--- _id: '6853' abstract: - lang: eng text: This monograph presents a short course in computational geometry and topology. In the first part the book covers Voronoi diagrams and Delaunay triangulations, then it presents the theory of alpha complexes which play a crucial role in biology. The central part of the book is the homology theory and their computation, including the theory of persistence which is indispensable for applications, e.g. shape reconstruction. The target audience comprises researchers and practitioners in mathematics, biology, neuroscience and computer science, but the book may also be beneficial to graduate students of these fields. alternative_title: - SpringerBriefs in Applied Sciences and Technology article_processing_charge: No author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 citation: ama: 'Edelsbrunner H. A Short Course in Computational Geometry and Topology. 1st ed. Cham: Springer Nature; 2014. doi:10.1007/978-3-319-05957-0' apa: 'Edelsbrunner, H. (2014). A Short Course in Computational Geometry and Topology (1st ed.). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-05957-0' chicago: 'Edelsbrunner, Herbert. A Short Course in Computational Geometry and Topology. 1st ed. SpringerBriefs in Applied Sciences and Technology. Cham: Springer Nature, 2014. https://doi.org/10.1007/978-3-319-05957-0.' ieee: 'H. Edelsbrunner, A Short Course in Computational Geometry and Topology, 1st ed. Cham: Springer Nature, 2014.' ista: 'Edelsbrunner H. 2014. A Short Course in Computational Geometry and Topology 1st ed., Cham: Springer Nature, IX, 110p.' mla: Edelsbrunner, Herbert. A Short Course in Computational Geometry and Topology. 1st ed., Springer Nature, 2014, doi:10.1007/978-3-319-05957-0. short: H. Edelsbrunner, A Short Course in Computational Geometry and Topology, 1st ed., Springer Nature, Cham, 2014. date_created: 2019-09-06T09:22:33Z date_published: 2014-01-01T00:00:00Z date_updated: 2022-03-04T07:47:54Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-319-05957-0 edition: '1' language: - iso: eng month: '01' oa_version: None page: IX, 110 place: Cham publication_identifier: eisbn: - 9-783-3190-5957-0 eissn: - 2191-5318 isbn: - 9-783-3190-5956-3 issn: - 2191-530X publication_status: published publisher: Springer Nature quality_controlled: '1' related_material: link: - description: available as eBook via catalog IST BookList relation: other url: https://koha.app.ist.ac.at/cgi-bin/koha/opac-detail.pl?biblionumber=356106 - description: available via catalog IST BookList relation: other url: https://koha.app.ist.ac.at/cgi-bin/koha/opac-detail.pl?biblionumber=373842 scopus_import: '1' series_title: SpringerBriefs in Applied Sciences and Technology status: public title: A Short Course in Computational Geometry and Topology type: book user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '10886' abstract: - lang: eng text: We propose a method for visualizing two-dimensional symmetric positive definite tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the heat kernel and was originally introduced as an isometry invariant shape signature. Each positive definite tensor field defines a Riemannian manifold by considering the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply the definition of the HKS. The resulting scalar quantity is used for the visualization of tensor fields. The HKS is closely related to the Gaussian curvature of the Riemannian manifold and the time parameter of the heat kernel allows a multiscale analysis in a natural way. In this way, the HKS represents field related scale space properties, enabling a level of detail analysis of tensor fields. This makes the HKS an interesting new scalar quantity for tensor fields, which differs significantly from usual tensor invariants like the trace or the determinant. A method for visualization and a numerical realization of the HKS for tensor fields is proposed in this chapter. To validate the approach we apply it to some illustrating simple examples as isolated critical points and to a medical diffusion tensor data set. acknowledgement: This research is partially supported by the TOPOSYS project FP7-ICT-318493-STREP. alternative_title: - Mathematics and Visualization article_processing_charge: No author: - first_name: Valentin full_name: Zobel, Valentin last_name: Zobel - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Ingrid full_name: Hotz, Ingrid last_name: Hotz citation: ama: 'Zobel V, Reininghaus J, Hotz I. Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. In: Topological Methods in Data Analysis and Visualization III . Springer; 2014:249-262. doi:10.1007/978-3-319-04099-8_16' apa: Zobel, V., Reininghaus, J., & Hotz, I. (2014). Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. In Topological Methods in Data Analysis and Visualization III (pp. 249–262). Springer. https://doi.org/10.1007/978-3-319-04099-8_16 chicago: Zobel, Valentin, Jan Reininghaus, and Ingrid Hotz. “Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature.” In Topological Methods in Data Analysis and Visualization III , 249–62. Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_16. ieee: V. Zobel, J. Reininghaus, and I. Hotz, “Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature,” in Topological Methods in Data Analysis and Visualization III , 2014, pp. 249–262. ista: Zobel V, Reininghaus J, Hotz I. 2014. Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. Topological Methods in Data Analysis and Visualization III . , Mathematics and Visualization, , 249–262. mla: Zobel, Valentin, et al. “Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature.” Topological Methods in Data Analysis and Visualization III , Springer, 2014, pp. 249–62, doi:10.1007/978-3-319-04099-8_16. short: V. Zobel, J. Reininghaus, I. Hotz, in:, Topological Methods in Data Analysis and Visualization III , Springer, 2014, pp. 249–262. date_created: 2022-03-18T13:05:39Z date_published: 2014-03-19T00:00:00Z date_updated: 2023-09-05T14:13:16Z day: '19' department: - _id: HeEd doi: 10.1007/978-3-319-04099-8_16 language: - iso: eng month: '03' oa_version: None page: 249-262 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' status: public title: Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature type: conference user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2014' ... --- _id: '10817' abstract: - lang: eng text: The Morse-Smale complex can be either explicitly or implicitly represented. Depending on the type of representation, the simplification of the Morse-Smale complex works differently. In the explicit representation, the Morse-Smale complex is directly simplified by explicitly reconnecting the critical points during the simplification. In the implicit representation, on the other hand, the Morse-Smale complex is given by a combinatorial gradient field. In this setting, the simplification changes the combinatorial flow, which yields an indirect simplification of the Morse-Smale complex. The topological complexity of the Morse-Smale complex is reduced in both representations. However, the simplifications generally yield different results. In this chapter, we emphasize properties of the two representations that cause these differences. We also provide a complexity analysis of the two schemes with respect to running time and memory consumption. acknowledgement: This research is supported and funded by the Digiteo unTopoVis project, the TOPOSYS project FP7-ICT-318493-STREP, and MPC-VCC. article_processing_charge: No author: - first_name: David full_name: Günther, David last_name: Günther - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Hans-Peter full_name: Seidel, Hans-Peter last_name: Seidel - first_name: Tino full_name: Weinkauf, Tino last_name: Weinkauf citation: ama: 'Günther D, Reininghaus J, Seidel H-P, Weinkauf T. Notes on the simplification of the Morse-Smale complex. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological Methods in Data Analysis and Visualization III. Mathematics and Visualization. Cham: Springer Nature; 2014:135-150. doi:10.1007/978-3-319-04099-8_9' apa: 'Günther, D., Reininghaus, J., Seidel, H.-P., & Weinkauf, T. (2014). Notes on the simplification of the Morse-Smale complex. In P.-T. Bremer, I. Hotz, V. Pascucci, & R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III. (pp. 135–150). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-04099-8_9' chicago: 'Günther, David, Jan Reininghaus, Hans-Peter Seidel, and Tino Weinkauf. “Notes on the Simplification of the Morse-Smale Complex.” In Topological Methods in Data Analysis and Visualization III., edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 135–50. Mathematics and Visualization. Cham: Springer Nature, 2014. https://doi.org/10.1007/978-3-319-04099-8_9.' ieee: 'D. Günther, J. Reininghaus, H.-P. Seidel, and T. Weinkauf, “Notes on the simplification of the Morse-Smale complex,” in Topological Methods in Data Analysis and Visualization III., P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer Nature, 2014, pp. 135–150.' ista: 'Günther D, Reininghaus J, Seidel H-P, Weinkauf T. 2014.Notes on the simplification of the Morse-Smale complex. In: Topological Methods in Data Analysis and Visualization III. , 135–150.' mla: Günther, David, et al. “Notes on the Simplification of the Morse-Smale Complex.” Topological Methods in Data Analysis and Visualization III., edited by Peer-Timo Bremer et al., Springer Nature, 2014, pp. 135–50, doi:10.1007/978-3-319-04099-8_9. short: D. Günther, J. Reininghaus, H.-P. Seidel, T. Weinkauf, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III., Springer Nature, Cham, 2014, pp. 135–150. date_created: 2022-03-04T08:33:57Z date_published: 2014-03-19T00:00:00Z date_updated: 2023-09-05T15:33:45Z day: '19' department: - _id: HeEd doi: 10.1007/978-3-319-04099-8_9 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 language: - iso: eng month: '03' oa_version: None page: 135-150 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 Nature quality_controlled: '1' scopus_import: '1' series_title: Mathematics and Visualization status: public title: Notes on the simplification of the Morse-Smale complex type: book_chapter user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 year: '2014' ... --- _id: '2255' abstract: - lang: eng text: Motivated by applications in biology, we present an algorithm for estimating the length of tube-like shapes in 3-dimensional Euclidean space. In a first step, we combine the tube formula of Weyl with integral geometric methods to obtain an integral representation of the length, which we approximate using a variant of the Koksma-Hlawka Theorem. In a second step, we use tools from computational topology to decrease the dependence on small perturbations of the shape. We present computational experiments that shed light on the stability and the convergence rate of our algorithm. author: - first_name: Herbert full_name: Edelsbrunner, Herbert id: 3FB178DA-F248-11E8-B48F-1D18A9856A87 last_name: Edelsbrunner orcid: 0000-0002-9823-6833 - first_name: Florian full_name: Pausinger, Florian id: 2A77D7A2-F248-11E8-B48F-1D18A9856A87 last_name: Pausinger orcid: 0000-0002-8379-3768 citation: ama: Edelsbrunner H, Pausinger F. Stable length estimates of tube-like shapes. Journal of Mathematical Imaging and Vision. 2014;50(1):164-177. doi:10.1007/s10851-013-0468-x apa: Edelsbrunner, H., & Pausinger, F. (2014). Stable length estimates of tube-like shapes. Journal of Mathematical Imaging and Vision. Springer. https://doi.org/10.1007/s10851-013-0468-x chicago: Edelsbrunner, Herbert, and Florian Pausinger. “Stable Length Estimates of Tube-like Shapes.” Journal of Mathematical Imaging and Vision. Springer, 2014. https://doi.org/10.1007/s10851-013-0468-x. ieee: H. Edelsbrunner and F. Pausinger, “Stable length estimates of tube-like shapes,” Journal of Mathematical Imaging and Vision, vol. 50, no. 1. Springer, pp. 164–177, 2014. ista: Edelsbrunner H, Pausinger F. 2014. Stable length estimates of tube-like shapes. Journal of Mathematical Imaging and Vision. 50(1), 164–177. mla: Edelsbrunner, Herbert, and Florian Pausinger. “Stable Length Estimates of Tube-like Shapes.” Journal of Mathematical Imaging and Vision, vol. 50, no. 1, Springer, 2014, pp. 164–77, doi:10.1007/s10851-013-0468-x. short: H. Edelsbrunner, F. Pausinger, Journal of Mathematical Imaging and Vision 50 (2014) 164–177. date_created: 2018-12-11T11:56:36Z date_published: 2014-09-01T00:00:00Z date_updated: 2023-09-07T11:41:25Z day: '01' ddc: - '000' department: - _id: HeEd doi: 10.1007/s10851-013-0468-x ec_funded: 1 file: - access_level: open_access checksum: 2f93f3e63a38a85cd4404d7953913b14 content_type: application/pdf creator: system date_created: 2018-12-12T10:16:18Z date_updated: 2020-07-14T12:45:35Z file_id: '5204' file_name: IST-2016-549-v1+1_2014-J-06-LengthEstimate.pdf file_size: 3941391 relation: main_file file_date_updated: 2020-07-14T12:45:35Z has_accepted_license: '1' intvolume: ' 50' issue: '1' language: - iso: eng month: '09' oa: 1 oa_version: Submitted Version page: 164 - 177 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Journal of Mathematical Imaging and Vision publication_identifier: issn: - '09249907' publication_status: published publisher: Springer publist_id: '4691' pubrep_id: '549' quality_controlled: '1' related_material: record: - id: '2843' relation: earlier_version status: public - id: '1399' relation: dissertation_contains status: public scopus_import: 1 status: public title: Stable length estimates of tube-like shapes type: journal_article user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87 volume: 50 year: '2014' ... --- _id: '10894' abstract: - lang: eng text: PHAT is a C++ library for the computation of persistent homology by matrix reduction. We aim for a simple generic design that decouples algorithms from data structures without sacrificing efficiency or user-friendliness. This makes PHAT a versatile platform for experimenting with algorithmic ideas and comparing them to state of the art implementations. article_processing_charge: No author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Michael full_name: Kerber, Michael last_name: Kerber - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Hubert full_name: Wagner, Hubert last_name: Wagner citation: ama: 'Bauer U, Kerber M, Reininghaus J, Wagner H. PHAT – Persistent Homology Algorithms Toolbox. In: ICMS 2014: International Congress on Mathematical Software. Vol 8592. LNCS. Berlin, Heidelberg: Springer Berlin Heidelberg; 2014:137-143. doi:10.1007/978-3-662-44199-2_24' apa: 'Bauer, U., Kerber, M., Reininghaus, J., & Wagner, H. (2014). PHAT – Persistent Homology Algorithms Toolbox. In ICMS 2014: International Congress on Mathematical Software (Vol. 8592, pp. 137–143). Berlin, Heidelberg: Springer Berlin Heidelberg. https://doi.org/10.1007/978-3-662-44199-2_24' chicago: 'Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “PHAT – Persistent Homology Algorithms Toolbox.” In ICMS 2014: International Congress on Mathematical Software, 8592:137–43. LNCS. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. https://doi.org/10.1007/978-3-662-44199-2_24.' ieee: 'U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “PHAT – Persistent Homology Algorithms Toolbox,” in ICMS 2014: International Congress on Mathematical Software, Seoul, South Korea, 2014, vol. 8592, pp. 137–143.' ista: 'Bauer U, Kerber M, Reininghaus J, Wagner H. 2014. PHAT – Persistent Homology Algorithms Toolbox. ICMS 2014: International Congress on Mathematical Software. ICMS: International Congress on Mathematical SoftwareLNCS vol. 8592, 137–143.' mla: 'Bauer, Ulrich, et al. “PHAT – Persistent Homology Algorithms Toolbox.” ICMS 2014: International Congress on Mathematical Software, vol. 8592, Springer Berlin Heidelberg, 2014, pp. 137–43, doi:10.1007/978-3-662-44199-2_24.' short: 'U. Bauer, M. Kerber, J. Reininghaus, H. Wagner, in:, ICMS 2014: International Congress on Mathematical Software, Springer Berlin Heidelberg, Berlin, Heidelberg, 2014, pp. 137–143.' conference: end_date: 2014-08-09 location: Seoul, South Korea name: 'ICMS: International Congress on Mathematical Software' start_date: 2014-08-05 date_created: 2022-03-21T07:12:16Z date_published: 2014-09-01T00:00:00Z date_updated: 2023-09-20T09:42:40Z day: '01' department: - _id: HeEd doi: 10.1007/978-3-662-44199-2_24 intvolume: ' 8592' language: - iso: eng month: '09' oa_version: None page: 137-143 place: Berlin, Heidelberg publication: 'ICMS 2014: International Congress on Mathematical Software' publication_identifier: eisbn: - '9783662441992' eissn: - 1611-3349 isbn: - '9783662441985' issn: - 0302-9743 publication_status: published publisher: Springer Berlin Heidelberg quality_controlled: '1' related_material: record: - id: '1433' relation: later_version status: public scopus_import: '1' series_title: LNCS status: public title: PHAT – Persistent Homology Algorithms Toolbox type: conference user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1 volume: 8592 year: '2014' ...