--- _id: '1930' abstract: - lang: eng text: (Figure Presented) Data acquisition, numerical inaccuracies, and sampling often introduce noise in measurements and simulations. Removing this noise is often necessary for efficient analysis and visualization of this data, yet many denoising techniques change the minima and maxima of a scalar field. For example, the extrema can appear or disappear, spatially move, and change their value. This can lead to wrong interpretations of the data, e.g., when the maximum temperature over an area is falsely reported being a few degrees cooler because the denoising method is unaware of these features. Recently, a topological denoising technique based on a global energy optimization was proposed, which allows the topology-controlled denoising of 2D scalar fields. While this method preserves the minima and maxima, it is constrained by the size of the data. We extend this work to large 2D data and medium-sized 3D data by introducing a novel domain decomposition approach. It allows processing small patches of the domain independently while still avoiding the introduction of new critical points. Furthermore, we propose an iterative refinement of the solution, which decreases the optimization energy compared to the previous approach and therefore gives smoother results that are closer to the input. We illustrate our technique on synthetic and real-world 2D and 3D data sets that highlight potential applications. acknowledgement: RTRA Digiteoproject; ERC grant; SNF award; Intel Doctoral Fellowship; MPC-VCC author: - first_name: David full_name: Günther, David last_name: Günther - first_name: Alec full_name: Jacobson, Alec last_name: Jacobson - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus - first_name: Hans full_name: Seidel, Hans last_name: Seidel - first_name: Olga full_name: Sorkine Hornung, Olga last_name: Sorkine Hornung - first_name: Tino full_name: Weinkauf, Tino last_name: Weinkauf citation: ama: Günther D, Jacobson A, Reininghaus J, Seidel H, Sorkine Hornung O, Weinkauf T. Fast and memory-efficient topological denoising of 2D and 3D scalar fields. IEEE Transactions on Visualization and Computer Graphics. 2014;20(12):2585-2594. doi:10.1109/TVCG.2014.2346432 apa: Günther, D., Jacobson, A., Reininghaus, J., Seidel, H., Sorkine Hornung, O., & Weinkauf, T. (2014). Fast and memory-efficient topological denoising of 2D and 3D scalar fields. IEEE Transactions on Visualization and Computer Graphics. IEEE. https://doi.org/10.1109/TVCG.2014.2346432 chicago: Günther, David, Alec Jacobson, Jan Reininghaus, Hans Seidel, Olga Sorkine Hornung, and Tino Weinkauf. “Fast and Memory-Efficient Topological Denoising of 2D and 3D Scalar Fields.” IEEE Transactions on Visualization and Computer Graphics. IEEE, 2014. https://doi.org/10.1109/TVCG.2014.2346432. ieee: D. Günther, A. Jacobson, J. Reininghaus, H. Seidel, O. Sorkine Hornung, and T. Weinkauf, “Fast and memory-efficient topological denoising of 2D and 3D scalar fields,” IEEE Transactions on Visualization and Computer Graphics, vol. 20, no. 12. IEEE, pp. 2585–2594, 2014. ista: Günther D, Jacobson A, Reininghaus J, Seidel H, Sorkine Hornung O, Weinkauf T. 2014. Fast and memory-efficient topological denoising of 2D and 3D scalar fields. IEEE Transactions on Visualization and Computer Graphics. 20(12), 2585–2594. mla: Günther, David, et al. “Fast and Memory-Efficient Topological Denoising of 2D and 3D Scalar Fields.” IEEE Transactions on Visualization and Computer Graphics, vol. 20, no. 12, IEEE, 2014, pp. 2585–94, doi:10.1109/TVCG.2014.2346432. short: D. Günther, A. Jacobson, J. Reininghaus, H. Seidel, O. Sorkine Hornung, T. Weinkauf, IEEE Transactions on Visualization and Computer Graphics 20 (2014) 2585–2594. date_created: 2018-12-11T11:54:46Z date_published: 2014-12-31T00:00:00Z date_updated: 2021-01-12T06:54:09Z day: '31' department: - _id: HeEd doi: 10.1109/TVCG.2014.2346432 intvolume: ' 20' issue: '12' language: - iso: eng month: '12' oa_version: None page: 2585 - 2594 publication: IEEE Transactions on Visualization and Computer Graphics publication_status: published publisher: IEEE publist_id: '5164' quality_controlled: '1' scopus_import: 1 status: public title: Fast and memory-efficient topological denoising of 2D and 3D scalar fields type: journal_article user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 volume: 20 year: '2014' ... --- _id: '2043' abstract: - lang: eng text: Persistent homology is a popular and powerful tool for capturing topological features of data. Advances in algorithms for computing persistent homology have reduced the computation time drastically – as long as the algorithm does not exhaust the available memory. Following up on a recently presented parallel method for persistence computation on shared memory systems [1], we demonstrate that a simple adaption of the standard reduction algorithm leads to a variant for distributed systems. Our algorithmic design ensures that the data is distributed over the nodes without redundancy; this permits the computation of much larger instances than on a single machine. Moreover, we observe that the parallelism at least compensates for the overhead caused by communication between nodes, and often even speeds up the computation compared to sequential and even parallel shared memory algorithms. In our experiments, we were able to compute the persistent homology of filtrations with more than a billion (109) elements within seconds on a cluster with 32 nodes using less than 6GB of memory per node. 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 orcid: 0000-0002-8030-9299 - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus citation: ama: 'Bauer U, Kerber M, Reininghaus J. Distributed computation of persistent homology. In: McGeoch C, Meyer U, eds. Proceedings of the Workshop on Algorithm Engineering and Experiments. Society of Industrial and Applied Mathematics; 2014:31-38. doi:10.1137/1.9781611973198.4' apa: 'Bauer, U., Kerber, M., & Reininghaus, J. (2014). Distributed computation of persistent homology. In C. McGeoch & U. Meyer (Eds.), Proceedings of the Workshop on Algorithm Engineering and Experiments (pp. 31–38). Portland, USA: Society of Industrial and Applied Mathematics. https://doi.org/10.1137/1.9781611973198.4' chicago: Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Distributed Computation of Persistent Homology.” In Proceedings of the Workshop on Algorithm Engineering and Experiments, edited by Catherine McGeoch and Ulrich Meyer, 31–38. Society of Industrial and Applied Mathematics, 2014. https://doi.org/10.1137/1.9781611973198.4. ieee: U. Bauer, M. Kerber, and J. Reininghaus, “Distributed computation of persistent homology,” in Proceedings of the Workshop on Algorithm Engineering and Experiments, Portland, USA, 2014, pp. 31–38. ista: 'Bauer U, Kerber M, Reininghaus J. 2014. Distributed computation of persistent homology. Proceedings of the Workshop on Algorithm Engineering and Experiments. ALENEX: Algorithm Engineering and Experiments, 31–38.' mla: Bauer, Ulrich, et al. “Distributed Computation of Persistent Homology.” Proceedings of the Workshop on Algorithm Engineering and Experiments, edited by Catherine McGeoch and Ulrich Meyer, Society of Industrial and Applied Mathematics, 2014, pp. 31–38, doi:10.1137/1.9781611973198.4. short: U. Bauer, M. Kerber, J. Reininghaus, in:, C. McGeoch, U. Meyer (Eds.), Proceedings of the Workshop on Algorithm Engineering and Experiments, Society of Industrial and Applied Mathematics, 2014, pp. 31–38. conference: end_date: 2014-01-05 location: Portland, USA name: 'ALENEX: Algorithm Engineering and Experiments' start_date: 2014-01-05 date_created: 2018-12-11T11:55:23Z date_published: 2014-01-01T00:00:00Z date_updated: 2021-01-12T06:54:56Z day: '01' department: - _id: HeEd doi: 10.1137/1.9781611973198.4 ec_funded: 1 editor: - first_name: Catherine full_name: ' McGeoch, Catherine' last_name: ' McGeoch' - first_name: Ulrich full_name: Meyer, Ulrich last_name: Meyer language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1310.0710 month: '01' oa: 1 oa_version: Submitted Version page: 31 - 38 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Workshop on Algorithm Engineering and Experiments publication_status: published publisher: Society of Industrial and Applied Mathematics publist_id: '5008' quality_controlled: '1' scopus_import: 1 status: public title: Distributed computation of persistent homology type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2044' abstract: - lang: eng text: We present a parallel algorithm for computing the persistent homology of a filtered chain complex. Our approach differs from the commonly used reduction algorithm by first computing persistence pairs within local chunks, then simplifying the unpaired columns, and finally applying standard reduction on the simplified matrix. The approach generalizes a technique by Günther et al., which uses discrete Morse Theory to compute persistence; we derive the same worst-case complexity bound in a more general context. The algorithm employs several practical optimization techniques, which are of independent interest. Our sequential implementation of the algorithm is competitive with state-of-the-art methods, and we further improve the performance through parallel computation. 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 orcid: 0000-0002-8030-9299 - first_name: Jan full_name: Reininghaus, Jan id: 4505473A-F248-11E8-B48F-1D18A9856A87 last_name: Reininghaus citation: ama: 'Bauer U, Kerber M, Reininghaus J. Clear and Compress: Computing Persistent Homology in Chunks. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological Methods in Data Analysis and Visualization III. Mathematics and Visualization. Springer; 2014:103-117. doi:10.1007/978-3-319-04099-8_7' apa: 'Bauer, U., Kerber, M., & Reininghaus, J. (2014). Clear and Compress: Computing Persistent Homology in Chunks. In P.-T. Bremer, I. Hotz, V. Pascucci, & R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III (pp. 103–117). Springer. https://doi.org/10.1007/978-3-319-04099-8_7' chicago: 'Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Clear and Compress: Computing Persistent Homology in Chunks.” In Topological Methods in Data Analysis and Visualization III, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 103–17. Mathematics and Visualization. Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_7.' ieee: 'U. Bauer, M. Kerber, and J. Reininghaus, “Clear and Compress: Computing Persistent Homology in Chunks,” in Topological Methods in Data Analysis and Visualization III, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Springer, 2014, pp. 103–117.' ista: 'Bauer U, Kerber M, Reininghaus J. 2014.Clear and Compress: Computing Persistent Homology in Chunks. In: Topological Methods in Data Analysis and Visualization III. , 103–117.' mla: 'Bauer, Ulrich, et al. “Clear and Compress: Computing Persistent Homology in Chunks.” Topological Methods in Data Analysis and Visualization III, edited by Peer-Timo Bremer et al., Springer, 2014, pp. 103–17, doi:10.1007/978-3-319-04099-8_7.' short: U. Bauer, M. Kerber, J. Reininghaus, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III, Springer, 2014, pp. 103–117. date_created: 2018-12-11T11:55:23Z date_published: 2014-03-19T00:00:00Z date_updated: 2021-01-12T06:54:56Z day: '19' department: - _id: HeEd doi: 10.1007/978-3-319-04099-8_7 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 main_file_link: - open_access: '1' url: http://arxiv.org/abs/1303.0477 month: '03' oa: 1 oa_version: Submitted Version page: 103 - 117 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_status: published publisher: Springer publist_id: '5007' quality_controlled: '1' scopus_import: 1 series_title: Mathematics and Visualization status: public title: 'Clear and Compress: Computing Persistent Homology in Chunks' type: book_chapter user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2153' abstract: - lang: eng text: 'We define a simple, explicit map sending a morphism f : M → N of pointwise finite dimensional persistence modules to a matching between the barcodes of M and N. Our main result is that, in a precise sense, the quality of this matching is tightly controlled by the lengths of the longest intervals in the barcodes of ker f and coker f . As an immediate corollary, we obtain a new proof of the algebraic stability theorem for persistence barcodes [5, 9], a fundamental result in the theory of persistent homology. In contrast to previous proofs, ours shows explicitly how a δ-interleaving morphism between two persistence modules induces a δ-matching between the barcodes of the two modules. Our main result also specializes to a structure theorem for submodules and quotients of persistence modules. Copyright is held by the owner/author(s).' 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: Lesnick, Michael last_name: Lesnick citation: ama: 'Bauer U, Lesnick M. Induced matchings of barcodes and the algebraic stability of persistence. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:355-364. doi:10.1145/2582112.2582168' apa: 'Bauer, U., & Lesnick, M. (2014). Induced matchings of barcodes and the algebraic stability of persistence. In Proceedings of the Annual Symposium on Computational Geometry (pp. 355–364). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582168' chicago: Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the Algebraic Stability of Persistence.” In Proceedings of the Annual Symposium on Computational Geometry, 355–64. ACM, 2014. https://doi.org/10.1145/2582112.2582168. ieee: U. Bauer and M. Lesnick, “Induced matchings of barcodes and the algebraic stability of persistence,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 355–364. ista: 'Bauer U, Lesnick M. 2014. Induced matchings of barcodes and the algebraic stability of persistence. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 355–364.' mla: Bauer, Ulrich, and Michael Lesnick. “Induced Matchings of Barcodes and the Algebraic Stability of Persistence.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–64, doi:10.1145/2582112.2582168. short: U. Bauer, M. Lesnick, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–364. conference: end_date: 2014-06-11 location: Kyoto, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2014-06-08 date_created: 2018-12-11T11:56:01Z date_published: 2014-06-01T00:00:00Z date_updated: 2021-01-12T06:55:38Z day: '01' department: - _id: HeEd doi: 10.1145/2582112.2582168 ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1311.3681 month: '06' oa: 1 oa_version: Submitted Version page: 355 - 364 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Annual Symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4853' quality_controlled: '1' scopus_import: 1 status: public title: Induced matchings of barcodes and the algebraic stability of persistence type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ... --- _id: '2156' abstract: - lang: eng text: We propose a metric for Reeb graphs, called the functional distortion distance. Under this distance, the Reeb graph is stable against small changes of input functions. At the same time, it remains discriminative at differentiating input functions. In particular, the main result is that the functional distortion distance between two Reeb graphs is bounded from below by the bottleneck distance between both the ordinary and extended persistence diagrams for appropriate dimensions. As an application of our results, we analyze a natural simplification scheme for Reeb graphs, and show that persistent features in Reeb graph remains persistent under simplification. Understanding the stability of important features of the Reeb graph under simplification is an interesting problem on its own right, and critical to the practical usage of Reeb graphs. Copyright is held by the owner/author(s). acknowledgement: National Science Foundation under grants CCF-1319406, CCF-1116258. author: - first_name: Ulrich full_name: Bauer, Ulrich id: 2ADD483A-F248-11E8-B48F-1D18A9856A87 last_name: Bauer orcid: 0000-0002-9683-0724 - first_name: Xiaoyin full_name: Ge, Xiaoyin last_name: Ge - first_name: Yusu full_name: Wang, Yusu last_name: Wang citation: ama: 'Bauer U, Ge X, Wang Y. Measuring distance between Reeb graphs. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:464-473. doi:10.1145/2582112.2582169' apa: 'Bauer, U., Ge, X., & Wang, Y. (2014). Measuring distance between Reeb graphs. In Proceedings of the Annual Symposium on Computational Geometry (pp. 464–473). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582169' chicago: Bauer, Ulrich, Xiaoyin Ge, and Yusu Wang. “Measuring Distance between Reeb Graphs.” In Proceedings of the Annual Symposium on Computational Geometry, 464–73. ACM, 2014. https://doi.org/10.1145/2582112.2582169. ieee: U. Bauer, X. Ge, and Y. Wang, “Measuring distance between Reeb graphs,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 464–473. ista: 'Bauer U, Ge X, Wang Y. 2014. Measuring distance between Reeb graphs. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 464–473.' mla: Bauer, Ulrich, et al. “Measuring Distance between Reeb Graphs.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–73, doi:10.1145/2582112.2582169. short: U. Bauer, X. Ge, Y. Wang, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–473. conference: end_date: 2014-06-11 location: Kyoto, Japan name: 'SoCG: Symposium on Computational Geometry' start_date: 2014-06-08 date_created: 2018-12-11T11:56:02Z date_published: 2014-06-01T00:00:00Z date_updated: 2021-01-12T06:55:39Z day: '01' department: - _id: HeEd doi: 10.1145/2582112.2582169 ec_funded: 1 language: - iso: eng main_file_link: - open_access: '1' url: http://arxiv.org/abs/1307.2839 month: '06' oa: 1 oa_version: Submitted Version page: 464 - 473 project: - _id: 255D761E-B435-11E9-9278-68D0E5697425 call_identifier: FP7 grant_number: '318493' name: Topological Complex Systems publication: Proceedings of the Annual Symposium on Computational Geometry publication_status: published publisher: ACM publist_id: '4850' quality_controlled: '1' scopus_import: 1 status: public title: Measuring distance between Reeb graphs type: conference user_id: 4435EBFC-F248-11E8-B48F-1D18A9856A87 year: '2014' ...