Definition of a Multifield - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 102

In practice the data mostly origin from simulations and measurements and are
only available on a finite subset of the domain. Due to the continuity assumptions
it is principally possible to evaluate the field over the entire domain using some
interpolation or approximation method. To account for the fact that data acquisition
in general involves further parameters, the continuous field domain is augmented by
an additional parameter space.

10.2.2 Definition of a Field

A field F is defined as a function

F : D → R .

• Domain D

The domain consists a Cartesian product of a finite-dimensional metrical space
and some parameter space ( D P ). Since for many applications time plays a dis-
tinguished role we further separate the temporal dimension ( D T ) from the spatial
dimensions ( D S ).

D S D T D P D = × × .

•

In the most general form the range R of a field F is defined as the Cartesian product
of a metric space R M and a set of categorical values .

R M

R = × .

The general case for R M are finite-dimensional spaces representing scalars, vectors
or tensors, which can be considered as subsets of

n

. But this
definition also allows for the more general case of function or distribution spaces.
Categorical values

R for some n ∈ N

are in general of discrete nature and include classifications

or binary markers.

10.2.3 Multifields

Based on this definition of fields a multifield M is defined as a set of fields

M ={ F 1 , F 2 ,..., F r } , r ∈ N

F i : D i → R i (10.1)

D i D i D i

= × ×

D i

R i R i = ×