Top

Published in:

2014 | OriginalPaper | Chapter

9. Mathematical Foundations of Uncertain Field Visualization

Authors : Gerik Scheuermann, Mario Hlawitschka, Christoph Garth, Hans Hagen

Published in: Scientific Visualization

Publisher: Springer London

Activate our intelligent search to find suitable subject content or patents.

search-config

AI-assisted search

Off

Abstract

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.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

inform now

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

inform now

previous chapter Fuzzy Fibers: Uncertainty in dMRI Tractography

next chapter Definition of a Multifield

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.

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.

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

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.

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 $.

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.

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.

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.

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.

Adler, R.: The Geometry of Random Fields. Wiley, Chichester (1981)

Adler, R., Taylor, J.: Random Fields and Geometry. Springer, New York (2007)

Adler, R., Taylor, J.: Topological complexity of smooth random functions. Lecture Notes in Mathematics, vol. 2019. Springer, Heidelberg (2011)

Doob, J.L.: Stochastic Processes. Wiley, New York (1953)

Pöthkow, K., Hege, H.-C: Positional uncertainty of isocontours: condition analysis and probabilistic measures. IEEE Trans. Vis. Comput. Graphics 17(10), 1393–1406 (2011)

Pöthkow, k., Weber, B., Hege, H.-C: Probabilistic marching cubes. Comput. Graphics Forum 30(3), 931–940 (2011)

Rasmussen, C.E., Williams, C.K.I.: Gaussian Processes for Machine Learning. The MIT Press, Cambridge (2006)

Title: Mathematical Foundations of Uncertain Field Visualization
Authors: Gerik Scheuermann
Mario Hlawitschka
Christoph Garth
Hans Hagen
Publisher: Springer London
Book: Scientific Visualization
Print ISBN: 978-1-4471-6496-8

Electronic ISBN: 978-1-4471-6497-5

Copyright Year: 2014
DOI: https://doi.org/10.1007/978-1-4471-6497-5_9

Springer Professional

Abstract

Please log in to get access to your license.

Dont have a licence yet? Then find out more about our products and how to get one now:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"

Premium Partner