@inbook{10817,
  abstract     = {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.},
  author       = {Günther, David and Reininghaus, Jan and Seidel, Hans-Peter and Weinkauf, Tino},
  booktitle    = {Topological Methods in Data Analysis and Visualization III.},
  editor       = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald},
  isbn         = {9783319040981},
  issn         = {2197-666X},
  pages        = {135--150},
  publisher    = {Springer Nature},
  title        = {{Notes on the simplification of the Morse-Smale complex}},
  doi          = {10.1007/978-3-319-04099-8_9},
  year         = {2014},
}

@inproceedings{10886,
  abstract     = {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.},
  author       = {Zobel, Valentin and Reininghaus, Jan and Hotz, Ingrid},
  booktitle    = {Topological Methods in Data Analysis and Visualization III },
  isbn         = {9783319040981},
  issn         = {2197-666X},
  pages        = {249--262},
  publisher    = {Springer},
  title        = {{Visualization of two-dimensional symmetric positive definite tensor fields using the heat kernel signature}},
  doi          = {10.1007/978-3-319-04099-8_16},
  year         = {2014},
}

@inbook{10893,
  abstract     = {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.},
  author       = {Kasten, Jens and Reininghaus, Jan and Reich, Wieland and Scheuermann, Gerik},
  booktitle    = {Topological Methods in Data Analysis and Visualization III },
  editor       = {Bremer, Peer-Timo and Hotz, Ingrid and Pascucci, Valerio and Peikert, Ronald},
  isbn         = {9783319040981},
  issn         = {2197-666X},
  pages        = {55--69},
  publisher    = {Springer},
  title        = {{Toward the extraction of saddle periodic orbits}},
  doi          = {10.1007/978-3-319-04099-8_4},
  volume       = {1},
  year         = {2014},
}