Advertisement

Visualizing Symmetric Indefinite 2D Tensor Fields Using the Heat Kernel Signature

  • Valentin ZobelEmail author
  • Jan Reininghaus
  • Ingrid Hotz
Conference paper
Part of the Mathematics and Visualization book series (MATHVISUAL)

Abstract

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.

Keywords

Riemannian Manifold Heat Kernel Diffusion Tensor Tensor Field Exponential Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Brannon, R.M.: Functional and structured tensor analysis for engineers. UNM Book Draft (2003). http://www.mech.utah.edu/~brannon/
  2. 2.
    Bronstein, M., Kokkinos, I.: Scale-invariant heat kernel signatures for non-rigid shape recognition. In: 2010 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), pp. 1704–1711 (2010)Google Scholar
  3. 3.
    Danielson, D.A.: Vectors and Tensors in Engineering and Physics, 2nd edn. Department of Mathematics, Naval Postgraduate School, Monterey, CA. Addison-Wesley, Reading, MA (1997) [Diss]Google Scholar
  4. 4.
    Dey, T., Li, K., Luo, C., Ranjan, P., Safa, I., Wang, Y.: Persistent heat signature for pose-oblivious matching of incomplete models. In: Computer Graphics Forum, vol. 29, pp. 1545–1554 (Wiley Online Library, 2010)Google Scholar
  5. 5.
    Hotz, I., Feng, L., Hagen, H., Hamann, B., Jeremic, B., Joy, K.I.: Physically based methods for tensor field visualization. In: VIS ’04: Proceedings of IEEE Visualization 2004, pp. 123–130. IEEE Computer Society Press, Los Alamitos (2004)Google Scholar
  6. 6.
    O’Donnell, L., Haker, S., Westin, C.F.: New approaches to estimation of white matter connectivity in diffusion tensor mri: elliptic pdes and geodesics in a tensor-warped space. In: Medical Image Computing and Computer-Assisted Intervention MICCAI 2002, pp. 459–466. Springer, Berlin (2002)Google Scholar
  7. 7.
    Ovsjanikov, M., Bronstein, A., Bronstein, M., Guibas, L.: Shape Google: a computer vision approach to isometry invariant shape retrieval. In: 2009 IEEE 12th International Conference on Computer Vision Workshops (ICCV Workshops), pp. 320–327 (2009)Google Scholar
  8. 8.
    Raviv, D., Bronstein, M., Bronstein, A., Kimmel, R.: Volumetric heat kernel signatures. In: Proceedings of the ACM Workshop on 3D Object Retrieval, pp. 39–44 (2010)Google Scholar
  9. 9.
    Rosenberg, S.: The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds. Cambridge University Press, Cambridge (1997)CrossRefzbMATHGoogle Scholar
  10. 10.
    Spira, A., Sochen, N., Kimmel, R.: A short time beltrami kernel for smoothing images and manifolds. IEEE Trans. Image Process. 16(6), 1628–1636 (2007)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Sun, J., Ovsjanikov, M., Guibas, L.: A concise and provably informative multi-scale signature based on heat diffusion. In: Proceedings of Eurographics Symposium on Geometry Processing (SGP) (2009)Google Scholar
  12. 12.
    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, pp. 249–262. Springer, Berlin (2014)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  1. 1.Leipzig UniversityLeipzigGermany
  2. 2.Institute of Science and Technology AustriaKlosterneuburgAustria
  3. 3.Linköping UniversityNorrköpingSweden

Personalised recommendations