Mathematical Foundations of Uncertain Field Visualization

  • Gerik Scheuermann
  • Mario Hlawitschka
  • Christoph Garth
  • Hans Hagen
Part of the Mathematics and Visualization book series (MATHVISUAL)


Uncertain field visualization is currently a hot topic as can be seen by the overview in this book. This article discusses a mathematical foundation for this research. To this purpose, we define uncertain fields as stochastic processes. Since uncertain field data is usually given in the form of value distributions on a finite set of positions in the domain, we show for the popular case of Gaussian distributions that the usual interpolation functions in visualization lead to Gaussian processes in a natural way. It is our intention that these remarks stimulate visualization research by providing a solid mathematical foundation for the modeling of uncertainty.


Probability Space Covariance Function Gaussian Process Multivariate Gaussian Distribution Continuous Domain 
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.


  1. 1.
    Adler, R.: The Geometry of Random Fields. Wiley, Chichester (1981)Google Scholar
  2. 2.
    Adler, R., Taylor, J.: Random Fields and Geometry. Springer, New York (2007)Google Scholar
  3. 3.
    Adler, R., Taylor, J.: Topological complexity of smooth random functions. Lecture Notes in Mathematics, vol. 2019. Springer, Heidelberg (2011)Google Scholar
  4. 4.
    Doob, J.L.: Stochastic Processes. Wiley, New York (1953)Google Scholar
  5. 5.
    Pöthkow, K., Hege, H.-C: Positional uncertainty of isocontours: condition analysis and probabilistic measures. IEEE Trans. Vis. Comput. Graphics 17(10), 1393–1406 (2011)Google Scholar
  6. 6.
    Pöthkow, k., Weber, B., Hege, H.-C: Probabilistic marching cubes. Comput. Graphics Forum 30(3), 931–940 (2011)Google Scholar
  7. 7.
    Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006)Google Scholar

Copyright information

© Springer-Verlag London 2014

Authors and Affiliations

  • Gerik Scheuermann
    • 1
  • Mario Hlawitschka
    • 1
  • Christoph Garth
    • 2
  • Hans Hagen
    • 2
  1. 1.University of LeipzigLeipzigGermany
  2. 2.TU KaiserslauternKaiserslauternGermany

Personalised recommendations