntVolume.pdf","creator":"system","checksum":"f8869ec110c35c852ef6a37425374af7","file_id":"4928","access_level":"open_access","file_size":248985,"content_type":"application/pdf","date_created":"2018-12-12T10:12:10Z","date_updated":"2020-07-14T12:45:10Z","relation":"main_file"}],"ec_funded":1,"scopus_import":"1","publisher":"Academic Press","status":"public","related_material":{"record":[{"id":"1399","relation":"dissertation_contains","status":"public"}]},"tmp":{"image":"/images/cc_by_nc_nd.png","legal_code_url":"https://creativecommons.org/licenses/by-nc-nd/4.0/legalcode","name":"Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)","short":"CC BY-NC-ND (4.0)"},"has_accepted_license":"1","day":"10","month":"01","department":[{"_id":"HeEd"}],"external_id":{"isi":["000375634100016"]},"volume":287,"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","citation":{"short":"H. Edelsbrunner, F. Pausinger, Advances in Mathematics 287 (2016) 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>.","ista":"Edelsbrunner H, Pausinger F. 2016. Approximation and convergence of the intrinsic volume. Advances in Mathematics. 287, 674–703.","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.","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>"},"publist_id":"5488","page":"674 - 703","date_created":"2018-12-11T11:53:20Z","type":"journal_article","article_processing_charge":"No","_id":"1662","ddc":["004"],"corr_author":"1","intvolume":"       287","pubrep_id":"774","doi":"10.1016/j.aim.2015.10.004","date_updated":"2026-07-29T10:08:33Z","title":"Approximation and convergence of the intrinsic volume","date_published":"2016-01-10T00:00:00Z","abstract":[{"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.","lang":"eng"}],"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.","oa":1,"publication":"Advances in Mathematics","publication_status":"published","language":[{"iso":"eng"}]},{"date_created":"2018-12-11T11:51:56Z","page":"3070 - 3078","type":"conference","author":[{"last_name":"Kwitt","first_name":"Roland","full_name":"Kwitt, Roland"},{"id":"4700A070-F248-11E8-B48F-1D18A9856A87","full_name":"Huber, Stefan","orcid":"0000-0002-8871-5814","first_name":"Stefan","last_name":"Huber"},{"first_name":"Marc","last_name":"Niethammer","full_name":"Niethammer, Marc"},{"full_name":"Lin, Weili","first_name":"Weili","last_name":"Lin"},{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","first_name":"Ulrich","last_name":"Bauer"}],"article_processing_charge":"No","_id":"1424","quality_controlled":"1","intvolume":"        28","year":"2015","oa_version":"Submitted Version","status":"public","publisher":"Neural Information Processing Systems Foundation","date_updated":"2025-06-03T11:41:36Z","date_published":"2015-12-01T00:00:00Z","title":"Statistical topological data analysis-A kernel perspective","day":"01","abstract":[{"lang":"eng","text":"We consider the problem of statistical computations with persistence diagrams, a summary representation of topological features in data. These diagrams encode persistent homology, a widely used invariant in topological data analysis. While several avenues towards a statistical treatment of the diagrams have been explored recently, we follow an alternative route that is motivated by the success of methods based on the embedding of probability measures into reproducing kernel Hilbert spaces. In fact, a positive definite kernel on persistence diagrams has recently been proposed, connecting persistent homology to popular kernel-based learning techniques such as support vector machines. However, important properties of that kernel enabling a principled use in the context of probability measure embeddings remain to be explored. Our contribution is to close this gap by proving universality of a variant of the original kernel, and to demonstrate its effective use in twosample hypothesis testing on synthetic as well as real-world data."}],"department":[{"_id":"HeEd"}],"acknowledgement":"This work was partially supported by the Austrian Science FUnd, project no. KLI 00012.","month":"12","volume":28,"main_file_link":[{"url":"https://papers.nips.cc/paper/5887-statistical-topological-data-analysis-a-kernel-perspective","open_access":"1"}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","conference":{"end_date":"2015-12-12","start_date":"2015-12-07","location":"Montreal, Canada","name":"NIPS: Neural Information Processing Systems"},"citation":{"mla":"Kwitt, Roland, et al. <i>Statistical Topological Data Analysis-A Kernel Perspective</i>. Vol. 28, Neural Information Processing Systems Foundation, 2015, pp. 3070–78.","short":"R. Kwitt, S. Huber, M. Niethammer, W. Lin, U. Bauer, in:, Neural Information Processing Systems Foundation, 2015, pp. 3070–3078.","ieee":"R. Kwitt, S. Huber, M. Niethammer, W. Lin, and U. Bauer, “Statistical topological data analysis-A kernel perspective,” presented at the NIPS: Neural Information Processing Systems, Montreal, Canada, 2015, vol. 28, pp. 3070–3078.","chicago":"Kwitt, Roland, Stefan Huber, Marc Niethammer, Weili Lin, and Ulrich Bauer. “Statistical Topological Data Analysis-A Kernel Perspective,” 28:3070–78. Neural Information Processing Systems Foundation, 2015.","ama":"Kwitt R, Huber S, Niethammer M, Lin W, Bauer U. Statistical topological data analysis-A kernel perspective. In: Vol 28. Neural Information Processing Systems Foundation; 2015:3070-3078.","apa":"Kwitt, R., Huber, S., Niethammer, M., Lin, W., &#38; Bauer, U. (2015). Statistical topological data analysis-A kernel perspective (Vol. 28, pp. 3070–3078). Presented at the NIPS: Neural Information Processing Systems, Montreal, Canada: Neural Information Processing Systems Foundation.","ista":"Kwitt R, Huber S, Niethammer M, Lin W, Bauer U. 2015. Statistical topological data analysis-A kernel perspective. NIPS: Neural Information Processing Systems, Advances in Neural Information Processing Systems, vol. 28, 3070–3078."},"publication_status":"published","alternative_title":["Advances in Neural Information Processing Systems"],"oa":1,"language":[{"iso":"eng"}],"publist_id":"5782"},{"year":"2015","oa_version":"Preprint","page":"4741 - 4748","date_created":"2018-12-11T11:52:17Z","type":"conference","article_processing_charge":"No","arxiv":1,"author":[{"first_name":"Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87","full_name":"Reininghaus, Jan"},{"last_name":"Huber","orcid":"0000-0002-8871-5814","first_name":"Stefan","full_name":"Huber, Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Bauer","first_name":"Ulrich","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87"},{"first_name":"Roland","last_name":"Kwitt","full_name":"Kwitt, Roland"}],"_id":"1483","month":"10","department":[{"_id":"HeEd"}],"conference":{"name":"CVPR: Computer Vision and Pattern Recognition","location":"Boston, MA, USA","start_date":"2015-06-07","end_date":"2015-06-12"},"external_id":{"arxiv":["1412.6821"]},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","main_file_link":[{"url":"http://arxiv.org/abs/1412.6821","open_access":"1"}],"oa":1,"publication_status":"published","citation":{"chicago":"Reininghaus, Jan, Stefan Huber, Ulrich Bauer, and Roland Kwitt. “A Stable Multi-Scale Kernel for Topological Machine Learning,” 4741–48. IEEE, 2015. <a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">https://doi.org/10.1109/CVPR.2015.7299106</a>.","ama":"Reininghaus J, Huber S, Bauer U, Kwitt R. A stable multi-scale kernel for topological machine learning. In: IEEE; 2015:4741-4748. doi:<a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">10.1109/CVPR.2015.7299106</a>","ieee":"J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt, “A stable multi-scale kernel for topological machine learning,” presented at the CVPR: Computer Vision and Pattern Recognition, Boston, MA, USA, 2015, pp. 4741–4748.","apa":"Reininghaus, J., Huber, S., Bauer, U., &#38; Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning (pp. 4741–4748). Presented at the CVPR: Computer Vision and Pattern Recognition, Boston, MA, USA: IEEE. <a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">https://doi.org/10.1109/CVPR.2015.7299106</a>","ista":"Reininghaus J, Huber S, Bauer U, Kwitt R. 2015. A stable multi-scale kernel for topological machine learning. CVPR: Computer Vision and Pattern Recognition, 4741–4748.","mla":"Reininghaus, Jan, et al. <i>A Stable Multi-Scale Kernel for Topological Machine Learning</i>. IEEE, 2015, pp. 4741–48, doi:<a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">10.1109/CVPR.2015.7299106</a>.","short":"J. Reininghaus, S. Huber, U. Bauer, R. Kwitt, in:, IEEE, 2015, pp. 4741–4748."},"publist_id":"5709","language":[{"iso":"eng"}],"scopus_import":"1","publisher":"IEEE","status":"public","publication_identifier":{"eisbn":["978-1-4673-6964-0 "]},"doi":"10.1109/CVPR.2015.7299106","date_updated":"2025-06-11T06:37:43Z","day":"14","title":"A stable multi-scale kernel for topological machine learning","date_published":"2015-10-14T00:00:00Z","abstract":[{"text":"Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.","lang":"eng"}]},{"project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7"}],"author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"id":"41B58C0C-F248-11E8-B48F-1D18A9856A87","full_name":"Iglesias Ham, Mabel","first_name":"Mabel","last_name":"Iglesias Ham"},{"last_name":"Kurlin","first_name":"Vitaliy","full_name":"Kurlin, Vitaliy"}],"year":"2015","oa_version":"Submitted Version","quality_controlled":"1","day":"01","status":"public","publisher":"Queen's University","ec_funded":1,"scopus_import":"1","citation":{"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.","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.","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.","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.","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.","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."},"publist_id":"5684","department":[{"_id":"HeEd"}],"month":"08","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":"2015-August","external_id":{"arxiv":["1505.03402"]},"conference":{"name":"CCCG: Canadian Conference on Computational Geometry","location":"Ontario, Canada","start_date":"2015-08-10","end_date":"2015-08-12"},"arxiv":1,"article_processing_charge":"No","_id":"1495","date_created":"2018-12-11T11:52:21Z","page":"128-135","type":"conference","date_published":"2015-08-01T00:00:00Z","title":"Relaxed disk packing","abstract":[{"lang":"eng","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. "}],"date_updated":"2025-06-11T06:38:01Z","publication":"Proceedings of the 27th Canadian Conference on Computational Geometry","publication_status":"published","oa":1,"language":[{"iso":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/abs/1505.03402"}]},{"conference":{"end_date":"2015-06-25","start_date":"2015-06-22","location":"Eindhoven, Netherlands","name":"SoCG: Symposium on Computational Geometry"},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","volume":34,"month":"06","department":[{"_id":"UlWa"},{"_id":"HeEd"}],"publist_id":"5667","citation":{"short":"P. Franek, M. Krcál, in:, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 842–856.","mla":"Franek, Peter, and Marek Krcál. <i>On Computability and Triviality of Well Groups</i>. Vol. 34, Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015, pp. 842–56, doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">10.4230/LIPIcs.SOCG.2015.842</a>.","ista":"Franek P, Krcál M. 2015. On computability and triviality of well groups. SoCG: Symposium on Computational Geometry, LIPIcs, vol. 34, 842–856.","apa":"Franek, P., &#38; Krcál, M. (2015). On computability and triviality of well groups (Vol. 34, pp. 842–856). Presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands: Schloss Dagstuhl - Leibniz-Zentrum für Informatik. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">https://doi.org/10.4230/LIPIcs.SOCG.2015.842</a>","ama":"Franek P, Krcál M. On computability and triviality of well groups. In: Vol 34. Schloss Dagstuhl - Leibniz-Zentrum für Informatik; 2015:842-856. doi:<a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">10.4230/LIPIcs.SOCG.2015.842</a>","chicago":"Franek, Peter, and Marek Krcál. “On Computability and Triviality of Well Groups,” 34:842–56. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2015. <a href=\"https://doi.org/10.4230/LIPIcs.SOCG.2015.842\">https://doi.org/10.4230/LIPIcs.SOCG.2015.842</a>.","ieee":"P. Franek and M. Krcál, “On computability and triviality of well groups,” presented at the SoCG: Symposium on Computational Geometry, Eindhoven, Netherlands, 2015, vol. 34, pp. 842–856."},"has_accepted_license":"1","ec_funded":1,"scopus_import":1,"status":"public","file":[{"creator":"system","file_name":"IST-2016-503-v1+1_32.pdf","checksum":"49eb5021caafaabe5356c65b9c5f8c9c","file_id":"5001","file_size":623563,"access_level":"open_access","content_type":"application/pdf","date_created":"2018-12-12T10:13:19Z","date_updated":"2020-07-14T12:44:59Z","relation":"main_file"}],"publisher":"Schloss Dagstuhl - Leibniz-Zentrum für Informatik","related_material":{"record":[{"status":"public","relation":"later_version","id":"1408"}]},"tmp":{"short":"CC BY (4.0)","name":"Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)","legal_code_url":"https://creativecommons.org/licenses/by/4.0/legalcode","image":"/images/cc_by.png"},"day":"11","quality_controlled":"1","oa_version":"Published Version","file_date_updated":"2020-07-14T12:44:59Z","year":"2015","project":[{"name":"International IST Postdoc Fellowship Programme","grant_number":"291734","_id":"25681D80-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"author":[{"orcid":"0000-0001-8878-8397","first_name":"Peter","last_name":"Franek","id":"473294AE-F248-11E8-B48F-1D18A9856A87","full_name":"Franek, Peter"},{"full_name":"Krcál, Marek","id":"33E21118-F248-11E8-B48F-1D18A9856A87","last_name":"Krcál","first_name":"Marek"}],"language":[{"iso":"eng"}],"oa":1,"alternative_title":["LIPIcs"],"publication_status":"published","date_updated":"2025-09-18T14:30:52Z","doi":"10.4230/LIPIcs.SOCG.2015.842","abstract":[{"lang":"eng","text":"The concept of well group in a special but important case captures homological properties of the zero set of a continuous map f from K to R^n on a compact space K that are invariant with respect to perturbations of f. The perturbations are arbitrary continuous maps within L_infty distance r from f for a given r &gt; 0. The main drawback of the approach is that the computability of well groups was shown only when dim K = n or n = 1. Our contribution to the theory of well groups is twofold: on the one hand we improve on the computability issue, but on the other hand we present a range of examples where the well groups are incomplete invariants, that is, fail to capture certain important robust properties of the zero set. For the first part, we identify a computable subgroup of the well group that is obtained by cap product with the pullback of the orientation of R^n by f. In other words, well groups can be algorithmically approximated from below. When f is smooth and dim K &lt; 2n-2, our approximation of the (dim K-n)th well group is exact. For the second part, we find examples of maps f, f' from K to R^n with all well groups isomorphic but whose perturbations have different zero sets. We discuss on a possible replacement of the well groups of vector valued maps by an invariant of a better descriptive power and computability status. "}],"title":"On computability and triviality of well groups","date_published":"2015-06-11T00:00:00Z","intvolume":"        34","pubrep_id":"503","type":"conference","page":"842 - 856","date_created":"2018-12-11T11:52:26Z","ddc":["510"],"_id":"1510"}]