--- _id: '10893' abstract: - lang: eng text: Saddle periodic orbits are an essential and stable part of the topological skeleton of a 3D vector field. Nevertheless, there is currently no efficient algorithm to robustly extract these features. In this chapter, we present a novel technique to extract saddle periodic orbits. Exploiting the analytic properties of such an orbit, we propose a scalar measure based on the finite-time Lyapunov exponent (FTLE) that indicates its presence. Using persistent homology, we can then extract the robust cycles of this field. These cycles thereby represent the saddle periodic orbits of the given vector field. We discuss the different existing FTLE approximation schemes regarding their applicability to this specific problem and propose an adapted version of FTLE called Normalized Velocity Separation. Finally, we evaluate our method using simple analytic vector field data. acknowledgement: First, we thank the reviewers of this paper for their ideas and critical comments. In addition, we thank Ronny Peikert and Filip Sadlo for a fruitful discussions. This research is supported by the European Commission under the TOPOSYS project FP7-ICT-318493-STREP, the European Social Fund (ESF App. No. 100098251), and the European Science Foundation under the ACAT Research Network Program. article_processing_charge: No author: - first_name: Jens full_name: Kasten, Jens last_name: Kasten - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Wieland full_name: Reich, Wieland last_name: Reich - first_name: Gerik full_name: Scheuermann, Gerik last_name: Scheuermann citation: ama: 'Kasten J, Reininghaus J, Reich W, Scheuermann G. Toward the extraction of saddle periodic orbits. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological Methods in Data Analysis and Visualization III . Vol 1. Mathematics and Visualization. Cham: Springer; 2014:55-69. doi:10.1007/978-3-319-04099-8_4' apa: 'Kasten, J., Reininghaus, J., Reich, W., & Scheuermann, G. (2014). Toward the extraction of saddle periodic orbits. In P.-T. Bremer, I. Hotz, V. Pascucci, & R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III (Vol. 1, pp. 55–69). Cham: Springer. https://doi.org/10.1007/978-3-319-04099-8_4' chicago: 'Kasten, Jens, Jan Reininghaus, Wieland Reich, and Gerik Scheuermann. “Toward the Extraction of Saddle Periodic Orbits.” In Topological Methods in Data Analysis and Visualization III , edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 1:55–69. Mathematics and Visualization. Cham: Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_4.' ieee: 'J. Kasten, J. Reininghaus, W. Reich, and G. Scheuermann, “Toward the extraction of saddle periodic orbits,” in Topological Methods in Data Analysis and Visualization III , vol. 1, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer, 2014, pp. 55–69.' ista: 'Kasten J, Reininghaus J, Reich W, Scheuermann G. 2014.Toward the extraction of saddle periodic orbits. In: Topological Methods in Data Analysis and Visualization III . vol. 1, 55–69.' mla: Kasten, Jens, et al. “Toward the Extraction of Saddle Periodic Orbits.” Topological Methods in Data Analysis and Visualization III , edited by Peer-Timo Bremer et al., vol. 1, Springer, 2014, pp. 55–69, doi:10.1007/978-3-319-04099-8_4. short: J. Kasten, J. Reininghaus, W. Reich, G. Scheuermann, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III , Springer, Cham, 2014, pp. 55–69. date_created: 2022-03-21T07:11:23Z date_published: 2014-03-19T00:00:00Z date_updated: 2022-06-21T12:01:47Z day: '19' department: - _id: HeEd doi: 10.1007/978-3-319-04099-8_4 ec_funded: 1 editor: - first_name: Peer-Timo full_name: Bremer, Peer-Timo last_name: Bremer - first_name: Ingrid full_name: Hotz, Ingrid last_name: Hotz - first_name: Valerio full_name: Pascucci, Valerio last_name: Pascucci - first_name: Ronald full_name: Peikert, Ronald last_name: Peikert intvolume: ' 1' language: - iso: eng month: '03' oa_version: None page: 55-69 place: Cham project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: 'Topological Methods in Data Analysis and Visualization III ' publication_identifier: eisbn: - '9783319040998' eissn: - 2197-666X isbn: - '9783319040981' issn: - 1612-3786 publication_status: published publisher: Springer quality_controlled: '1' scopus_import: '1' series_title: Mathematics and Visualization status: public title: Toward the extraction of saddle periodic orbits type: book_chapter user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 1 year: '2014' ... --- _id: '1816' abstract: - lang: eng text: Watermarking techniques for vector graphics dislocate vertices in order to embed imperceptible, yet detectable, statistical features into the input data. The embedding process may result in a change of the topology of the input data, e.g., by introducing self-intersections, which is undesirable or even disastrous for many applications. In this paper we present a watermarking framework for two-dimensional vector graphics that employs conventional watermarking techniques but still provides the guarantee that the topology of the input data is preserved. The geometric part of this framework computes so-called maximum perturbation regions (MPR) of vertices. We propose two efficient algorithms to compute MPRs based on Voronoi diagrams and constrained triangulations. Furthermore, we present two algorithms to conditionally correct the watermarked data in order to increase the watermark embedding capacity and still guarantee topological correctness. While we focus on the watermarking of input formed by straight-line segments, one of our approaches can also be extended to circular arcs. We conclude the paper by demonstrating and analyzing the applicability of our framework in conjunction with two well-known watermarking techniques. acknowledgement: 'Work by Martin Held and Stefan Huber was supported by Austrian Science Fund (FWF): L367-N15 and P25816-N15.' author: - first_name: Stefan full_name: Huber, Stefan id: 4700A070-F248-11E8-B48F-1D18A9856A87 last_name: Huber orcid: 0000-0002-8871-5814 - first_name: Martin full_name: Held, Martin last_name: Held - first_name: Peter full_name: Meerwald, Peter last_name: Meerwald - first_name: Roland full_name: Kwitt, Roland last_name: Kwitt citation: ama: Huber S, Held M, Meerwald P, Kwitt R. Topology-preserving watermarking of vector graphics. International Journal of Computational Geometry and Applications. 2014;24(1):61-86. doi:10.1142/S0218195914500034 apa: Huber, S., Held, M., Meerwald, P., & Kwitt, R. (2014). Topology-preserving watermarking of vector graphics. International Journal of Computational Geometry and Applications. World Scientific Publishing. https://doi.org/10.1142/S0218195914500034 chicago: Huber, Stefan, Martin Held, Peter Meerwald, and Roland Kwitt. “Topology-Preserving Watermarking of Vector Graphics.” International Journal of Computational Geometry and Applications. World Scientific Publishing, 2014. https://doi.org/10.1142/S0218195914500034. ieee: S. Huber, M. Held, P. Meerwald, and R. Kwitt, “Topology-preserving watermarking of vector graphics,” International Journal of Computational Geometry and Applications, vol. 24, no. 1. World Scientific Publishing, pp. 61–86, 2014. ista: Huber S, Held M, Meerwald P, Kwitt R. 2014. Topology-preserving watermarking of vector graphics. International Journal of Computational Geometry and Applications. 24(1), 61–86. mla: Huber, Stefan, et al. “Topology-Preserving Watermarking of Vector Graphics.” International Journal of Computational Geometry and Applications, vol. 24, no. 1, World Scientific Publishing, 2014, pp. 61–86, doi:10.1142/S0218195914500034. short: S. Huber, M. Held, P. Meerwald, R. Kwitt, International Journal of Computational Geometry and Applications 24 (2014) 61–86. date_created: 2018-12-11T11:54:10Z date_published: 2014-03-16T00:00:00Z date_updated: 2021-01-12T06:53:23Z day: '16' ddc: - '000' department: - _id: HeEd doi: 10.1142/S0218195914500034 file: - access_level: open_access checksum: be45c133ab4d43351260e21beaa8f4b1 content_type: application/pdf creator: system date_created: 2018-12-12T10:08:43Z date_updated: 2020-07-14T12:45:17Z file_id: '4704' file_name: IST-2016-443-v1+1_S0218195914500034.pdf file_size: 991734 relation: main_file file_date_updated: 2020-07-14T12:45:17Z has_accepted_license: '1' intvolume: ' 24' issue: '1' language: - iso: eng month: '03' oa: 1 oa_version: Published Version page: 61 - 86 publication: International Journal of Computational Geometry and Applications publication_status: published publisher: World Scientific Publishing publist_id: '5290' pubrep_id: '443' quality_controlled: '1' scopus_import: 1 status: public title: Topology-preserving watermarking of vector graphics tmp: image: /images/cc_by.png legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0) short: CC BY (4.0) type: journal_article user_id: 3E5EF7F0-F248-11E8-B48F-1D18A9856A87 volume: 24 year: '2014' ... --- _id: '1842' abstract: - lang: eng text: We prove polynomial upper bounds of geometric Ramsey numbers of pathwidth-2 outerplanar triangulations in both convex and general cases. We also prove that the geometric Ramsey numbers of the ladder graph on 2n vertices are bounded by O(n3) and O(n10), in the convex and general case, respectively. We then apply similar methods to prove an (Formula presented.) upper bound on the Ramsey number of a path with n ordered vertices. acknowledgement: Marek Krčál was supported by the ERC Advanced Grant No. 267165. author: - first_name: Josef full_name: Cibulka, Josef last_name: Cibulka - first_name: Pu full_name: Gao, Pu last_name: Gao - first_name: Marek full_name: Krcál, Marek id: 33E21118-F248-11E8-B48F-1D18A9856A87 last_name: Krcál - first_name: Tomáš full_name: Valla, Tomáš last_name: Valla - first_name: Pavel full_name: Valtr, Pavel last_name: Valtr citation: ama: Cibulka J, Gao P, Krcál M, Valla T, Valtr P. On the geometric ramsey number of outerplanar graphs. Discrete & Computational Geometry. 2014;53(1):64-79. doi:10.1007/s00454-014-9646-x apa: Cibulka, J., Gao, P., Krcál, M., Valla, T., & Valtr, P. (2014). On the geometric ramsey number of outerplanar graphs. Discrete & Computational Geometry. Springer. https://doi.org/10.1007/s00454-014-9646-x chicago: Cibulka, Josef, Pu Gao, Marek Krcál, Tomáš Valla, and Pavel Valtr. “On the Geometric Ramsey Number of Outerplanar Graphs.” Discrete & Computational Geometry. Springer, 2014. https://doi.org/10.1007/s00454-014-9646-x. ieee: J. Cibulka, P. Gao, M. Krcál, T. Valla, and P. Valtr, “On the geometric ramsey number of outerplanar graphs,” Discrete & Computational Geometry, vol. 53, no. 1. Springer, pp. 64–79, 2014. ista: Cibulka J, Gao P, Krcál M, Valla T, Valtr P. 2014. On the geometric ramsey number of outerplanar graphs. Discrete & Computational Geometry. 53(1), 64–79. mla: Cibulka, Josef, et al. “On the Geometric Ramsey Number of Outerplanar Graphs.” Discrete & Computational Geometry, vol. 53, no. 1, Springer, 2014, pp. 64–79, doi:10.1007/s00454-014-9646-x. short: J. Cibulka, P. Gao, M. Krcál, T. Valla, P. Valtr, Discrete & Computational Geometry 53 (2014) 64–79. date_created: 2018-12-11T11:54:18Z date_published: 2014-11-14T00:00:00Z date_updated: 2021-01-12T06:53:33Z day: '14' department: - _id: UlWa - _id: HeEd doi: 10.1007/s00454-014-9646-x intvolume: ' 53' issue: '1' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1310.7004 month: '11' oa: 1 oa_version: Submitted Version page: 64 - 79 publication: Discrete & Computational Geometry publication_status: published publisher: Springer publist_id: '5260' scopus_import: 1 status: public title: On the geometric ramsey number of outerplanar graphs type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 53 year: '2014' ... --- _id: '1876' abstract: - lang: eng text: We study densities of functionals over uniformly bounded triangulations of a Delaunay set of vertices, and prove that the minimum is attained for the Delaunay triangulation if this is the case for finite sets. article_processing_charge: No article_type: original author: - first_name: Nikolai full_name: Dolbilin, Nikolai last_name: Dolbilin - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Alexey full_name: Glazyrin, Alexey last_name: Glazyrin - first_name: Oleg full_name: Musin, Oleg last_name: Musin citation: ama: Dolbilin N, Edelsbrunner H, Glazyrin A, Musin O. Functionals on triangulations of delaunay sets. Moscow Mathematical Journal. 2014;14(3):491-504. doi:10.17323/1609-4514-2014-14-3-491-504 apa: Dolbilin, N., Edelsbrunner, H., Glazyrin, A., & Musin, O. (2014). Functionals on triangulations of delaunay sets. Moscow Mathematical Journal. Independent University of Moscow. https://doi.org/10.17323/1609-4514-2014-14-3-491-504 chicago: Dolbilin, Nikolai, Herbert Edelsbrunner, Alexey Glazyrin, and Oleg Musin. “Functionals on Triangulations of Delaunay Sets.” Moscow Mathematical Journal. Independent University of Moscow, 2014. https://doi.org/10.17323/1609-4514-2014-14-3-491-504. ieee: N. Dolbilin, H. Edelsbrunner, A. Glazyrin, and O. Musin, “Functionals on triangulations of delaunay sets,” Moscow Mathematical Journal, vol. 14, no. 3. Independent University of Moscow, pp. 491–504, 2014. ista: Dolbilin N, Edelsbrunner H, Glazyrin A, Musin O. 2014. Functionals on triangulations of delaunay sets. Moscow Mathematical Journal. 14(3), 491–504. mla: Dolbilin, Nikolai, et al. “Functionals on Triangulations of Delaunay Sets.” Moscow Mathematical Journal, vol. 14, no. 3, Independent University of Moscow, 2014, pp. 491–504, doi:10.17323/1609-4514-2014-14-3-491-504. short: N. Dolbilin, H. Edelsbrunner, A. Glazyrin, O. Musin, Moscow Mathematical Journal 14 (2014) 491–504. date_created: 2018-12-11T11:54:29Z date_published: 2014-07-01T00:00:00Z date_updated: 2022-03-03T11:47:09Z day: '01' department: - _id: HeEd doi: 10.17323/1609-4514-2014-14-3-491-504 external_id: arxiv: - '1211.7053' intvolume: ' 14' issue: '3' language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1211.7053 month: '07' oa: 1 oa_version: Submitted Version page: 491 - 504 publication: Moscow Mathematical Journal publication_identifier: issn: - '16093321' publication_status: published publisher: Independent University of Moscow publist_id: '5220' quality_controlled: '1' scopus_import: '1' status: public title: Functionals on triangulations of delaunay sets type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 14 year: '2014' ... --- _id: '1929' abstract: - lang: eng text: We propose an algorithm for the generalization of cartographic objects that can be used to represent maps on different scales. acknowledgement: We would like to offer our special thanks to students of the Department of Mathematics of Demidov Yaroslavl State University A. A. Gorokhov and V. N. Knyazev for participation in developing the program and assistance in preparation of test data. This work was supported by grant 11.G34.31.0053 from the government of the Russian Federation. article_processing_charge: No article_type: original author: - first_name: V V full_name: Alexeev, V V last_name: Alexeev - first_name: V G full_name: Bogaevskaya, V G last_name: Bogaevskaya - first_name: M M full_name: Preobrazhenskaya, M M last_name: Preobrazhenskaya - first_name: A Y full_name: Ukhalov, A Y last_name: Ukhalov - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Olga full_name: Yakimova, Olga last_name: Yakimova citation: ama: Alexeev VV, Bogaevskaya VG, Preobrazhenskaya MM, Ukhalov AY, Edelsbrunner H, Yakimova O. An algorithm for cartographic generalization that preserves global topology. Journal of Mathematical Sciences. 2014;203(6):754-760. doi:10.1007/s10958-014-2165-8 apa: Alexeev, V. V., Bogaevskaya, V. G., Preobrazhenskaya, M. M., Ukhalov, A. Y., Edelsbrunner, H., & Yakimova, O. (2014). An algorithm for cartographic generalization that preserves global topology. Journal of Mathematical Sciences. Springer. https://doi.org/10.1007/s10958-014-2165-8 chicago: Alexeev, V V, V G Bogaevskaya, M M Preobrazhenskaya, A Y Ukhalov, Herbert Edelsbrunner, and Olga Yakimova. “An Algorithm for Cartographic Generalization That Preserves Global Topology.” Journal of Mathematical Sciences. Springer, 2014. https://doi.org/10.1007/s10958-014-2165-8. ieee: V. V. Alexeev, V. G. Bogaevskaya, M. M. Preobrazhenskaya, A. Y. Ukhalov, H. Edelsbrunner, and O. Yakimova, “An algorithm for cartographic generalization that preserves global topology,” Journal of Mathematical Sciences, vol. 203, no. 6. Springer, pp. 754–760, 2014. ista: Alexeev VV, Bogaevskaya VG, Preobrazhenskaya MM, Ukhalov AY, Edelsbrunner H, Yakimova O. 2014. An algorithm for cartographic generalization that preserves global topology. Journal of Mathematical Sciences. 203(6), 754–760. mla: Alexeev, V. V., et al. “An Algorithm for Cartographic Generalization That Preserves Global Topology.” Journal of Mathematical Sciences, vol. 203, no. 6, Springer, 2014, pp. 754–60, doi:10.1007/s10958-014-2165-8. short: V.V. Alexeev, V.G. Bogaevskaya, M.M. Preobrazhenskaya, A.Y. Ukhalov, H. Edelsbrunner, O. Yakimova, Journal of Mathematical Sciences 203 (2014) 754–760. date_created: 2018-12-11T11:54:46Z date_published: 2014-11-16T00:00:00Z date_updated: 2022-05-24T10:39:06Z day: '16' department: - _id: HeEd doi: 10.1007/s10958-014-2165-8 intvolume: ' 203' issue: '6' language: - iso: eng month: '11' oa_version: None page: 754 - 760 publication: Journal of Mathematical Sciences publication_identifier: eissn: - 1573-8795 issn: - 1072-3374 publication_status: published publisher: Springer publist_id: '5165' quality_controlled: '1' scopus_import: '1' status: public title: An algorithm for cartographic generalization that preserves global topology type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 203 year: '2014' ...