---
_id: '10817'
abstract:
- lang: eng
  text: The Morse-Smale complex can be either explicitly or implicitly represented.
    Depending on the type of representation, the simplification of the Morse-Smale
    complex works differently. In the explicit representation, the Morse-Smale complex
    is directly simplified by explicitly reconnecting the critical points during the
    simplification. In the implicit representation, on the other hand, the Morse-Smale
    complex is given by a combinatorial gradient field. In this setting, the simplification
    changes the combinatorial flow, which yields an indirect simplification of the
    Morse-Smale complex. The topological complexity of the Morse-Smale complex is
    reduced in both representations. However, the simplifications generally yield
    different results. In this chapter, we emphasize properties of the two representations
    that cause these differences. We also provide a complexity analysis of the two
    schemes with respect to running time and memory consumption.
acknowledgement: This research is supported and funded by the Digiteo unTopoVis project,
  the TOPOSYS project FP7-ICT-318493-STREP, and MPC-VCC.
article_processing_charge: No
author:
- first_name: David
  full_name: Günther, David
  last_name: Günther
- first_name: Jan
  full_name: Reininghaus, Jan
  id: 4505473A-F248-11E8-B48F-1D18A9856A87
  last_name: Reininghaus
- first_name: Hans-Peter
  full_name: Seidel, Hans-Peter
  last_name: Seidel
- first_name: Tino
  full_name: Weinkauf, Tino
  last_name: Weinkauf
citation:
  ama: 'Günther D, Reininghaus J, Seidel H-P, Weinkauf T. Notes on the simplification
    of the Morse-Smale complex. In: Bremer P-T, Hotz I, Pascucci V, Peikert R, eds.
    <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>'
  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>'
  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>.'
  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.'
  ista: 'Günther D, Reininghaus J, Seidel H-P, Weinkauf T. 2014.Notes on the simplification
    of the Morse-Smale complex. In: Topological Methods in Data Analysis and Visualization
    III. , 135–150.'
  mla: Günther, David, et al. “Notes on the Simplification of the Morse-Smale Complex.”
    <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.
date_created: 2022-03-04T08:33:57Z
date_published: 2014-03-19T00:00:00Z
date_updated: 2023-09-05T15:33:45Z
day: '19'
department:
- _id: HeEd
doi: 10.1007/978-3-319-04099-8_9
ec_funded: 1
editor:
- first_name: Peer-Timo
  full_name: Bremer, Peer-Timo
  last_name: Bremer
- first_name: Ingrid
  full_name: Hotz, Ingrid
  last_name: Hotz
- first_name: Valerio
  full_name: Pascucci, Valerio
  last_name: Pascucci
- first_name: Ronald
  full_name: Peikert, Ronald
  last_name: Peikert
language:
- iso: eng
month: '03'
oa_version: None
page: 135-150
place: Cham
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '318493'
  name: Topological Complex Systems
publication: Topological Methods in Data Analysis and Visualization III.
publication_identifier:
  eisbn:
  - '9783319040998'
  eissn:
  - 2197-666X
  isbn:
  - '9783319040981'
  issn:
  - 1612-3786
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
series_title: Mathematics and Visualization
status: public
title: Notes on the simplification of the Morse-Smale complex
type: book_chapter
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2014'
...
---
_id: '10886'
abstract:
- lang: eng
  text: We propose a method for visualizing two-dimensional symmetric positive definite
    tensor fields using the Heat Kernel Signature (HKS). The HKS is derived from the
    heat kernel and was originally introduced as an isometry invariant shape signature.
    Each positive definite tensor field defines a Riemannian manifold by considering
    the tensor field as a Riemannian metric. On this Riemmanian manifold we can apply
    the definition of the HKS. The resulting scalar quantity is used for the visualization
    of tensor fields. The HKS is closely related to the Gaussian curvature of the
    Riemannian manifold and the time parameter of the heat kernel allows a multiscale
    analysis in a natural way. In this way, the HKS represents field related scale
    space properties, enabling a level of detail analysis of tensor fields. This makes
    the HKS an interesting new scalar quantity for tensor fields, which differs significantly
    from usual tensor invariants like the trace or the determinant. A method for visualization
    and a numerical realization of the HKS for tensor fields is proposed in this chapter.
    To validate the approach we apply it to some illustrating simple examples as isolated
    critical points and to a medical diffusion tensor data set.
acknowledgement: This research is partially supported by the TOPOSYS project FP7-ICT-318493-STREP.
alternative_title:
- Mathematics and Visualization
article_processing_charge: No
author:
- first_name: Valentin
  full_name: Zobel, Valentin
  last_name: Zobel
- first_name: Jan
  full_name: Reininghaus, Jan
  id: 4505473A-F248-11E8-B48F-1D18A9856A87
  last_name: Reininghaus
- first_name: Ingrid
  full_name: Hotz, Ingrid
  last_name: Hotz
citation:
  ama: 'Zobel V, Reininghaus J, Hotz I. Visualization of two-dimensional symmetric
    positive definite tensor fields using the heat kernel signature. In: <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>'
  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>
  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>.
  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.
  ista: Zobel V, Reininghaus J, Hotz I. 2014. Visualization of two-dimensional symmetric
    positive definite tensor fields using the heat kernel signature. Topological Methods
    in Data Analysis and Visualization III . , Mathematics and Visualization, , 249–262.
  mla: Zobel, Valentin, et al. “Visualization of Two-Dimensional Symmetric Positive
    Definite Tensor Fields Using the Heat Kernel Signature.” <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.
date_created: 2022-03-18T13:05:39Z
date_published: 2014-03-19T00:00:00Z
date_updated: 2023-09-05T14:13:16Z
day: '19'
department:
- _id: HeEd
doi: 10.1007/978-3-319-04099-8_16
language:
- iso: eng
month: '03'
oa_version: None
page: 249-262
publication: 'Topological Methods in Data Analysis and Visualization III '
publication_identifier:
  eisbn:
  - '9783319040998'
  eissn:
  - 2197-666X
  isbn:
  - '9783319040981'
  issn:
  - 1612-3786
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
status: public
title: Visualization of two-dimensional symmetric positive definite tensor fields
  using the heat kernel signature
type: conference
user_id: c635000d-4b10-11ee-a964-aac5a93f6ac1
year: '2014'
...
---
_id: '10893'
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.
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.
article_processing_charge: No
author:
- first_name: Jens
  full_name: Kasten, Jens
  last_name: Kasten
- first_name: Jan
  full_name: Reininghaus, Jan
  id: 4505473A-F248-11E8-B48F-1D18A9856A87
  last_name: Reininghaus
- first_name: Wieland
  full_name: Reich, Wieland
  last_name: Reich
- first_name: Gerik
  full_name: Scheuermann, Gerik
  last_name: Scheuermann
citation:
  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>'
  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>'
  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>.'
  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.'
  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.'
  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>.
  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.
date_created: 2022-03-21T07:11:23Z
date_published: 2014-03-19T00:00:00Z
date_updated: 2022-06-21T12:01:47Z
day: '19'
department:
- _id: HeEd
doi: 10.1007/978-3-319-04099-8_4
ec_funded: 1
editor:
- first_name: Peer-Timo
  full_name: Bremer, Peer-Timo
  last_name: Bremer
- first_name: Ingrid
  full_name: Hotz, Ingrid
  last_name: Hotz
- first_name: Valerio
  full_name: Pascucci, Valerio
  last_name: Pascucci
- first_name: Ronald
  full_name: Peikert, Ronald
  last_name: Peikert
intvolume: '         1'
language:
- iso: eng
month: '03'
oa_version: None
page: 55-69
place: Cham
project:
- _id: 255D761E-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '318493'
  name: Topological Complex Systems
publication: 'Topological Methods in Data Analysis and Visualization III '
publication_identifier:
  eisbn:
  - '9783319040998'
  eissn:
  - 2197-666X
  isbn:
  - '9783319040981'
  issn:
  - 1612-3786
publication_status: published
publisher: Springer
quality_controlled: '1'
scopus_import: '1'
series_title: Mathematics and Visualization
status: public
title: Toward the extraction of saddle periodic orbits
type: book_chapter
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 1
year: '2014'
...