Image Processing Reference
In-Depth Information
tensor-valued variables, up to several dozens [
1
]. Therefore, it is not surprising that
multifield visualization is crucial for the understanding of flow data. This also holds
for the class of feature-based visualization methods, as some of the physically most
relevant features of flows rely on several physical quantities in their definition. For
example, some of the classical vortex detectors are based on the combination of
pressure and vorticity [
2
] or of velocity and vorticity [
18
]. Since vorticity can be
derived from the velocity field using a numerical gradient estimation technique, the
classification of such a method as a multifield technique might be questionable.
However, the velocity gradient tensor is typically an original result of a CFD solver.
In the case of vortex methods [
7
], it is even so that vorticity is an original variable,
while velocity is a derived one.
Multifield feature detectors can be realized in a number of ways. An efficient class
of methods requires only
local information
, i.e., the field values at a given point and,
possibly, some low-order derivatives of them. One such approach is reduction to a
scalar field, followed by an
isosurface
extraction. It was used by Levy et al. in their
definition of a vortex as a region where
helicity
[
18
] is high. Helicity is defined as the
scalar product of velocity and vorticity, and often normalized helicity is used where
vectors are normalized.
Multifields lend themselves also to
predictor-corrector
approaches. The vortex
detector developed by Banks and Singer [
2
] is based on the assumption that the vortex
axis roughly follows the vorticity vector while passing through pressure minima in
its cross section. Therefore, it is realized by taking steps along the vorticity vector
alternating with steps along the pressure gradient in the orthogonal space. A variant
of this method, proposed by Stegmaier et al. [
32
], replaced pressure by the
λ
2
vortex
indicator [
17
]. Both of these methods not only compute the vortex axis as a curve, but
they also provide the vortex hull, which is obtained by a radial search in the normal
plane of the vortex axis.
In many cases, a feature defined by a pair of variables can be expressed by the
par-
allel vectors operator
[
25
], which is a generic approach to extract line-like features
from a pair of two- or three-dimensional vector fields. This way, the aforementioned
methods [
2
,
18
] can be reformulated. Further examples are the criteria by Sujudi
and Haimes [
33
] for vortex axes in steady flow and its extension to unsteady flow
by Fuchs et al. [
11
]. Here, the vortex axis is defined as the locus where velocity and
acceleration are parallel vectors, subject to the additional condition that the velocity
gradient has a pair of complex conjugate eigenvalues.
Related to the parallel vectors technique is the extraction of
creases
(ridges and
valleys). Examples are the definition of vortex axes as valley lines of pressure by
Miura and Kida [
20
] and the definition of
Lagrangian coherent structures
as ridge
surfaces of the
finite-time Lyapunov exponent
, by Haller [
13
]. Extraction of creases
requires first and second derivatives of a scalar field. Therefore, this is not a typical
multifield method, but rather a method based on derived fields.
Fields of a multifield often represent different physical quantities. An interesting
technique for comparingmultiple scalar fields is to compare their gradients. For
k
n
scalar fields in
n
-space, the comparison measure proposed by Edelsbrunner et al. [
10
]
is the volume of the parallelotope spanned by the
k
gradient vectors. A feature can
≤
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