Mathematical Foundations of Uncertain Field Visualization

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.

1. 1.

   In general, we only need a complete probability space, i.e. some set \(\varOmega \) with a \(\sigma \)-algebra and a probability measure on this \(\sigma \)-algebra. Completeness means that any subset of a set with measure zero must be in the \(\sigma \)-algebra. One can construct a complete probability space from an arbitrary probability space by adding elements to the \(\sigma \)-algebra and defining the measure on these elements accordingly [4, Suppl. 2] without any change of practical relevance.

2. 2.

   The Borelalgebra is the smallest \(\sigma \)-algebra that contains all open and closed subsets. This ensures in our case that we can measure the probability for all subsets of interest in practical cases.

3. 3.

   A map is measurable if each preimage of a measurable set is measurable

4. 4.

   The case \(v=1\) means a scalar, \(v=d, d=2,3\) means a vector and the case \(v=d \times d =d^2,\) \(d=2,3\) describes a second order tensor.

5. 5.

   This condition removes subtle measurement problems without imposing restrictions of practical relevance, see Adler and Taylor [2, p. 8]. The concept was originally introduced by Doob [4] in his book on stochastic processes. In essence, it demands a dense countable subset \(D \subset P\), and a fixed null set \( N {\in } {\mathbb {S}}\) with \( {\mathbb {P}}(N)=0 \) such that for any closed \(B \subset {\mathbb {R}}^d\) and open \(I \subset P\)

   $$\{\omega | f(x,\omega ) \in B \forall x \in I\} \varDelta \{\omega | f(x,\omega ) \in B \forall x \in I \cap D\} \subset N $$

   with symmetric set difference \(\varDelta \).

6. 6.

   According to Doob [4, I.5, II.1] and going back to theorems by Kolmogorov, one needs to define probability distribution functions

   $$\begin{aligned} F_{x_1, \ldots ,x_n}(a_1, \ldots ,a_n) = {\mathbb {P}}(|x_1| \le a_1, \ldots ,|x_n| \le a_n) \end{aligned}$$

   for arbitrary finite tuples \((x_1, \ldots ,x_n)\) of points in D, such that the following rather obvious two consistency conditions hold for all finite subsets of points \(\{x_1, \ldots ,x_n\}\) and value bounds \(a_1, \ldots ,a_n \in \mathbb {R}\):

   $$ F_{x_1, \ldots ,x_n}(a_1, \ldots ,a_n) = F_{x_{\alpha _1}, \ldots ,x_{\alpha _n}} (a_{\alpha _1}, \ldots ,a_{\alpha _n})\quad \forall \text{ permutations } \alpha $$

   and

   $$ F_{x_1, \ldots ,x_m}(a_1, \ldots ,a_m) = \lim _{\lambda _j \rightarrow \infty , j\,=\,m+1, \ldots ,n} F_{x_1, \ldots ,x_n}(a_1, \ldots ,a_n)\quad \forall m < n $$

   We will use multivariate Gaussian distributions for this purpose in the next sections. This footnote illustrates that other distributions are possible.

7. 7.

   A normal distribution on \(\mathbb {R}\) is defined by a probability density function

   $$ \phi (x) = \frac{1}{\sqrt{2 \pi }\sigma } \exp ^{-\frac{(x-\mu )^2}{2\sigma ^2}} .$$

   \(\mu \) is the mean of the distribution and \(\sigma \) the standard deviation.

8. 8.

   In praxis, the covariances are either given or have to be estimated from several given sample fields. Obviously, this estimation might be a challenge in its own right as the number of positions is almost certainly larger than the number of sample fields. Pöthkow et al. [6] made some comments in this direction.

9. 9.

   In praxis, there will be eigenvalues very close to zero in the estimated covariance matrix which one might want to set to zero. Again, this is an obvious challenge outside the scope of this article.

Authors and Affiliations

1. University of Leipzig, Leipzig, Germany

   Gerik Scheuermann & Mario Hlawitschka

2. TU Kaiserslautern, Kaiserslautern, Germany

   Christoph Garth & Hans Hagen

Corresponding author

Correspondence to Gerik Scheuermann .

Editors and Affiliations

1. School of Computing Scientific Computing and Imaging Institu, University of Utah, Salt Lake City, Utah, USA

   Charles D. Hansen

2. e-Research Centre, University of Oxford, Oxford, United Kingdom

   Min Chen

3. School of Computing Scientific Computing and Imaging Institu, University of Utah, Salt Lake City, Utah, USA

   Christopher R. Johnson

4. Department of Computer Science, Stony Brook University, New York, New York, USA

   Arie E. Kaufman

5. Fachbereich Informatik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

   Hans Hagen

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