[{"intvolume":" 20","month":"12","scopus_import":1,"quality_controlled":"1","publisher":"IEEE","acknowledgement":"RTRA Digiteoproject; ERC grant; SNF award; Intel Doctoral Fellowship; MPC-VCC","oa_version":"None","abstract":[{"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.","lang":"eng"}],"date_created":"2018-12-11T11:54:46Z","issue":"12","doi":"10.1109/TVCG.2014.2346432","volume":20,"date_published":"2014-12-31T00:00:00Z","page":"2585 - 2594","language":[{"iso":"eng"}],"publication":"IEEE Transactions on Visualization and Computer Graphics","day":"31","publication_status":"published","year":"2014","status":"public","type":"journal_article","_id":"1930","title":"Fast and memory-efficient topological denoising of 2D and 3D scalar fields","department":[{"_id":"HeEd"}],"author":[{"first_name":"David","last_name":"Günther","full_name":"Günther, David"},{"first_name":"Alec","last_name":"Jacobson","full_name":"Jacobson, Alec"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan"},{"last_name":"Seidel","full_name":"Seidel, Hans","first_name":"Hans"},{"first_name":"Olga","last_name":"Sorkine Hornung","full_name":"Sorkine Hornung, Olga"},{"first_name":"Tino","last_name":"Weinkauf","full_name":"Weinkauf, Tino"}],"publist_id":"5164","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"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.","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.","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","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","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.","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.","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."},"date_updated":"2021-01-12T06:54:09Z"},{"publication_status":"published","language":[{"iso":"eng"}],"ec_funded":1,"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."}],"oa_version":"Submitted Version","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1310.0710"}],"scopus_import":1,"month":"01","date_updated":"2021-01-12T06:54:56Z","department":[{"_id":"HeEd"}],"_id":"2043","conference":{"name":"ALENEX: Algorithm Engineering and Experiments","location":"Portland, USA","end_date":"2014-01-05","start_date":"2014-01-05"},"type":"conference","status":"public","year":"2014","publication":"Proceedings of the Workshop on Algorithm Engineering and Experiments","day":"01","page":"31 - 38","date_created":"2018-12-11T11:55:23Z","date_published":"2014-01-01T00:00:00Z","doi":"10.1137/1.9781611973198.4","oa":1,"quality_controlled":"1","publisher":"Society of Industrial and Applied Mathematics","citation":{"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.","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.","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.","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","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","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."},"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","publist_id":"5008","author":[{"last_name":"Bauer","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich"},{"last_name":"Kerber","full_name":"Kerber, Michael","orcid":"0000-0002-8030-9299","first_name":"Michael"},{"first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87","full_name":"Reininghaus, Jan","last_name":"Reininghaus"}],"editor":[{"full_name":" McGeoch, Catherine","last_name":" McGeoch","first_name":"Catherine"},{"full_name":"Meyer, Ulrich","last_name":"Meyer","first_name":"Ulrich"}],"title":"Distributed computation of persistent homology","project":[{"name":"Topological Complex Systems","grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}]},{"_id":"2044","series_title":"Mathematics and Visualization","status":"public","type":"book_chapter","date_updated":"2021-01-12T06:54:56Z","department":[{"_id":"HeEd"}],"oa_version":"Submitted Version","abstract":[{"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.","lang":"eng"}],"month":"03","main_file_link":[{"url":"http://arxiv.org/abs/1303.0477","open_access":"1"}],"scopus_import":1,"language":[{"iso":"eng"}],"publication_status":"published","ec_funded":1,"project":[{"name":"Topological Complex Systems","grant_number":"318493","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","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","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.","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.","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.","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.","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."},"title":"Clear and Compress: Computing Persistent Homology in Chunks","editor":[{"first_name":"Peer-Timo","last_name":"Bremer","full_name":"Bremer, Peer-Timo"},{"first_name":"Ingrid","full_name":"Hotz, Ingrid","last_name":"Hotz"},{"first_name":"Valerio","last_name":"Pascucci","full_name":"Pascucci, Valerio"},{"full_name":"Peikert, Ronald","last_name":"Peikert","first_name":"Ronald"}],"publist_id":"5007","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer"},{"first_name":"Michael","orcid":"0000-0002-8030-9299","full_name":"Kerber, Michael","last_name":"Kerber"},{"first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87","full_name":"Reininghaus, Jan","last_name":"Reininghaus"}],"oa":1,"quality_controlled":"1","publisher":"Springer","publication":"Topological Methods in Data Analysis and Visualization III","day":"19","year":"2014","date_created":"2018-12-11T11:55:23Z","date_published":"2014-03-19T00:00:00Z","doi":"10.1007/978-3-319-04099-8_7","page":"103 - 117"},{"date_created":"2018-12-11T11:56:01Z","ec_funded":1,"doi":"10.1145/2582112.2582168","date_published":"2014-06-01T00:00:00Z","page":"355 - 364","publication":"Proceedings of the Annual Symposium on Computational Geometry","language":[{"iso":"eng"}],"day":"01","year":"2014","publication_status":"published","month":"06","main_file_link":[{"url":"http://arxiv.org/abs/1311.3681","open_access":"1"}],"oa":1,"scopus_import":1,"publisher":"ACM","quality_controlled":"1","oa_version":"Submitted Version","abstract":[{"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).","lang":"eng"}],"title":"Induced matchings of barcodes and the algebraic stability of persistence","department":[{"_id":"HeEd"}],"publist_id":"4853","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","last_name":"Bauer","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724"},{"full_name":"Lesnick, Michael","last_name":"Lesnick","first_name":"Michael"}],"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"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.","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","short":"U. Bauer, M. Lesnick, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 355–364.","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.","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.","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."},"date_updated":"2021-01-12T06:55:38Z","project":[{"grant_number":"318493","name":"Topological Complex Systems","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"status":"public","conference":{"start_date":"2014-06-08","end_date":"2014-06-11","location":"Kyoto, Japan","name":"SoCG: Symposium on Computational Geometry"},"type":"conference","_id":"2153"},{"department":[{"_id":"HeEd"}],"date_updated":"2021-01-12T06:55:39Z","type":"conference","conference":{"name":"SoCG: Symposium on Computational Geometry","location":"Kyoto, Japan","end_date":"2014-06-11","start_date":"2014-06-08"},"status":"public","_id":"2156","ec_funded":1,"publication_status":"published","language":[{"iso":"eng"}],"scopus_import":1,"main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1307.2839"}],"month":"06","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)."}],"oa_version":"Submitted Version","publist_id":"4850","author":[{"orcid":"0000-0002-9683-0724","full_name":"Bauer, Ulrich","last_name":"Bauer","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich"},{"first_name":"Xiaoyin","last_name":"Ge","full_name":"Ge, Xiaoyin"},{"first_name":"Yusu","last_name":"Wang","full_name":"Wang, Yusu"}],"title":"Measuring distance between Reeb graphs","citation":{"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.","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.","short":"U. Bauer, X. Ge, Y. Wang, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 464–473.","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","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","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.","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."},"user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","project":[{"grant_number":"318493","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","call_identifier":"FP7"}],"page":"464 - 473","date_published":"2014-06-01T00:00:00Z","doi":"10.1145/2582112.2582169","date_created":"2018-12-11T11:56:02Z","year":"2014","day":"01","publication":"Proceedings of the Annual Symposium on Computational Geometry","quality_controlled":"1","publisher":"ACM","oa":1,"acknowledgement":"National Science Foundation under grants CCF-1319406, CCF-1116258."},{"project":[{"grant_number":"318493","name":"Topological Complex Systems","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425"}],"title":"The morse theory of Čech and Delaunay filtrations","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","full_name":"Bauer, Ulrich","orcid":"0000-0002-9683-0724","last_name":"Bauer"},{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","last_name":"Edelsbrunner"}],"publist_id":"4851","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"mla":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–90, doi:10.1145/2582112.2582167.","apa":"Bauer, U., & Edelsbrunner, H. (2014). The morse theory of Čech and Delaunay filtrations. In Proceedings of the Annual Symposium on Computational Geometry (pp. 484–490). Kyoto, Japan: ACM. https://doi.org/10.1145/2582112.2582167","ama":"Bauer U, Edelsbrunner H. The morse theory of Čech and Delaunay filtrations. In: Proceedings of the Annual Symposium on Computational Geometry. ACM; 2014:484-490. doi:10.1145/2582112.2582167","short":"U. Bauer, H. Edelsbrunner, in:, Proceedings of the Annual Symposium on Computational Geometry, ACM, 2014, pp. 484–490.","ieee":"U. Bauer and H. Edelsbrunner, “The morse theory of Čech and Delaunay filtrations,” in Proceedings of the Annual Symposium on Computational Geometry, Kyoto, Japan, 2014, pp. 484–490.","chicago":"Bauer, Ulrich, and Herbert Edelsbrunner. “The Morse Theory of Čech and Delaunay Filtrations.” In Proceedings of the Annual Symposium on Computational Geometry, 484–90. ACM, 2014. https://doi.org/10.1145/2582112.2582167.","ista":"Bauer U, Edelsbrunner H. 2014. The morse theory of Čech and Delaunay filtrations. Proceedings of the Annual Symposium on Computational Geometry. SoCG: Symposium on Computational Geometry, 484–490."},"publisher":"ACM","quality_controlled":"1","oa":1,"acknowledgement":"This research is partially supported by ESF under the ACAT Research Network Programme, and by the Russian Government under mega project 11.G34.31.0053","doi":"10.1145/2582112.2582167","date_published":"2014-06-01T00:00:00Z","date_created":"2018-12-11T11:56:01Z","page":"484 - 490","day":"01","publication":"Proceedings of the Annual Symposium on Computational Geometry","year":"2014","status":"public","type":"conference","conference":{"start_date":"2014-06-08","location":"Kyoto, Japan","end_date":"2014-06-11","name":"SoCG: Symposium on Computational Geometry"},"_id":"2155","department":[{"_id":"HeEd"}],"date_updated":"2021-01-12T06:55:38Z","month":"06","scopus_import":1,"main_file_link":[{"url":"http://arxiv.org/abs/1312.1231","open_access":"1"}],"oa_version":"Submitted Version","abstract":[{"text":"Given a finite set of points in Rn and a positive radius, we study the Čech, Delaunay-Čech, alpha, and wrap complexes as instances of a generalized discrete Morse theory. We prove that the latter three complexes are simple-homotopy equivalent. Our results have applications in topological data analysis and in the reconstruction of shapes from sampled data. Copyright is held by the owner/author(s).","lang":"eng"}],"ec_funded":1,"language":[{"iso":"eng"}],"publication_status":"published"},{"publist_id":"4805","author":[{"id":"3FB178DA-F248-11E8-B48F-1D18A9856A87","first_name":"Herbert","full_name":"Edelsbrunner, Herbert","orcid":"0000-0002-9823-6833","last_name":"Edelsbrunner"},{"first_name":"Salman","id":"4BDBD4F2-F248-11E8-B48F-1D18A9856A87","last_name":"Parsa","full_name":"Parsa, Salman"}],"title":"On the computational complexity of betti numbers reductions from matrix rank","department":[{"_id":"HeEd"}],"citation":{"mla":"Edelsbrunner, Herbert, and Salman Parsa. “On the Computational Complexity of Betti Numbers Reductions from Matrix Rank.” Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2014, pp. 152–60, doi:10.1137/1.9781611973402.11.","short":"H. Edelsbrunner, S. Parsa, in:, Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2014, pp. 152–160.","ieee":"H. Edelsbrunner and S. Parsa, “On the computational complexity of betti numbers reductions from matrix rank,” in Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, Portland, USA, 2014, pp. 152–160.","ama":"Edelsbrunner H, Parsa S. On the computational complexity of betti numbers reductions from matrix rank. In: Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. SIAM; 2014:152-160. doi:10.1137/1.9781611973402.11","apa":"Edelsbrunner, H., & Parsa, S. (2014). On the computational complexity of betti numbers reductions from matrix rank. In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms (pp. 152–160). Portland, USA: SIAM. https://doi.org/10.1137/1.9781611973402.11","chicago":"Edelsbrunner, Herbert, and Salman Parsa. “On the Computational Complexity of Betti Numbers Reductions from Matrix Rank.” In Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms, 152–60. SIAM, 2014. https://doi.org/10.1137/1.9781611973402.11.","ista":"Edelsbrunner H, Parsa S. 2014. On the computational complexity of betti numbers reductions from matrix rank. Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms. SODA: Symposium on Discrete Algorithms, 152–160."},"date_updated":"2021-01-12T06:55:48Z","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","type":"conference","conference":{"end_date":"2014-01-07","location":"Portland, USA","start_date":"2014-01-05","name":"SODA: Symposium on Discrete Algorithms"},"status":"public","_id":"2177","page":"152 - 160","doi":"10.1137/1.9781611973402.11","date_published":"2014-01-01T00:00:00Z","date_created":"2018-12-11T11:56:09Z","publication_status":"published","year":"2014","day":"01","publication":"Proceedings of the Annual ACM-SIAM Symposium on Discrete Algorithms","language":[{"iso":"eng"}],"publisher":"SIAM","scopus_import":1,"quality_controlled":"1","month":"01","abstract":[{"lang":"eng","text":"We give evidence for the difficulty of computing Betti numbers of simplicial complexes over a finite field. We do this by reducing the rank computation for sparse matrices with to non-zero entries to computing Betti numbers of simplicial complexes consisting of at most a constant times to simplices. Together with the known reduction in the other direction, this implies that the two problems have the same computational complexity."}],"oa_version":"None"},{"volume":61,"issue":"3","language":[{"iso":"eng"}],"publication_status":"published","intvolume":" 61","month":"05","main_file_link":[{"url":"http://arxiv.org/abs/1105.6257","open_access":"1"}],"scopus_import":1,"oa_version":"Preprint","abstract":[{"text":"Given topological spaces X,Y, a fundamental problem of algebraic topology is understanding the structure of all continuous maps X→ Y. We consider a computational version, where X,Y are given as finite simplicial complexes, and the goal is to compute [X,Y], that is, all homotopy classes of suchmaps.We solve this problem in the stable range, where for some d ≥ 2, we have dim X ≤ 2d-2 and Y is (d-1)-connected; in particular, Y can be the d-dimensional sphere Sd. The algorithm combines classical tools and ideas from homotopy theory (obstruction theory, Postnikov systems, and simplicial sets) with algorithmic tools from effective algebraic topology (locally effective simplicial sets and objects with effective homology). In contrast, [X,Y] is known to be uncomputable for general X,Y, since for X = S1 it includes a well known undecidable problem: testing triviality of the fundamental group of Y. In follow-up papers, the algorithm is shown to run in polynomial time for d fixed, and extended to other problems, such as the extension problem, where we are given a subspace A ⊂ X and a map A→ Y and ask whether it extends to a map X → Y, or computing the Z2-index-everything in the stable range. Outside the stable range, the extension problem is undecidable.","lang":"eng"}],"department":[{"_id":"UlWa"},{"_id":"HeEd"}],"date_updated":"2021-01-12T06:55:50Z","status":"public","type":"journal_article","_id":"2184","date_created":"2018-12-11T11:56:12Z","date_published":"2014-05-01T00:00:00Z","doi":"10.1145/2597629","publication":"Journal of the ACM","day":"01","year":"2014","oa":1,"publisher":"ACM","quality_controlled":"1","acknowledgement":"The research by M. K. was supported by project GAUK 49209. The research by M. K. was also supported by project 1M0545 by the Ministry of Education of the Czech Republic and by Center of Excellence { Inst. for Theor. Comput. Sci., Prague (project P202/12/G061 of GACR). The research by U. W. was supported by the Swiss National Science Foundation (SNF Projects 200021-125309, 200020-138230, and PP00P2-138948).","title":"Computing all maps into a sphere","author":[{"first_name":"Martin","last_name":"Čadek","full_name":"Čadek, Martin"},{"last_name":"Krcál","full_name":"Krcál, Marek","first_name":"Marek","id":"33E21118-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Matoušek, Jiří","last_name":"Matoušek","first_name":"Jiří"},{"last_name":"Sergeraert","full_name":"Sergeraert, Francis","first_name":"Francis"},{"first_name":"Lukáš","full_name":"Vokřínek, Lukáš","last_name":"Vokřínek"},{"last_name":"Wagner","orcid":"0000-0002-1494-0568","full_name":"Wagner, Uli","first_name":"Uli","id":"36690CA2-F248-11E8-B48F-1D18A9856A87"}],"publist_id":"4797","user_id":"4435EBFC-F248-11E8-B48F-1D18A9856A87","citation":{"ieee":"M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, and U. Wagner, “Computing all maps into a sphere,” Journal of the ACM, vol. 61, no. 3. ACM, 2014.","short":"M. Čadek, M. Krcál, J. Matoušek, F. Sergeraert, L. Vokřínek, U. Wagner, Journal of the ACM 61 (2014).","ama":"Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. Computing all maps into a sphere. Journal of the ACM. 2014;61(3). doi:10.1145/2597629","apa":"Čadek, M., Krcál, M., Matoušek, J., Sergeraert, F., Vokřínek, L., & Wagner, U. (2014). Computing all maps into a sphere. Journal of the ACM. ACM. https://doi.org/10.1145/2597629","mla":"Čadek, Martin, et al. “Computing All Maps into a Sphere.” Journal of the ACM, vol. 61, no. 3, 17, ACM, 2014, doi:10.1145/2597629.","ista":"Čadek M, Krcál M, Matoušek J, Sergeraert F, Vokřínek L, Wagner U. 2014. Computing all maps into a sphere. Journal of the ACM. 61(3), 17.","chicago":"Čadek, Martin, Marek Krcál, Jiří Matoušek, Francis Sergeraert, Lukáš Vokřínek, and Uli Wagner. “Computing All Maps into a Sphere.” Journal of the ACM. ACM, 2014. https://doi.org/10.1145/2597629."},"article_number":"17 "},{"publication_status":"published","language":[{"iso":"eng"}],"file":[{"access_level":"open_access","relation":"main_file","content_type":"application/pdf","file_id":"5232","checksum":"1d4a046f1af945c407c5c4d411d4c5e4","creator":"system","date_updated":"2020-07-14T12:45:52Z","file_size":435320,"date_created":"2018-12-12T10:16:43Z","file_name":"IST-2016-544-v1+1_2012-P-11-PHTheoryPractice.pdf"}],"month":"01","abstract":[{"lang":"eng","text":"Persistent homology is a recent grandchild of homology that has found use in\r\nscience and engineering as well as in mathematics. This paper surveys the method as well\r\nas the applications, neglecting completeness in favor of highlighting ideas and directions."}],"oa_version":"Submitted Version","file_date_updated":"2020-07-14T12:45:52Z","department":[{"_id":"HeEd"}],"date_updated":"2021-01-12T07:00:36Z","ddc":["000"],"conference":{"name":"ECM: European Congress of Mathematics","start_date":"2012-07-02","end_date":"2012-07-07","location":"Kraków, Poland"},"type":"conference","pubrep_id":"544","status":"public","_id":"2905","page":"31 - 50","date_created":"2018-12-11T12:00:16Z","date_published":"2014-01-01T00:00:00Z","doi":"10.4171/120-1/3","year":"2014","has_accepted_license":"1","day":"01","oa":1,"publisher":"European Mathematical Society Publishing House","quality_controlled":"1","acknowledgement":"This research is partially supported by NSF under grant DBI-0820624, by ESF under the Research Networking Programme, and by the Russian Government Project 11.G34.31.0053.","article_processing_charge":"No","publist_id":"3842","author":[{"last_name":"Edelsbrunner","orcid":"0000-0002-9823-6833","full_name":"Edelsbrunner, Herbert","first_name":"Herbert","id":"3FB178DA-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Morozovy, Dmitriy","last_name":"Morozovy","first_name":"Dmitriy"}],"title":"Persistent homology: Theory and practice","citation":{"mla":"Edelsbrunner, Herbert, and Dmitriy Morozovy. Persistent Homology: Theory and Practice. European Mathematical Society Publishing House, 2014, pp. 31–50, doi:10.4171/120-1/3.","ama":"Edelsbrunner H, Morozovy D. Persistent homology: Theory and practice. In: European Mathematical Society Publishing House; 2014:31-50. doi:10.4171/120-1/3","apa":"Edelsbrunner, H., & Morozovy, D. (2014). Persistent homology: Theory and practice (pp. 31–50). Presented at the ECM: European Congress of Mathematics, Kraków, Poland: European Mathematical Society Publishing House. https://doi.org/10.4171/120-1/3","short":"H. Edelsbrunner, D. Morozovy, in:, European Mathematical Society Publishing House, 2014, pp. 31–50.","ieee":"H. Edelsbrunner and D. Morozovy, “Persistent homology: Theory and practice,” presented at the ECM: European Congress of Mathematics, Kraków, Poland, 2014, pp. 31–50.","chicago":"Edelsbrunner, Herbert, and Dmitriy Morozovy. “Persistent Homology: Theory and Practice,” 31–50. European Mathematical Society Publishing House, 2014. https://doi.org/10.4171/120-1/3.","ista":"Edelsbrunner H, Morozovy D. 2014. Persistent homology: Theory and practice. ECM: European Congress of Mathematics, 31–50."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87"},{"citation":{"apa":"Biedl, T., Huber, S., & Palfrader, P. (2014). Planar matchings for weighted straight skeletons. In 25th International Symposium, ISAAC 2014 (Vol. 8889, pp. 117–127). Jeonju, Korea: Springer Nature. https://doi.org/10.1007/978-3-319-13075-0_10","ama":"Biedl T, Huber S, Palfrader P. Planar matchings for weighted straight skeletons. In: 25th International Symposium, ISAAC 2014. Vol 8889. Springer Nature; 2014:117-127. doi:10.1007/978-3-319-13075-0_10","short":"T. Biedl, S. Huber, P. Palfrader, in:, 25th International Symposium, ISAAC 2014, Springer Nature, 2014, pp. 117–127.","ieee":"T. Biedl, S. Huber, and P. Palfrader, “Planar matchings for weighted straight skeletons,” in 25th International Symposium, ISAAC 2014, Jeonju, Korea, 2014, vol. 8889, pp. 117–127.","mla":"Biedl, Therese, et al. “Planar Matchings for Weighted Straight Skeletons.” 25th International Symposium, ISAAC 2014, vol. 8889, Springer Nature, 2014, pp. 117–27, doi:10.1007/978-3-319-13075-0_10.","ista":"Biedl T, Huber S, Palfrader P. 2014. Planar matchings for weighted straight skeletons. 25th International Symposium, ISAAC 2014. ISAAC: International Symposium on Algorithms and Computation, LNCS, vol. 8889, 117–127.","chicago":"Biedl, Therese, Stefan Huber, and Peter Palfrader. “Planar Matchings for Weighted Straight Skeletons.” In 25th International Symposium, ISAAC 2014, 8889:117–27. Springer Nature, 2014. https://doi.org/10.1007/978-3-319-13075-0_10."},"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Therese","last_name":"Biedl","full_name":"Biedl, Therese"},{"orcid":"0000-0002-8871-5814","full_name":"Huber, Stefan","last_name":"Huber","first_name":"Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Palfrader, Peter","last_name":"Palfrader","first_name":"Peter"}],"article_processing_charge":"No","title":"Planar matchings for weighted straight skeletons","year":"2014","day":"08","publication":"25th International Symposium, ISAAC 2014","page":"117-127","date_published":"2014-11-08T00:00:00Z","doi":"10.1007/978-3-319-13075-0_10","date_created":"2022-03-21T07:09:03Z","acknowledgement":"T. Biedl was supported by NSERC and the Ross and Muriel Cheriton Fellowship. P. Palfrader was supported by Austrian Science Fund (FWF): P25816-N15.","quality_controlled":"1","publisher":"Springer Nature","date_updated":"2023-02-23T12:20:55Z","department":[{"_id":"HeEd"}],"_id":"10892","type":"conference","conference":{"start_date":"2014-12-15","location":"Jeonju, Korea","end_date":"2014-12-17","name":"ISAAC: International Symposium on Algorithms and Computation"},"status":"public","publication_identifier":{"eisbn":["9783319130750"],"eissn":["1611-3349"],"isbn":["9783319130743"],"issn":["0302-9743"]},"publication_status":"published","language":[{"iso":"eng"}],"volume":8889,"related_material":{"record":[{"relation":"later_version","status":"public","id":"481"}]},"abstract":[{"lang":"eng","text":"In this paper, we introduce planar matchings on directed pseudo-line arrangements, which yield a planar set of pseudo-line segments such that only matching-partners are adjacent. By translating the planar matching problem into a corresponding stable roommates problem we show that such matchings always exist.\r\nUsing our new framework, we establish, for the first time, a complete, rigorous definition of weighted straight skeletons, which are based on a so-called wavefront propagation process. We present a generalized and unified approach to treat structural changes in the wavefront that focuses on the restoration of weak planarity by finding planar matchings."}],"oa_version":"None","scopus_import":"1","alternative_title":["LNCS"],"month":"11","intvolume":" 8889"}]