Definition of a Multifield - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 101

multiple quantities, which are somehow related to each other. Other examples are
imaging data acquired with different modalities or multiple runs of one simulation
exploring a complex parameter space.
Applying a side-by-side visualization using established methods for the single
fields might give a first insight into the data. But such an approach might be over-
whelming for the user and, even worse, it ignores the information resulting from
the interrelation of the different fields. The need for more integrated visualization
methods for such cases have lead to the development of a scientific area focusing
on “multifields”, whereby the term is used in very different contexts for different
applications.
Some instances of multifield visualization go back to the 20th century, e.g., visual-
ization involving different quantities of computational fluids dynamics (CFD) results
or volumetric scans using different modalities. The term multifield visualization has
been introduced in 2001 by Johnson et al. [ 1 ] who describe it as “an area ripe for
research [...] in which a scientist could visualize combinations of the above fields in
such a way as to see the interactions of the fields”. A recent overview of multifield
visualization is given by Obermaier in his PhD thesis [ 2 ].
To start a proper analysis of the current state of this area, the first step is to agree
on a basic definition for “multifields”. There are many possibilities for a definition
of such fields, which might be more or less appropriate for different applications.
10.2 Definitions
The purpose of this section is to define multifields in a way that is general enough
to cover most of its usages, distilling the shared properties, which we consider to be
the most important.
10.2.1 Fields
As a first step towards a definition of “multifields” as collections of somehow inter-
related fields, it is important to agree on a formal definition of “fields”.
In visualization, the term “field” has been adopted as it is used in physics where it
describes a quantity which is associated to each point in space-time. Therefore, con-
fusion with other meanings of the term, such as for algebraic structures, is unlikely.
The prevalent representatives in visualization are scalar fields, vector fields and ten-
sor fields. But the notion of fields has also been extended to functions or distributions
linked to points in space-time. Essential for the understanding of fields is the assump-
tion of some continuity of the underlying space (domain) as well as the described
phenomenon (range). This is in contrast to data inherently defined over a discrete
space or abstract data. The continuity assumptions together with distance measures
facilitates the use of analysis methods based on derivatives.