Feature Analysis in Multifields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 188

Rautek et al. [ 30 ] used fuzzy logic on multiple derived properties such as gradient
and curvature. Because this implicitly uses ranges in each, the effect of this is to
apply blocks or Gaussian ellipsoids in the histograms to define features, even if they
are not shown directly. Johnson and Huang [ 19 ] compute histograms separately for
individual properties, and compare themusing boolean predicates to identify features
of interest.
Song et al. [ 38 ] provided a volume-renderer for meteorological data that incorpo-
rate an equation parser so that scientists could compute derived properties at runtime
in order to identify features. In a similar vein, Glatter et al. [ 14 ] assumed that the
underlying numerical data could be represented in textual form, then used grep-style
pattern matching to identify features. Gosink et al. [ 15 ] also took a similar approach,
giving it the name of Query-Driven Visualization.
Maciejewski et al. [ 27 ] combined this with clustering to define features as arbitrary
shapes in the isovalue-gradient histogram. It thus becomes clear that it is feasible,
although not necessarily desirable, to discuss clustering in the context of n-D his-
tograms.
More recently, Lindholm et al. [ 26 ] observed that small spatial neighborhoods
in medical data normally intersect only a few distinct materials. Thus, by identify-
ing ellipsoidal Gaussian peaks in local histograms, material types are identified and
mapped to transfer functions. These peaks are mapped manually, except in simple
cases, where they can be detected by iterative peak detection, or in established work-
flows, where templates from other data sets can be adapted. Similarly, Correa and
Ma [ 7 ] used manual sums of Gaussians to define features in 2D histograms.
18.4 Manifold Features
We have just seen that features can be detected in either the domain of the function or
in its range. Interestingly, this allows us to see in retrospect that multiple researchers
have converged on similar solutions: defining features to be compact sets (preferably
peaks) in n-D histograms or n-D feature space.
More recently, work that relates n-D feature space or n-D histograms back to the
domain started with the observation by Carr et al. [ 5 ] that statistics of data sampled
from a continuous function are directly related to geometric properties of contours
in the domain. Subsequent work by Scheidegger et al. [ 36 ] refined the formulation
to include a missing gradient factor. Simultaneously, Bachthaler and Weiskopf [ 1 ]
showed that the observation could be extended to multifields, and that doing so
resulted in the continuous scatterplot : i.e. that multi-dimensional histograms are
discrete approximations of a projection of the function manifold to the range.
Subsequently, Lehman and Theisel [ 24 ] observed that a prominent feature of
continuous scatterplots was the presence of visible edges, or boundary curves , and
extracted these curves with Canny edge-detection. This ties in with recent work
by Duffy et al. [ 9 ] which formalises the relationship of histogram statistics with the
functionmanifold and GeometricMeasure Theory [ 13 ], the underlyingmathematical