Derived Fields - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 146

Fig. 14.12 Example tensors and their corresponding vector field patterns [ 28 ]. © IEEE Reprinted,
with permission, from IEEE Transactions on Visualization and Computer Graphics 15(1)

Fig. 14.13 The tensor decomposition in Eq. ( 14.4 ) can be adapted to symmetric tensors. In this
example the symmetric tensor is the curvature tensor in the surface. Note that this tensor decom-
position can lead to surface classification and feature extraction [ 15 ]. © IEEE Reprinted, with
permission, from IEEE Transactions on Visualization and Computer Graphics 18(6)

The decomposition in Eq. ( 14.4 ) can also be used to symmetric tensor fields. In
this case γ r = 0 and the tensor can be rewritten as:

10
01

cos

θ sin θ

ρ φ + ρ φ

sin cos (14.5)

sin θ − cos θ

2
d

where

s again is the tensor magnitude. Like Eq. ( 14.4 ), this equation
is a special case of Eq. ( 14.2 ) where one of the three components disappears (the
anti-symmetric component). Nieser et al. [ 15 ] have applied this to the curvature
tensor to extract surface features for remeshing purposes. Figure 14.13 illustrates this

ρ = γ + γ