On the (Un)Suitability of Strict Feature Definitions for Uncertain Data - Scientific Visualization: Uncertainty, Multifield, Biomedical, and Scalable Visualization - page 45

in many applications. For example, vortex core lines can be found in a flow data
set as lines where the Q -criterion becomes maximal [ 18 ], or pressure minimal [ 10 ].
Strongest particle separation in an unsteady flow is denoted by surfaces where the
Finite Time Lyapunov Exponent (FTLE) becomes maximal [ 7 ].
There are two types of approaches for extracting extremal structures: the local
analysis due to ridges/valleys and the global point of view by means of topology.

4.2.3.1 Filtering of Ridges and Valleys

The extraction of ridges and valleys requires derivatives of the examined scalar field
f . The commonly used Height Ridge definition [ 2 ] builds on the first and second
derivatives of f , i.e., the gradient g and the Hessian H . As elegantly formulated by
Peikert and Sadlo [ 14 ], ridge lines in a 2D scalar field are found as a subset of the zero
contour of the derived field d

. Noise in the original data as well as its
amplification in the derivatives usually cause a wealth of spurious extraction results
for ridge and valley lines. Therefore, filtering of extraction results ismandatory. Many
filtering criteria have been proposed in the literature to quantify the importance of
ridges: the feature strength of a ridge, the height of a ridge, the angle between the
gradient and a ridge line segment, or the length/area of a connected component [ 14 ,
17 ].

= det ( g | Hg )

4.2.3.2 Filtering of Topological Structures

Topology provides a different means for extracting extremal structures. The Morse-
Smale (MS) complex of a 2D scalar field f is comprised of points and lines, which
provide a segmentation of the domain intomonotone cells [ 9 ], i.e., regions in which f
behaves monotonically increasing from a local minimum to a local maximum. Each
cell is cornered by critical points (a minimum, a maximum, and saddle points). The
boundaries between cells are provided by separation lines—so-called separatrices .
They are extremal lines—the topological analog to ridges/valleys.
Two types of approaches exist to extract the MS complex. The continuous
approach [ 8 , 20 ] builds on the gradient g and Hessian H of f .Noisein f and
noise amplification in g and H pose a numerical challenge for this approach just as
much as for ridges. The discrete approach due to Forman's discrete Morse theory [ 4 ]
works on sampled data only, but does not require any derivatives or other numerical
computations, since it describes the MS complex in a purely combinatorial fashion.
So while noise is less of a problem due to the exclusion of derivatives, spurious
extraction results still show up because of the noise level in the original data f .
Hence, filtering is necessary. A well-accepted filtering criterion for critical points
is persistence due to Edelsbrunner et al. [ 3 ]. The separation lines of a 2D scalar
field can be filtered using a closely related measure called separatrix persistence
[ 21 ], which determines the feature strength of a separatrix or parts thereof. It was
originally introduced to filter salient edges on surfaces meshes. Figure 4.2 shows an