Feature-Based Visualization of Multifields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 181

now be defined to consist of the places where all gradients are similar with respect
to this measure. In the given application example, a pair of scalar quantities from a
combustion simulation was used, and the resulting features represent the frame front.
Nagaraj et al. [ 22 ] use as an alternative comparison measure the norm of the matrix
composed of the gradient vectors.
Feature definitions can involve, besides physical quantities, also geometric infor-
mation. An example are the spatial coordinates, which can serve to define features
restricted to regions-of-interest. A less trivial information, that deserves to be called
a “field” within the multifield, is the wall distance , the distance of each data point to
the nearest point on a solid boundary. Wall distance is required for some turbulence
models, therefore it is provided as output by most CFD solvers. It can be used to
focus on features that are either close to walls or distant from walls. Furthermore,
wall distance is needed for estimating limits, such as the wall-shear stress , a quantity
which is of importance, e.g., for the study of aneurysms in medical data [ 23 ]. The
wall-shear stress is amenable to 2D vector field topology , and the latter can be com-
bined with the 3D topology of the velocity field, thus making vector field topology
a two-field method [ 26 ].

17.3.3 Statistical Features

In multifield data sets with homogeneous field types, i.e., collections of scalar fields,
the existence of a common dimensionality allows the application of statistical meth-
ods for visualization purposes. Especially in this context of the visualization of sta-
tistics of fields, data is often called multi-variate data ,see[ 5 ]. While most features
defined in this manner are representatives of the group of combined features, as is the
case for multifield feature detectors or operators, some techniques compute statistics
over time or space rather than different fields.
The most basic geometric feature in a scalar field is an isocontour or, in a three-
dimensional setting, an isosurface. Their mathematical and computational simplic-
ity makes them prime for feature analysis in sets of scalar fields. Gosink et al. [ 12 ]
perform statistical multifield visualization on multiple scalar fields by extracting
representative principle isosurfaces of one of the scalar fields and color it accord-
ing to correlation of a pair of two other scalar fields present in the data set. These
principle isosurfaces correspond to contours of the most frequently occurring scalar
values. The required statistics for frequency determination are gathered by scalar-
value histogram computation. The variation of a collection of scalar fields is another
statistical quantity that can contribute to multifield visualization. In the work by
Nagaraj et al. [ 21 ] relevant isovalues are identified by computing and plotting the
variation density function of multiple scalar fields in a data set. Their method is able
to compare n k scalar fields in a k -dimensional data set and helps in identify-

ing characteristic isosurfaces in sets of scalar fields. Two scalar field comparison
methods that extend to n dimensions are presented by Sauber et al. [ 30 ], namely
gradient similarity and local correlation coefficient . The large number of correlation

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