Visualization of Two-Dimensional Symmetric Positive Definite Tensor Fields Using the Heat Kernel Signature

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


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.


Riemannian Manifold Heat Kernel Gaussian Curvature Neumann Boundary Condition Tensor Field 
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.



This research is partially supported by the TOPOSYS project FP7-ICT-318493-STREP.


  1. 1.
    M. Bronstein, I. Kokkinos, Scale-invariant heat kernel signatures for non-rigid shape recognition, in IEEE Conference on Computer Vision and Pattern Recognition (CVPR), 2010, San Francisco (IEEE, 2010), pp. 1704–1711Google Scholar
  2. 2.
    T. Delmarcelle, L. Hesselink, The topology of symmetric, second-order tensor fields, in Proceedings of the Conference on Visualization’94, Washington, DC (IEEE, 1994), pp. 140–147Google Scholar
  3. 3.
    M. Desbrun, E. Kanso, Y. Tong, Discrete differential forms for computational modeling, in SIGGRAPH ’06: ACM SIGGRAPH 2006 Courses (ACM, New York, 2006), pp. 39–54. doi:
  4. 4.
    T. Dey, K. Li, C. Luo, P. Ranjan, I. Safa, Y. Wang, Persistent heat signature for pose-oblivious matching of incomplete models, in Computer Graphics Forum, vol. 29 (Wiley Online Library, 2010), pp. 1545–1554Google Scholar
  5. 5.
    M. Ovsjanikov, A. Bronstein, M. Bronstein, L. Guibas, Shape google: a computer vision approach to isometry invariant shape retrieval, in IEEE 12th International Conference on Computer Vision Workshops (ICCV Workshops), 2009, Kyoto (IEEE, 2009), pp. 320–327Google Scholar
  6. 6.
    D. Raviv, M. Bronstein, A. Bronstein, R. Kimmel, Volumetric heat kernel signatures, in Proceedings of the ACM Workshop on 3D Object Retrieval, Firenze (ACM, 2010), pp. 39–44Google Scholar
  7. 7.
    S. Rosenberg, The Laplacian on a Riemannian Manifold: An Introduction to Analysis on Manifolds (Cambridge University Press, Cambridge, 1997)Google Scholar
  8. 8.
    J. Sun, M. Ovsjanikov, L. Guibas, A concise and provably informative multi-scale signature based on heat diffusion, in Proceedings of Eurographics Symposium on Geometry Processing (SGP), Berlin, 2009Google Scholar
  9. 9.
    H. Zhang, O. van Kaick, R. Dyer, Spectral mesh processing, in Computer Graphics Forum (Wiley, 2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Zuse Institute BerlinBerlinGermany
  2. 2.Institute of Science and Technology AustriaKlosterneuburgAustria
  3. 3.German Aerospace Center (DLR)BraunschweigGermany

Personalised recommendations