Derived Fields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 133

14.2.2.1 Definitions and Properties

Sauber et al. [ 22 ] introduce a gradient similarity measure (GSIM) between two
gradient fields f j that assumes high values when the gradients have similar

magnitude and direction. The measure is defined at each point as

∇ f i and ∇

r

s ( ∇ f i , ∇ f j ) = ( s d ( ∇ f i , ∇ f j ) · s m ( ∇ f i , ∇ f j )) , where

2

f i

∇ ∇ f j

s d ( ∇ f i , ∇ f j ) = , and

∇ f i ·∇ f j

∇ f i ·∇ f j

s m ( ∇ f i , ∇ f j ) = 4

2 .

( ∇ f i +∇ f j )

In the above expression, s d represents the similarity in gradient direction, s m rep-
resents the similarity in gradient magnitude, and the exponent r is a parameter that
determines the sensitivity of themeasure. The fields are normalized to have a common
range before computing gradients.
Edelsbrunner et al. [ 3 ] define a derived field that assumes high values when the
gradients are orthogonal to each other. The derived field, denoted by κ , is essentially

the length of the cross product between the two gradients vectors.
While the two fields are different in the sense that GSIM measures similarity
whereas

measures dissimilarity, both derived fields have many similarities besides
the fact that both are based on gradient comparison. Both GSIM and

κ

depend
on the scale and length of the gradients, are pointwise comparisons, and do not
distinguish between positive and negative correlation. The similarities imply that
both techniques are applicable to the same data sets. Gosink et al. [ 5 ] also compute
correlation between gradient fields to study the interactions between the different
pairs of scalar fields in multifield data. The inner product of the gradients of two
fields of interest is computed over principle level sets of a third field. They employ
this approach to study combustion in methane and hydrogen. The correlation field
proposed by Gosink et al. is similar to GSIM described above with the difference
being the domain over which the correlation field is computed.

κ

14.2.2.2 Applications

Figure 14.2 shows the derived field GSIM for two pairs of quantities measured in
the simulation of hurricane Isabel. The transfer function assigns non-zero opacity to
regions with values of GSIM greater than 0.9. Patterns in the derived field can help
in the analysis of various phenomena like fronts in the hurricane.
A visualization of helps in the study of the different phases in a combustion

simulation as shown in Fig. 14.3 . Three time steps are shown: the ignition, burn-
ing, and the final phase. The flame front is tracked by regions with large values of
κ

κ

computed for the scalar field pair pr og and H 2 , which represent the progress of