Mathematical Foundations of Uncertain Field Visualization - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 89

Chapter 9
Mathematical Foundations of Uncertain Field
Visualization

Gerik Scheuermann, Mario Hlawitschka, Christoph Garth and Hans Hagen

Abstract Uncertain field visualization is currently a hot topic as can be seen by
the overview in this topic. 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.

9.1 Introduction

The visualization of uncertain field data has attracted a lot of attention in recent
time. As practically no measured or simulated data is exact, visualization research
attempts to incorporate uncertainty in the images presented to the user. Despite
this undebated need, there has been only slow progress towards this goal. There
are many field visualization methods without an extension taking uncertainty into
account. We think that a major reason for this fact is a lack of knowledge regarding
the necessary mathematical description of uncertainty in the case of fields. As we
argue in this article, stochastic processes are a viable tool to describe uncertain
functions over continuous domains. Since stochastic processes are usually not part

( B ) · M. Hlawitschka
University of Leipzig, Leipzig, Germany
e-mail: [e-mail]
M. Hlawitschka
e-mail: [e-mail]
H. Hagen

G. Scheuermann

C. Garth
TU Kaiserslautern, Kaiserslautern, Germany
e-mail: [e-mail]
C. Garth
e-mail: [e-mail]

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