[{"publication_identifier":{"isbn":["9783319040981"],"eissn":["2197-666X"],"eisbn":["9783319040998"],"issn":["1612-3786"]},"month":"03","project":[{"call_identifier":"FP7","name":"Topological Complex Systems","_id":"255D761E-B435-11E9-9278-68D0E5697425","grant_number":"318493"}],"quality_controlled":"1","doi":"10.1007/978-3-319-04099-8_4","language":[{"iso":"eng"}],"place":"Cham","ec_funded":1,"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.","year":"2014","publisher":"Springer","editor":[{"full_name":"Bremer, Peer-Timo","first_name":"Peer-Timo","last_name":"Bremer"},{"last_name":"Hotz","first_name":"Ingrid","full_name":"Hotz, Ingrid"},{"last_name":"Pascucci","first_name":"Valerio","full_name":"Pascucci, Valerio"},{"full_name":"Peikert, Ronald","first_name":"Ronald","last_name":"Peikert"}],"department":[{"_id":"HeEd"}],"publication_status":"published","author":[{"full_name":"Kasten, Jens","last_name":"Kasten","first_name":"Jens"},{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","last_name":"Reininghaus","full_name":"Reininghaus, Jan"},{"full_name":"Reich, Wieland","first_name":"Wieland","last_name":"Reich"},{"full_name":"Scheuermann, Gerik","last_name":"Scheuermann","first_name":"Gerik"}],"volume":1,"date_created":"2022-03-21T07:11:23Z","date_updated":"2022-06-21T12:01:47Z","scopus_import":"1","series_title":"Mathematics and Visualization","article_processing_charge":"No","day":"19","citation":{"mla":"Kasten, Jens, et al. “Toward the Extraction of Saddle Periodic Orbits.” Topological Methods in Data Analysis and Visualization III , edited by Peer-Timo Bremer et al., vol. 1, Springer, 2014, pp. 55–69, doi:10.1007/978-3-319-04099-8_4.","short":"J. Kasten, J. Reininghaus, W. Reich, G. Scheuermann, in:, P.-T. Bremer, I. Hotz, V. Pascucci, R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III , Springer, Cham, 2014, pp. 55–69.","chicago":"Kasten, Jens, Jan Reininghaus, Wieland Reich, and Gerik Scheuermann. “Toward the Extraction of Saddle Periodic Orbits.” In Topological Methods in Data Analysis and Visualization III , edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 1:55–69. Mathematics and Visualization. Cham: Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_4.","ama":"Kasten J, Reininghaus J, Reich W, Scheuermann G. Toward the extraction of saddle periodic orbits. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological Methods in Data Analysis and Visualization III . Vol 1. Mathematics and Visualization. Cham: Springer; 2014:55-69. doi:10.1007/978-3-319-04099-8_4","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.","apa":"Kasten, J., Reininghaus, J., Reich, W., & Scheuermann, G. (2014). Toward the extraction of saddle periodic orbits. In P.-T. Bremer, I. Hotz, V. Pascucci, & R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III (Vol. 1, pp. 55–69). Cham: Springer. https://doi.org/10.1007/978-3-319-04099-8_4","ieee":"J. Kasten, J. Reininghaus, W. Reich, and G. Scheuermann, “Toward the extraction of saddle periodic orbits,” in Topological Methods in Data Analysis and Visualization III , vol. 1, P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer, 2014, pp. 55–69."},"publication":"Topological Methods in Data Analysis and Visualization III ","page":"55-69","date_published":"2014-03-19T00:00:00Z","type":"book_chapter","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."}],"user_id":"2DF688A6-F248-11E8-B48F-1D18A9856A87","_id":"10893","intvolume":" 1","title":"Toward the extraction of saddle periodic orbits","status":"public","oa_version":"None"},{"doi":"10.1007/978-3-319-04099-8_16","date_published":"2014-03-19T00:00:00Z","language":[{"iso":"eng"}],"citation":{"chicago":"Zobel, Valentin, Jan Reininghaus, and Ingrid Hotz. “Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature.” In Topological Methods in Data Analysis and Visualization III , 249–62. Springer, 2014. https://doi.org/10.1007/978-3-319-04099-8_16.","short":"V. Zobel, J. Reininghaus, I. Hotz, in:, Topological Methods in Data Analysis and Visualization III , Springer, 2014, pp. 249–262.","mla":"Zobel, Valentin, et al. “Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature.” Topological Methods in Data Analysis and Visualization III , Springer, 2014, pp. 249–62, doi:10.1007/978-3-319-04099-8_16.","apa":"Zobel, V., Reininghaus, J., & Hotz, I. (2014). Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. In Topological Methods in Data Analysis and Visualization III (pp. 249–262). Springer. https://doi.org/10.1007/978-3-319-04099-8_16","ieee":"V. Zobel, J. Reininghaus, and I. Hotz, “Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature,” in Topological Methods in Data Analysis and Visualization III , 2014, pp. 249–262.","ista":"Zobel V, Reininghaus J, Hotz I. 2014. Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. Topological Methods in Data Analysis and Visualization III . , Mathematics and Visualization, , 249–262.","ama":"Zobel V, Reininghaus J, Hotz I. Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature. In: Topological Methods in Data Analysis and Visualization III . Springer; 2014:249-262. doi:10.1007/978-3-319-04099-8_16"},"publication":"Topological Methods in Data Analysis and Visualization III ","page":"249-262","quality_controlled":"1","article_processing_charge":"No","publication_identifier":{"issn":["1612-3786"],"eisbn":["9783319040998"],"isbn":["9783319040981"],"eissn":["2197-666X"]},"month":"03","day":"19","scopus_import":"1","author":[{"full_name":"Zobel, Valentin","first_name":"Valentin","last_name":"Zobel"},{"full_name":"Reininghaus, Jan","first_name":"Jan","last_name":"Reininghaus","id":"4505473A-F248-11E8-B48F-1D18A9856A87"},{"last_name":"Hotz","first_name":"Ingrid","full_name":"Hotz, Ingrid"}],"oa_version":"None","date_created":"2022-03-18T13:05:39Z","date_updated":"2023-09-05T14:13:16Z","_id":"10886","acknowledgement":"This research is partially supported by the TOPOSYS project FP7-ICT-318493-STREP.","year":"2014","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","publisher":"Springer","department":[{"_id":"HeEd"}],"status":"public","publication_status":"published","title":"Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature","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."}],"type":"conference","alternative_title":["Mathematics and Visualization"]},{"department":[{"_id":"HeEd"}],"publisher":"Springer Nature","editor":[{"full_name":"Bremer, Peer-Timo","first_name":"Peer-Timo","last_name":"Bremer"},{"last_name":"Hotz","first_name":"Ingrid","full_name":"Hotz, Ingrid"},{"last_name":"Pascucci","first_name":"Valerio","full_name":"Pascucci, Valerio"},{"full_name":"Peikert, Ronald","last_name":"Peikert","first_name":"Ronald"}],"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.","year":"2014","date_updated":"2023-09-05T15:33:45Z","date_created":"2022-03-04T08:33:57Z","author":[{"full_name":"Günther, David","last_name":"Günther","first_name":"David"},{"id":"4505473A-F248-11E8-B48F-1D18A9856A87","first_name":"Jan","last_name":"Reininghaus","full_name":"Reininghaus, Jan"},{"full_name":"Seidel, Hans-Peter","last_name":"Seidel","first_name":"Hans-Peter"},{"first_name":"Tino","last_name":"Weinkauf","full_name":"Weinkauf, Tino"}],"place":"Cham","ec_funded":1,"project":[{"grant_number":"318493","_id":"255D761E-B435-11E9-9278-68D0E5697425","name":"Topological Complex Systems","call_identifier":"FP7"}],"quality_controlled":"1","language":[{"iso":"eng"}],"doi":"10.1007/978-3-319-04099-8_9","publication_identifier":{"isbn":["9783319040981"],"eissn":["2197-666X"],"eisbn":["9783319040998"],"issn":["1612-3786"]},"month":"03","title":"Notes on the simplification of the Morse-Smale complex","status":"public","_id":"10817","user_id":"c635000d-4b10-11ee-a964-aac5a93f6ac1","oa_version":"None","type":"book_chapter","abstract":[{"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.","lang":"eng"}],"page":"135-150","citation":{"chicago":"Günther, David, Jan Reininghaus, Hans-Peter Seidel, and Tino Weinkauf. “Notes on the Simplification of the Morse-Smale Complex.” In Topological Methods in Data Analysis and Visualization III., edited by Peer-Timo Bremer, Ingrid Hotz, Valerio Pascucci, and Ronald Peikert, 135–50. Mathematics and Visualization. Cham: Springer Nature, 2014. https://doi.org/10.1007/978-3-319-04099-8_9.","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.","mla":"Günther, David, et al. “Notes on the Simplification of the Morse-Smale Complex.” Topological Methods in Data Analysis and Visualization III., edited by Peer-Timo Bremer et al., Springer Nature, 2014, pp. 135–50, doi:10.1007/978-3-319-04099-8_9.","ieee":"D. Günther, J. Reininghaus, H.-P. Seidel, and T. Weinkauf, “Notes on the simplification of the Morse-Smale complex,” in Topological Methods in Data Analysis and Visualization III., P.-T. Bremer, I. Hotz, V. Pascucci, and R. Peikert, Eds. Cham: Springer Nature, 2014, pp. 135–150.","apa":"Günther, D., Reininghaus, J., Seidel, H.-P., & Weinkauf, T. (2014). Notes on the simplification of the Morse-Smale complex. In P.-T. Bremer, I. Hotz, V. Pascucci, & R. Peikert (Eds.), Topological Methods in Data Analysis and Visualization III. (pp. 135–150). Cham: Springer Nature. https://doi.org/10.1007/978-3-319-04099-8_9","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.","ama":"Günther D, Reininghaus J, Seidel H-P, Weinkauf T. Notes on the simplification of the Morse-Smale complex. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds. Topological Methods in Data Analysis and Visualization III. Mathematics and Visualization. Cham: Springer Nature; 2014:135-150. doi:10.1007/978-3-319-04099-8_9"},"publication":"Topological Methods in Data Analysis and Visualization III.","date_published":"2014-03-19T00:00:00Z","series_title":"Mathematics and Visualization","scopus_import":"1","article_processing_charge":"No","day":"19"}]