Derived Fields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 148

Higher-order tensors, i.e., tensors of a rank larger than two, are of great interests
to scientists and engineers in many application domains. For example, general rela-
tivity deals with higher-order tensors. Elasticity tensor, a fourth-order tensor, relates
the strain tensor (deformation) to the stress tensor (force). The spatial gradient of an
N -th order tensor is an N 1-th order tensor. This has been used by Delmarcelle and

Hesselink to classify degenerate points for symmetric second-order tensors [ 2 ]. A
special class of higher-order tensors have also been used to describe rotational sym-
metries on surfaces [ 16 ], with applications in pen-and-ink sketching [ 7 ], remesh-
ing [ 15 , 20 ], and regular texture and geometry synthesis on surfaces [ 15 ].
There have been a number of decomposition methods [ 12 ]. However, physical
interpretation of these decompositions as well as effective analysis and visualization
is still lacking. The only prominent work available at this point is [ 8 ].

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14.5 Conclusions

In this chapter we have examined applications and existing techniques on multi-
field visualization based on the notion of derived fields. The derived fields play an
important role in understanding relationships between multiple input fields. In addi-
tion, for a single input field, multiple derived fields can be generated as a result of
decomposition which have the potential of providing insights on the input field.
There are a number of future research directions that we believe are important
and can have major impact on multifield visualization:
1. Adaptation of pairwise similarity/dissimilarity as well as global alignment and
dependency measures to a set of heterogeneous fields.
2. A detailed study of the sensitivity of the derived fields to the mesh that represents
the domain. This study will be particularly useful if the input fields are specified
on different meshes representing a common domain.
3. An effective interface using derived fields that supports identification of (a) impor-
tant fields that can further analyzed in detail and (b) redundant fields that can be
discarded from further studies.
4. Integrating the derived fields with existing techniques for interactive exploration
like query-based visualization framework [ 5 ], focus+context, and show and brush
for visual analysis.

Acknowledgments Natarajan acknowledges support from the Department of Science and Tech-
nology, India under grants SR/S3/EECE/048/2007 and SR/S3/EECE/0086/2012. Zhang is partially
supported by National Science Foundation awards IIS-0546881, CCF-0830808 and IIS-0917308.