Mathematical Foundations of Uncertain Field Visualization - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 95

9.5 General Interpolation

We turn now to a realistic interpolation scenario. We consider some closed domain
D

d . We assume that we are given N positions p 1

p N D . At these posi-

tions, we are given N uncertain v -dimensional values with normal distributions, say

⊂ R ,..., ∈

W i

i

C i ∼ N v (μ , ), ∀ i = 1 ,..., N

where C i ∈ R ( v × v ) denotes the covariances between the dimensions at a single

position. We still assume that the N values are independent. Our interpolationmethod
is given by N (deterministic) weight functions

p j φ i : D → R , ∀ i = 1 ,..., N with φ i ( ) = δ ij

with Kronecker . This is the typical case in finite element formulations and for

almost all grid based field data in visualization.
We define our probability space via

δ

N × v , Borelalgebra

Ω = R B (Ω) and

⎛
⎝

⎞
⎠ ,

⎛
⎝

⎞
⎠

1

C 1

μ

.
μ

. . . C N

P ∼ N (μ, C ), μ = C =

N

as probability measure. Our uncertain field f is defined as

N

v D i

f : Ω → ( R ) , f (ω, x ) = f ω ( x ) = f x (ω) = ( f (ω)) x =

1 ω

φ i ( x ).

i =

Fixing position x ∈ D , we get a random variable

v

f x : Ω → R

that describes the distribution of values at that position as a Gaussian distribution

N

j v × v

f x ∼ N (μ( x ), C ( x )), μ( x ) =

1 μ

φ j ( x ), C ( x ) ∈ R ,

i =

N

C kl φ

2

C kl (

) =

i (

).

x x

i = 1

Looking at the whole uncertain field again, we have the expectation function