[{"oa":1,"acknowledgement":"The authors acknowledge funding of the German Re-\r\nsearch  Foundation  (DFG)  via  the  Collaborative  Re-\r\nsearch  Center  (SFB  557)  \\Control  of  Complex  Turbu-\r\nlent  Shear  Flows\"  and  the  Emmy  Noether  Program.\r\nFurther  funding  was  provided  by  the  Zuse  Institute\r\nBerlin  (ZIB),  the  DFG-CNRS  research  group  \\Noise\r\nGeneration in Turbulent Flows\" (2003{2010), the Chaire\r\nd'Excellence 'Closed-loop control of turbulent shear  ows\r\nusing reduced-order models' (TUCOROM) of the French\r\nAgence Nationale de la Recherche (ANR), and the Eu-\r\nropean  Social  Fund  (ESF  App.   No.   100098251).   We\r\nthank  the  Ambrosys  Ltd.  Society  for  Complex  Sys-\r\ntems  Management  and  the  Bernd  R.  Noack  Cybernet-\r\nics  Foundation  for  additional  support.   A  part  of  this\r\nwork was performed using HPC resources from GENCI-[CCRT/CINES/IDRIS]  supported  by  the  Grant  2011-\r\n[x2011020912","status":"public","publisher":"Polish Academy of Sciences Publishing House","type":"journal_article","day":"01","main_file_link":[{"open_access":"1","url":"http://am.ippt.pan.pl/am/article/viewFile/v68p55/pdf"}],"issue":"1","year":"2016","oa_version":"Published Version","date_updated":"2021-01-12T06:49:09Z","date_published":"2016-01-01T00:00:00Z","page":"55 - 80","intvolume":"        68","scopus_import":1,"user_id":"3E5EF7F0-F248-11E8-B48F-1D18A9856A87","author":[{"first_name":"Jens","last_name":"Kasten","full_name":"Kasten, Jens"},{"last_name":"Reininghaus","full_name":"Reininghaus, Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan"},{"full_name":"Hotz, Ingrid","last_name":"Hotz","first_name":"Ingrid"},{"full_name":"Hege, Hans","last_name":"Hege","first_name":"Hans"},{"first_name":"Bernd","last_name":"Noack","full_name":"Noack, Bernd"},{"first_name":"Guillaume","last_name":"Daviller","full_name":"Daviller, Guillaume"},{"last_name":"Morzyński","full_name":"Morzyński, Marek","first_name":"Marek"}],"publication":"Archives of Mechanics","title":"Acceleration feature points of unsteady shear flows","quality_controlled":"1","month":"01","_id":"1216","volume":68,"citation":{"chicago":"Kasten, Jens, Jan Reininghaus, Ingrid Hotz, Hans Hege, Bernd Noack, Guillaume Daviller, and Marek Morzyński. “Acceleration Feature Points of Unsteady Shear Flows.” <i>Archives of Mechanics</i>. Polish Academy of Sciences Publishing House, 2016.","ista":"Kasten J, Reininghaus J, Hotz I, Hege H, Noack B, Daviller G, Morzyński M. 2016. Acceleration feature points of unsteady shear flows. Archives of Mechanics. 68(1), 55–80.","ama":"Kasten J, Reininghaus J, Hotz I, et al. Acceleration feature points of unsteady shear flows. <i>Archives of Mechanics</i>. 2016;68(1):55-80.","apa":"Kasten, J., Reininghaus, J., Hotz, I., Hege, H., Noack, B., Daviller, G., &#38; Morzyński, M. (2016). Acceleration feature points of unsteady shear flows. <i>Archives of Mechanics</i>. Polish Academy of Sciences Publishing House.","ieee":"J. Kasten <i>et al.</i>, “Acceleration feature points of unsteady shear flows,” <i>Archives of Mechanics</i>, vol. 68, no. 1. Polish Academy of Sciences Publishing House, pp. 55–80, 2016.","mla":"Kasten, Jens, et al. “Acceleration Feature Points of Unsteady Shear Flows.” <i>Archives of Mechanics</i>, vol. 68, no. 1, Polish Academy of Sciences Publishing House, 2016, pp. 55–80.","short":"J. Kasten, J. Reininghaus, I. Hotz, H. Hege, B. Noack, G. Daviller, M. Morzyński, Archives of Mechanics 68 (2016) 55–80."},"abstract":[{"text":"A framework fo r extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance is proposed in this paper. The minima of the acceleration magnitude (a superset of acceleration zeros) are extracted and discriminated into vortices and saddle points, based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These feature points are tracked in time with a robust algorithm for tracking features. Thus, a space-time hierarchy of the minima is built and vortex merging events are detected. We apply the acceleration feature extraction strategy to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of the pressure field and minima of λ2.","lang":"eng"}],"publication_status":"published","publist_id":"6118","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:50:46Z"},{"type":"conference","main_file_link":[{"open_access":"1","url":"http://arxiv.org/abs/1412.6821"}],"day":"14","publisher":"IEEE","status":"public","publication_identifier":{"eisbn":["978-1-4673-6964-0 "]},"oa":1,"author":[{"first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87","last_name":"Reininghaus","full_name":"Reininghaus, Jan"},{"last_name":"Huber","full_name":"Huber, Stefan","first_name":"Stefan","id":"4700A070-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-8871-5814"},{"orcid":"0000-0002-9683-0724","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","full_name":"Bauer, Ulrich","last_name":"Bauer"},{"last_name":"Kwitt","full_name":"Kwitt, Roland","first_name":"Roland"}],"scopus_import":"1","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"4741 - 4748","date_published":"2015-10-14T00:00:00Z","oa_version":"Preprint","year":"2015","date_updated":"2025-06-11T06:37:43Z","abstract":[{"text":"Topological data analysis offers a rich source of valuable information to study vision problems. Yet, so far we lack a theoretically sound connection to popular kernel-based learning techniques, such as kernel SVMs or kernel PCA. In this work, we establish such a connection by designing a multi-scale kernel for persistence diagrams, a stable summary representation of topological features in data. We show that this kernel is positive definite and prove its stability with respect to the 1-Wasserstein distance. Experiments on two benchmark datasets for 3D shape classification/retrieval and texture recognition show considerable performance gains of the proposed method compared to an alternative approach that is based on the recently introduced persistence landscapes.","lang":"eng"}],"article_processing_charge":"No","conference":{"start_date":"2015-06-07","name":"CVPR: Computer Vision and Pattern Recognition","end_date":"2015-06-12","location":"Boston, MA, USA"},"citation":{"ieee":"J. Reininghaus, S. Huber, U. Bauer, and R. Kwitt, “A stable multi-scale kernel for topological machine learning,” presented at the CVPR: Computer Vision and Pattern Recognition, Boston, MA, USA, 2015, pp. 4741–4748.","mla":"Reininghaus, Jan, et al. <i>A Stable Multi-Scale Kernel for Topological Machine Learning</i>. IEEE, 2015, pp. 4741–48, doi:<a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">10.1109/CVPR.2015.7299106</a>.","short":"J. Reininghaus, S. Huber, U. Bauer, R. Kwitt, in:, IEEE, 2015, pp. 4741–4748.","chicago":"Reininghaus, Jan, Stefan Huber, Ulrich Bauer, and Roland Kwitt. “A Stable Multi-Scale Kernel for Topological Machine Learning,” 4741–48. IEEE, 2015. <a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">https://doi.org/10.1109/CVPR.2015.7299106</a>.","ista":"Reininghaus J, Huber S, Bauer U, Kwitt R. 2015. A stable multi-scale kernel for topological machine learning. CVPR: Computer Vision and Pattern Recognition, 4741–4748.","ama":"Reininghaus J, Huber S, Bauer U, Kwitt R. A stable multi-scale kernel for topological machine learning. In: IEEE; 2015:4741-4748. doi:<a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">10.1109/CVPR.2015.7299106</a>","apa":"Reininghaus, J., Huber, S., Bauer, U., &#38; Kwitt, R. (2015). A stable multi-scale kernel for topological machine learning (pp. 4741–4748). Presented at the CVPR: Computer Vision and Pattern Recognition, Boston, MA, USA: IEEE. <a href=\"https://doi.org/10.1109/CVPR.2015.7299106\">https://doi.org/10.1109/CVPR.2015.7299106</a>"},"_id":"1483","title":"A stable multi-scale kernel for topological machine learning","month":"10","external_id":{"arxiv":["1412.6821"]},"department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:52:17Z","doi":"10.1109/CVPR.2015.7299106","language":[{"iso":"eng"}],"publist_id":"5709","arxiv":1,"publication_status":"published"},{"doi":"10.1007/978-3-319-15090-1_13","publist_id":"5640","alternative_title":["Mathematics and Visualization"],"_id":"1531","title":"Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature","month":"01","article_processing_charge":"No","citation":{"short":"V. Zobel, J. Reininghaus, I. Hotz, in:, I. Hotz, T. Schultz (Eds.), Visualization and Processing of Higher Order Descriptors for Multi-Valued Data, 1st ed., Springer, 2015, pp. 257–267.","mla":"Zobel, Valentin, et al. “Visualizing Symmetric Indefinite 2D Tensor Fields Using The Heat Kernel Signature.” <i>Visualization and Processing of Higher Order Descriptors for Multi-Valued Data</i>, edited by Ingrid Hotz and Thomas Schultz, 1st ed., vol. 40, Springer, 2015, pp. 257–67, doi:<a href=\"https://doi.org/10.1007/978-3-319-15090-1_13\">10.1007/978-3-319-15090-1_13</a>.","ieee":"V. Zobel, J. Reininghaus, and I. Hotz, “Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature,” in <i>Visualization and Processing of Higher Order Descriptors for Multi-Valued Data</i>, 1st ed., vol. 40, I. Hotz and T. Schultz, Eds. Springer, 2015, pp. 257–267.","ama":"Zobel V, Reininghaus J, Hotz I. Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature. In: Hotz I, Schultz T, eds. <i>Visualization and Processing of Higher Order Descriptors for Multi-Valued Data</i>. Vol 40. 1st ed. Springer; 2015:257-267. doi:<a href=\"https://doi.org/10.1007/978-3-319-15090-1_13\">10.1007/978-3-319-15090-1_13</a>","apa":"Zobel, V., Reininghaus, J., &#38; Hotz, I. (2015). Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature. In I. Hotz &#38; T. Schultz (Eds.), <i>Visualization and Processing of Higher Order Descriptors for Multi-Valued Data</i> (1st ed., Vol. 40, pp. 257–267). Springer. <a href=\"https://doi.org/10.1007/978-3-319-15090-1_13\">https://doi.org/10.1007/978-3-319-15090-1_13</a>","chicago":"Zobel, Valentin, Jan Reininghaus, and Ingrid Hotz. “Visualizing Symmetric Indefinite 2D Tensor Fields Using The Heat Kernel Signature.” In <i>Visualization and Processing of Higher Order Descriptors for Multi-Valued Data</i>, edited by Ingrid Hotz and Thomas Schultz, 1st ed., 40:257–67. Springer, 2015. <a href=\"https://doi.org/10.1007/978-3-319-15090-1_13\">https://doi.org/10.1007/978-3-319-15090-1_13</a>.","ista":"Zobel V, Reininghaus J, Hotz I. 2015.Visualizing symmetric indefinite 2D tensor fields using The Heat Kernel Signature. In: Visualization and Processing of Higher Order Descriptors for Multi-Valued Data. Mathematics and Visualization, vol. 40, 257–267."},"date_published":"2015-01-01T00:00:00Z","year":"2015","oa_version":"None","date_updated":"2022-06-10T09:50:14Z","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","page":"257 - 267","edition":"1","status":"public","type":"book_chapter","publisher":"Springer","day":"01","publication_status":"published","department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:52:33Z","language":[{"iso":"eng"}],"volume":40,"quality_controlled":"1","abstract":[{"lang":"eng","text":"The Heat Kernel Signature (HKS) is a scalar quantity which is derived from the heat kernel of a given shape. Due to its robustness, isometry invariance, and multiscale nature, it has been successfully applied in many geometric applications. From a more general point of view, the HKS can be considered as a descriptor of the metric of a Riemannian manifold. Given a symmetric positive definite tensor field we may interpret it as the metric of some Riemannian manifold and thereby apply the HKS to visualize and analyze the given tensor data. In this paper, we propose a generalization of this approach that enables the treatment of indefinite tensor fields, like the stress tensor, by interpreting them as a generator of a positive definite tensor field. To investigate the usefulness of this approach we consider the stress tensor from the two-point-load model example and from a mechanical work piece."}],"editor":[{"first_name":"Ingrid","last_name":"Hotz","full_name":"Hotz, Ingrid"},{"last_name":"Schultz","full_name":"Schultz, Thomas","first_name":"Thomas"}],"publication":"Visualization and Processing of Higher Order Descriptors for Multi-Valued Data","author":[{"first_name":"Valentin","last_name":"Zobel","full_name":"Zobel, Valentin"},{"first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87","last_name":"Reininghaus","full_name":"Reininghaus, Jan"},{"full_name":"Hotz, Ingrid","last_name":"Hotz","first_name":"Ingrid"}],"scopus_import":"1","intvolume":"        40","publication_identifier":{"isbn":["978-3-319-15089-5"]}},{"type":"conference","day":"01","publisher":"Society for Industrial and Applied Mathematics","status":"public","page":"31 - 38","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","oa_version":"Submitted Version","year":"2014","date_updated":"2025-06-11T08:03:07Z","project":[{"name":"Topological Complex Systems","call_identifier":"FP7","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493"}],"date_published":"2014-01-01T00:00:00Z","citation":{"mla":"Bauer, Ulrich, et al. “Distributed Computation of Persistent Homology.” <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>, edited by Catherine  McGeoch and Ulrich Meyer, Society for Industrial and Applied Mathematics, 2014, pp. 31–38, doi:<a href=\"https://doi.org/10.1137/1.9781611973198.4\">10.1137/1.9781611973198.4</a>.","short":"U. Bauer, M. Kerber, J. Reininghaus, in:, C.  McGeoch, U. Meyer (Eds.), Proceedings of the Workshop on Algorithm Engineering and Experiments, Society for Industrial and Applied Mathematics, 2014, pp. 31–38.","ieee":"U. Bauer, M. Kerber, and J. Reininghaus, “Distributed computation of persistent homology,” in <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>, Portland, USA, 2014, pp. 31–38.","ama":"Bauer U, Kerber M, Reininghaus J. Distributed computation of persistent homology. In:  McGeoch C, Meyer U, eds. <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>. Society for Industrial and Applied Mathematics; 2014:31-38. doi:<a href=\"https://doi.org/10.1137/1.9781611973198.4\">10.1137/1.9781611973198.4</a>","apa":"Bauer, U., Kerber, M., &#38; Reininghaus, J. (2014). Distributed computation of persistent homology. In C.  McGeoch &#38; U. Meyer (Eds.), <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i> (pp. 31–38). Portland, USA: Society for Industrial and Applied Mathematics. <a href=\"https://doi.org/10.1137/1.9781611973198.4\">https://doi.org/10.1137/1.9781611973198.4</a>","chicago":"Bauer, Ulrich, Michael Kerber, and Jan Reininghaus. “Distributed Computation of Persistent Homology.” In <i>Proceedings of the Workshop on Algorithm Engineering and Experiments</i>, edited by Catherine  McGeoch and Ulrich Meyer, 31–38. Society for Industrial and Applied Mathematics, 2014. <a href=\"https://doi.org/10.1137/1.9781611973198.4\">https://doi.org/10.1137/1.9781611973198.4</a>.","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."},"article_processing_charge":"No","conference":{"end_date":"2014-01-05","location":"Portland, USA","start_date":"2014-01-05","name":"ALENEX: Algorithm Engineering and Experiments"},"title":"Distributed computation of persistent homology","month":"01","_id":"2043","doi":"10.1137/1.9781611973198.4","publist_id":"5008","arxiv":1,"main_file_link":[{"url":"http://arxiv.org/abs/1310.0710","open_access":"1"}],"oa":1,"ec_funded":1,"scopus_import":"1","publication":"Proceedings of the Workshop on Algorithm Engineering and Experiments","author":[{"full_name":"Bauer, Ulrich","last_name":"Bauer","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","orcid":"0000-0002-9683-0724"},{"first_name":"Michael","full_name":"Kerber, Michael","last_name":"Kerber","orcid":"0000-0002-8030-9299"},{"last_name":"Reininghaus","full_name":"Reininghaus, Jan","first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87"}],"editor":[{"full_name":" McGeoch, Catherine","last_name":" McGeoch","first_name":"Catherine"},{"first_name":"Ulrich","last_name":"Meyer","full_name":"Meyer, Ulrich"}],"abstract":[{"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.","lang":"eng"}],"quality_controlled":"1","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"external_id":{"arxiv":["1310.0710"]},"date_created":"2018-12-11T11:55:23Z","publication_status":"published"},{"status":"public","type":"book_chapter","day":"19","publisher":"Springer","date_updated":"2025-06-11T07:56:57Z","oa_version":"Submitted Version","year":"2014","project":[{"_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493","call_identifier":"FP7","name":"Topological Complex Systems"}],"date_published":"2014-03-19T00:00:00Z","page":"103 - 117","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","month":"03","title":"Clear and Compress: Computing Persistent Homology in Chunks","_id":"2044","citation":{"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.","mla":"Bauer, Ulrich, et al. “Clear and Compress: Computing Persistent Homology in Chunks.” <i>Topological Methods in Data Analysis and Visualization III</i>, edited by Peer-Timo Bremer et al., Springer, 2014, pp. 103–17, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">10.1007/978-3-319-04099-8_7</a>.","ieee":"U. Bauer, M. Kerber, and J. Reininghaus, “Clear and Compress: Computing Persistent Homology in Chunks,” in <i>Topological Methods in Data Analysis and Visualization III</i>, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Springer, 2014, pp. 103–117.","apa":"Bauer, U., Kerber, M., &#38; Reininghaus, J. (2014). Clear and Compress: Computing Persistent Homology in Chunks. In P.-T. Bremer, I. Hotz, V. Pascucci, &#38; R. Peikert (Eds.), <i>Topological Methods in Data Analysis and Visualization III</i> (pp. 103–117). Springer. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">https://doi.org/10.1007/978-3-319-04099-8_7</a>","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. <i>Topological Methods in Data Analysis and Visualization III</i>. Mathematics and Visualization. Springer; 2014:103-117. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">10.1007/978-3-319-04099-8_7</a>","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 <i>Topological Methods in Data Analysis and Visualization III</i>, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 103–17. Mathematics and Visualization. Springer, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_7\">https://doi.org/10.1007/978-3-319-04099-8_7</a>."},"article_processing_charge":"No","arxiv":1,"doi":"10.1007/978-3-319-04099-8_7","publist_id":"5007","oa":1,"series_title":"Mathematics and Visualization","main_file_link":[{"url":"http://arxiv.org/abs/1303.0477","open_access":"1"}],"ec_funded":1,"scopus_import":"1","author":[{"orcid":"0000-0002-9683-0724","first_name":"Ulrich","id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","last_name":"Bauer","full_name":"Bauer, Ulrich"},{"orcid":"0000-0002-8030-9299","last_name":"Kerber","full_name":"Kerber, Michael","first_name":"Michael"},{"last_name":"Reininghaus","full_name":"Reininghaus, Jan","first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87"}],"publication":"Topological Methods in Data Analysis and Visualization III","quality_controlled":"1","editor":[{"full_name":"Bremer, Peer-Timo","last_name":"Bremer","first_name":"Peer-Timo"},{"full_name":"Hotz, Ingrid","last_name":"Hotz","first_name":"Ingrid"},{"full_name":"Pascucci, Valerio","last_name":"Pascucci","first_name":"Valerio"},{"first_name":"Ronald","full_name":"Peikert, Ronald","last_name":"Peikert"}],"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."}],"publication_status":"published","corr_author":"1","language":[{"iso":"eng"}],"date_created":"2018-12-11T11:55:23Z","external_id":{"arxiv":["1303.0477"]},"department":[{"_id":"HeEd"}]},{"user_id":"317138e5-6ab7-11ef-aa6d-ffef3953e345","page":"2585 - 2594","date_published":"2014-12-31T00:00:00Z","oa_version":"None","year":"2014","date_updated":"2025-09-29T12:11:45Z","day":"31","type":"journal_article","publisher":"IEEE","issue":"12","status":"public","doi":"10.1109/TVCG.2014.2346432","publist_id":"5164","article_processing_charge":"No","citation":{"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,” <i>IEEE Transactions on Visualization and Computer Graphics</i>, vol. 20, no. 12. IEEE, pp. 2585–2594, 2014.","mla":"Günther, David, et al. “Fast and Memory-Efficient Topological Denoising of 2D and 3D Scalar Fields.” <i>IEEE Transactions on Visualization and Computer Graphics</i>, vol. 20, no. 12, IEEE, 2014, pp. 2585–94, doi:<a href=\"https://doi.org/10.1109/TVCG.2014.2346432\">10.1109/TVCG.2014.2346432</a>.","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.","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.” <i>IEEE Transactions on Visualization and Computer Graphics</i>. IEEE, 2014. <a href=\"https://doi.org/10.1109/TVCG.2014.2346432\">https://doi.org/10.1109/TVCG.2014.2346432</a>.","apa":"Günther, D., Jacobson, A., Reininghaus, J., Seidel, H., Sorkine Hornung, O., &#38; Weinkauf, T. (2014). Fast and memory-efficient topological denoising of 2D and 3D scalar fields. <i>IEEE Transactions on Visualization and Computer Graphics</i>. IEEE. <a href=\"https://doi.org/10.1109/TVCG.2014.2346432\">https://doi.org/10.1109/TVCG.2014.2346432</a>","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. <i>IEEE Transactions on Visualization and Computer Graphics</i>. 2014;20(12):2585-2594. doi:<a href=\"https://doi.org/10.1109/TVCG.2014.2346432\">10.1109/TVCG.2014.2346432</a>"},"_id":"1930","title":"Fast and memory-efficient topological denoising of 2D and 3D scalar fields","month":"12","scopus_import":"1","publication":"IEEE Transactions on Visualization and Computer Graphics","author":[{"last_name":"Günther","full_name":"Günther, David","first_name":"David"},{"full_name":"Jacobson, Alec","last_name":"Jacobson","first_name":"Alec"},{"last_name":"Reininghaus","full_name":"Reininghaus, Jan","first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87"},{"full_name":"Seidel, Hans","last_name":"Seidel","first_name":"Hans"},{"first_name":"Olga","full_name":"Sorkine Hornung, Olga","last_name":"Sorkine Hornung"},{"full_name":"Weinkauf, Tino","last_name":"Weinkauf","first_name":"Tino"}],"intvolume":"        20","isi":1,"acknowledgement":"RTRA Digiteoproject; ERC grant; SNF award; Intel Doctoral Fellowship; MPC-VCC","external_id":{"isi":["000344991700104"]},"department":[{"_id":"HeEd"}],"date_created":"2018-12-11T11:54:46Z","language":[{"iso":"eng"}],"publication_status":"published","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"}],"volume":20,"quality_controlled":"1"},{"article_processing_charge":"No","citation":{"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.","chicago":"Günther, David, Jan Reininghaus, Hans-Peter Seidel, and Tino Weinkauf. “Notes on the Simplification of the Morse-Smale Complex.” In <i>Topological Methods in Data Analysis and Visualization III.</i>, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 135–50. Mathematics and Visualization. Cham: Springer Nature, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">https://doi.org/10.1007/978-3-319-04099-8_9</a>.","apa":"Günther, D., Reininghaus, J., Seidel, H.-P., &#38; Weinkauf, T. (2014). Notes on the simplification of the Morse-Smale complex. In P.-T. Bremer, I. Hotz, V. Pascucci, &#38; R. Peikert (Eds.), <i>Topological Methods in Data Analysis and Visualization III.</i> (pp. 135–150). Cham: Springer Nature. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">https://doi.org/10.1007/978-3-319-04099-8_9</a>","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. <i>Topological Methods in Data Analysis and Visualization III.</i> Mathematics and Visualization. Cham: Springer Nature; 2014:135-150. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">10.1007/978-3-319-04099-8_9</a>","ieee":"D. Günther, J. Reininghaus, H.-P. Seidel, and T. Weinkauf, “Notes on the simplification of the Morse-Smale complex,” in <i>Topological Methods in Data Analysis and Visualization III.</i>, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer Nature, 2014, pp. 135–150.","mla":"Günther, David, et al. “Notes on the Simplification of the Morse-Smale Complex.” <i>Topological Methods in Data Analysis and Visualization III.</i>, edited by Peer-Timo Bremer et al., Springer Nature, 2014, pp. 135–50, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_9\">10.1007/978-3-319-04099-8_9</a>.","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."},"_id":"10817","title":"Notes on the simplification of the Morse-Smale complex","month":"03","doi":"10.1007/978-3-319-04099-8_9","publisher":"Springer Nature","type":"book_chapter","day":"19","status":"public","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","place":"Cham","page":"135-150","project":[{"grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","call_identifier":"FP7"}],"date_published":"2014-03-19T00:00:00Z","year":"2014","oa_version":"None","date_updated":"2025-04-15T08:37:54Z","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."}],"editor":[{"last_name":"Bremer","full_name":"Bremer, Peer-Timo","first_name":"Peer-Timo"},{"first_name":"Ingrid","full_name":"Hotz, Ingrid","last_name":"Hotz"},{"first_name":"Valerio","full_name":"Pascucci, Valerio","last_name":"Pascucci"},{"full_name":"Peikert, Ronald","last_name":"Peikert","first_name":"Ronald"}],"quality_controlled":"1","department":[{"_id":"HeEd"}],"date_created":"2022-03-04T08:33:57Z","language":[{"iso":"eng"}],"publication_status":"published","acknowledgement":"This research is supported and funded by the Digiteo unTopoVis project, the TOPOSYS project FP7-ICT-318493-STREP, and MPC-VCC.","publication_identifier":{"isbn":["9783319040981"],"eisbn":["9783319040998"],"eissn":["2197-666X"],"issn":["1612-3786"]},"series_title":"Mathematics and Visualization","author":[{"first_name":"David","full_name":"Günther, David","last_name":"Günther"},{"first_name":"Jan","id":"4505473A-F248-11E8-B48F-1D18A9856A87","full_name":"Reininghaus, Jan","last_name":"Reininghaus"},{"last_name":"Seidel","full_name":"Seidel, Hans-Peter","first_name":"Hans-Peter"},{"full_name":"Weinkauf, Tino","last_name":"Weinkauf","first_name":"Tino"}],"publication":"Topological Methods in Data Analysis and Visualization III.","scopus_import":"1","ec_funded":1},{"title":"Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature","quality_controlled":"1","month":"03","_id":"10886","citation":{"mla":"Zobel, Valentin, et al. “Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature.” <i>Topological Methods in Data Analysis and Visualization III </i>, Springer, 2014, pp. 249–62, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_16\">10.1007/978-3-319-04099-8_16</a>.","short":"V. Zobel, J. Reininghaus, I. Hotz, in:, Topological Methods in Data Analysis and Visualization III , Springer, 2014, pp. 249–262.","ieee":"V. Zobel, J. Reininghaus, and I. Hotz, “Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature,” in <i>Topological Methods in Data Analysis and Visualization III </i>, 2014, pp. 249–262.","apa":"Zobel, V., Reininghaus, J., &#38; Hotz, I. (2014). Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. In <i>Topological Methods in Data Analysis and Visualization III </i> (pp. 249–262). Springer. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_16\">https://doi.org/10.1007/978-3-319-04099-8_16</a>","ama":"Zobel V, Reininghaus J, Hotz I. Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. In: <i>Topological Methods in Data Analysis and Visualization III </i>. Springer; 2014:249-262. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_16\">10.1007/978-3-319-04099-8_16</a>","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.","chicago":"Zobel, Valentin, Jan Reininghaus, and Ingrid Hotz. “Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature.” In <i>Topological Methods in Data Analysis and Visualization III </i>, 249–62. Springer, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_16\">https://doi.org/10.1007/978-3-319-04099-8_16</a>."},"article_processing_charge":"No","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."}],"publication_status":"published","doi":"10.1007/978-3-319-04099-8_16","language":[{"iso":"eng"}],"alternative_title":["Mathematics and Visualization"],"department":[{"_id":"HeEd"}],"date_created":"2022-03-18T13:05:39Z","publication_identifier":{"eisbn":["9783319040998"],"isbn":["9783319040981"],"issn":["1612-3786"],"eissn":["2197-666X"]},"acknowledgement":"This research is partially supported by the TOPOSYS project FP7-ICT-318493-STREP.","status":"public","day":"19","publisher":"Springer","type":"conference","year":"2014","oa_version":"None","date_updated":"2023-09-05T14:13:16Z","date_published":"2014-03-19T00:00:00Z","page":"249-262","publication":"Topological Methods in Data Analysis and Visualization III ","author":[{"first_name":"Valentin","last_name":"Zobel","full_name":"Zobel, Valentin"},{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","full_name":"Reininghaus, Jan","last_name":"Reininghaus"},{"full_name":"Hotz, Ingrid","last_name":"Hotz","first_name":"Ingrid"}],"user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","scopus_import":"1"},{"publication_status":"published","language":[{"iso":"eng"}],"department":[{"_id":"HeEd"}],"date_created":"2022-03-21T07:11:23Z","quality_controlled":"1","volume":1,"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","full_name":"Pascucci, Valerio","last_name":"Pascucci"},{"last_name":"Peikert","full_name":"Peikert, Ronald","first_name":"Ronald"}],"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."}],"ec_funded":1,"intvolume":"         1","publication":"Topological Methods in Data Analysis and Visualization III ","scopus_import":"1","author":[{"full_name":"Kasten, Jens","last_name":"Kasten","first_name":"Jens"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan"},{"first_name":"Wieland","full_name":"Reich, Wieland","last_name":"Reich"},{"last_name":"Scheuermann","full_name":"Scheuermann, Gerik","first_name":"Gerik"}],"publication_identifier":{"isbn":["9783319040981"],"eisbn":["9783319040998"],"issn":["1612-3786"],"eissn":["2197-666X"]},"series_title":"Mathematics and Visualization","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.","doi":"10.1007/978-3-319-04099-8_4","title":"Toward the extraction of saddle periodic orbits","month":"03","_id":"10893","citation":{"apa":"Kasten, J., Reininghaus, J., Reich, W., &#38; Scheuermann, G. (2014). Toward the extraction of saddle periodic orbits. In P.-T. Bremer, I. Hotz, V. Pascucci, &#38; R. Peikert (Eds.), <i>Topological Methods in Data Analysis and Visualization III </i> (Vol. 1, pp. 55–69). Cham: Springer. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">https://doi.org/10.1007/978-3-319-04099-8_4</a>","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. <i>Topological Methods in Data Analysis and Visualization III </i>. Vol 1. Mathematics and Visualization. Cham: Springer; 2014:55-69. doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">10.1007/978-3-319-04099-8_4</a>","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.","chicago":"Kasten, Jens, Jan Reininghaus, Wieland Reich, and Gerik Scheuermann. “Toward the Extraction of Saddle Periodic Orbits.” In <i>Topological Methods in Data Analysis and Visualization III </i>, edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 1:55–69. Mathematics and Visualization. Cham: Springer, 2014. <a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">https://doi.org/10.1007/978-3-319-04099-8_4</a>.","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.","mla":"Kasten, Jens, et al. “Toward the Extraction of Saddle Periodic Orbits.” <i>Topological Methods in Data Analysis and Visualization III </i>, edited by Peer-Timo Bremer et al., vol. 1, Springer, 2014, pp. 55–69, doi:<a href=\"https://doi.org/10.1007/978-3-319-04099-8_4\">10.1007/978-3-319-04099-8_4</a>.","ieee":"J. Kasten, J. Reininghaus, W. Reich, and G. Scheuermann, “Toward the extraction of saddle periodic orbits,” in <i>Topological Methods in Data Analysis and Visualization III </i>, vol. 1, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer, 2014, pp. 55–69."},"article_processing_charge":"No","oa_version":"None","year":"2014","date_updated":"2025-04-15T08:37:54Z","project":[{"call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493"}],"date_published":"2014-03-19T00:00:00Z","page":"55-69","user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","place":"Cham","status":"public","day":"19","type":"book_chapter","publisher":"Springer"},{"scopus_import":"1","publication":"ICMS 2014: International Congress on Mathematical Software","author":[{"id":"2ADD483A-F248-11E8-B48F-1D18A9856A87","first_name":"Ulrich","full_name":"Bauer, Ulrich","last_name":"Bauer","orcid":"0000-0002-9683-0724"},{"first_name":"Michael","last_name":"Kerber","full_name":"Kerber, Michael"},{"full_name":"Reininghaus, Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan"},{"last_name":"Wagner","full_name":"Wagner, Hubert","first_name":"Hubert"}],"intvolume":"      8592","publication_identifier":{"isbn":["9783662441985"],"eisbn":["9783662441992"],"issn":["0302-9743"],"eissn":["1611-3349"]},"series_title":"LNCS","date_created":"2022-03-21T07:12:16Z","department":[{"_id":"HeEd"}],"language":[{"iso":"eng"}],"publication_status":"published","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."}],"volume":8592,"quality_controlled":"1","place":"Berlin, Heidelberg","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","page":"137-143","date_published":"2014-09-01T00:00:00Z","date_updated":"2025-10-01T07:39:50Z","year":"2014","oa_version":"None","type":"conference","day":"01","publisher":"Springer Berlin Heidelberg","status":"public","doi":"10.1007/978-3-662-44199-2_24","conference":{"name":"ICMS: International Congress on Mathematical Software","start_date":"2014-08-05","location":"Seoul, South Korea","end_date":"2014-08-09"},"article_processing_charge":"No","related_material":{"record":[{"id":"1433","status":"public","relation":"later_version"}]},"citation":{"chicago":"Bauer, Ulrich, Michael Kerber, Jan Reininghaus, and Hubert Wagner. “PHAT – Persistent Homology Algorithms Toolbox.” In <i>ICMS 2014: International Congress on Mathematical Software</i>, 8592:137–43. LNCS. Berlin, Heidelberg: Springer Berlin Heidelberg, 2014. <a href=\"https://doi.org/10.1007/978-3-662-44199-2_24\">https://doi.org/10.1007/978-3-662-44199-2_24</a>.","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.","ama":"Bauer U, Kerber M, Reininghaus J, Wagner H. PHAT – Persistent Homology Algorithms Toolbox. In: <i>ICMS 2014: International Congress on Mathematical Software</i>. Vol 8592. LNCS. Berlin, Heidelberg: Springer Berlin Heidelberg; 2014:137-143. doi:<a href=\"https://doi.org/10.1007/978-3-662-44199-2_24\">10.1007/978-3-662-44199-2_24</a>","apa":"Bauer, U., Kerber, M., Reininghaus, J., &#38; Wagner, H. (2014). PHAT – Persistent Homology Algorithms Toolbox. In <i>ICMS 2014: International Congress on Mathematical Software</i> (Vol. 8592, pp. 137–143). Berlin, Heidelberg: Springer Berlin Heidelberg. <a href=\"https://doi.org/10.1007/978-3-662-44199-2_24\">https://doi.org/10.1007/978-3-662-44199-2_24</a>","ieee":"U. Bauer, M. Kerber, J. Reininghaus, and H. Wagner, “PHAT – Persistent Homology Algorithms Toolbox,” in <i>ICMS 2014: International Congress on Mathematical Software</i>, Seoul, South Korea, 2014, vol. 8592, pp. 137–143.","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.","mla":"Bauer, Ulrich, et al. “PHAT – Persistent Homology Algorithms Toolbox.” <i>ICMS 2014: International Congress on Mathematical Software</i>, vol. 8592, Springer Berlin Heidelberg, 2014, pp. 137–43, doi:<a href=\"https://doi.org/10.1007/978-3-662-44199-2_24\">10.1007/978-3-662-44199-2_24</a>."},"_id":"10894","month":"09","title":"PHAT – Persistent Homology Algorithms Toolbox"}]