Feature Analysis in Multifields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 189

theory of integration. As part of the connection between integration, mensuration and
statistics, these authors identified the role in integration of the multiplicity function
which measures the number of distinct input values mapping to a single output value.
From this viewpoint, it then becomes apparent that the boundary curves detected by
Lehman and Theisel [ 24 ] are the folds in the manifold when projected into the
range. It then follows that this form of feature detection is related to domain-based
topological analysis.
Nagaraj and Natarajan [ 28 ] have also recently defined a variation density function
for multifields: while the details have not been explored, it seems likely that this
function is also connected to mensuration.
Heinrich andWeiskopf [ 18 ] have also extended continuous scatterplots to parallel
coordinates: by implication feature-recognition techniques are likely to develop in
this area as well.
In summary, then, three broad trends are visible in recent work in this area: that
transfer function papers often rely in practice on human perception to detect features
in multi-dimensional histograms, that theoretical work is providing strong linkages
between the multi-dimensional histograms and the underlying multifield, and that
feature recognition methods are increasingly exploiting these linkages to identify
significant features.
18.5 Feature Overlap
Since feature detection algorithms now exist in scalar and vector fields, one simple
approach is to detect features separately in each field, then overlap them spatially to
see how well they match.
Woodring and Shen [ 41 ] allowed the user to perform set operations on
(iso-)surfaces defined by individual properties. Similarly, Navrátil et al. [ 29 ] mapped
isosurfaces of different properties to different colours. In each case, we can see that
isovalued-features (i.e. isosurfaces) are in effect being overlapped either logically or
visually.
Schneider et al. [ 37 ] took the next step by using contour trees to recognize features
of each of two fields separately. For each pair of features, the spatial overlap was
computed as a similarity measure. An interface then showed all pairings above a
similarity threshold as a bipartite graph for user selection. Heine et al. [ 17 ] then
extended this to arbitrary numbers of fields, showing strong correlations as cliques.
18.6 Joint Feature Analysis
We have noted above that scalar topological analysis can be applied to one or more
properties of the multifield. The obvious next question is whether there exist forms
of topological analysis that can be applied directly to the multifield. To date, several