Mathematical Foundations of Uncertain Field Visualization - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 90

of the standard curriculum in computer science and sometimes even mathematics,
visualization researchers are not very familiar with this non-trivial subject.
In many cases, the field visualization problem consists of a finite set of given
positions where the field value is known. It shall be noted that this holds for scalar,
vector, and tensor fields. Before most field visualization methods are applied, an
interpolation of these values is defined creating a continuous field over the whole
continuous domain. The uncertain field visualization problem is very similar: One
is given a finite set of positions with a (known or estimated) distribution of the
(unknown) field value at each position. We consider the prominent case of Gaussian
distributions in this article and show that all the well-known interpolation methods in
visualization can be used in this case to define the uncertain field over the continuous
domain as a Gaussian process. This rarely known fact emphasizes the potential of
stochastic processes as model for uncertain fields in visualization research.

9.2 Stochastic Processes

We want to describe a (scalar, vector or tensor) field over some closed domain
D

d

⊂ R , d = 1 , 2 , or 3, that depends on some unknown (typically high dimensional)

parameter ω ∈ Ω . The whole uncertainty is contained in this parameter: If we know

u ,

the parameter ω , we know the field. To keep things simple, we assume that Ω = R

but that is not necessary. 1 In addition, we assume that

Ω

, is known i.e. the number

and type of parameters that determine our field.
In a first step, we need a probability measure on Ω .As Ω contains an uncountable

u ,theBore-

number of elements, we use a σ -algebra S on Ω . Because of Ω = R

u

is a natural choice. 2 Furthermore, we need a probability measure

lalgebra B ( R )

P : S →[ 0 , 1 ] . As usually, this means that the probability for ω ∈ A ⊂ Ω is

P ( .Again, we assume that this probability measure is known .

In our second step, we define a random variable

A ) ∈[ 0 , 1 ]

v

X : Ω → R

as measurable 3 map where the

v v

. Essen-
tially, this is a usual (i.e. deterministic) function, assigning each (unknown) parameter

σ -algebra on R is the Borelalgebra B ( R )

1 In general, we only need a complete probability space, i.e. some set

Ω with a σ -algebra and a

probability measure on this σ -algebra. Completeness means that any subset of a set with measure

zeromustbeinthe -algebra. One can construct a complete probability space from an arbitrary

probability space by adding elements to the

σ

-algebra and defining the measure on these elements
accordingly [ 4 , Suppl. 2] without any change of practical relevance.
2 The Borelalgebra is the smallest

σ

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

σ