Image Processing Reference
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vector field topology by Otto et al. [ 11 - 13 ], where the uncertainty is incorporated
into the integration of stream lines by means of a Monte Carlo process. This leads to
a notion of critical points as distributions rather than distinct points as in the classic
case. In order to reflect the uncertainty inherent to brain diffusion MRI data, Schultz
et al. [ 19 ] introduced fuzzy versions of topological features for tensor fields based
on probabilistic tractography.
Strict definitions such as topology usually come as a part of a larger theoretical
framework with a number of properties and guarantees. In the beginning, newly
developed fuzzy analogs replace usually only certain parts of such a theoretical
framework. The development effort to rebuild thewhole framework in a fuzzymanner
may be very high. For the field of topology, it is for example unclear whether the
Morse inequalities still hold for uncertain topology. This leaves room for future
research, but it also shows that fuzzy analogs are usually not a full replacement for
the corresponding strict feature definitions—at least in the beginning.
Another example for a fuzzy analog is the Probabilistic Marching Cubes method
by Pthkow et al. [ 15 , 16 ], where the positional uncertainty of isocontours is evaluated.
Again, it remains a task for future research how this relates for example to contour
trees.
4.2.2 Aggregation of Features
As already mentioned, strict feature definitions usually produce a wealth of struc-
tures in the presence of noise or uncertainty. Aggregating features using statistical
methods may help in some applications to reveal the most dominating trends in a
data set. This has been done by Garth et al. [ 5 ] for tracked critical points in unsteady
3D flows. These features are curves in 4D and often prone to noise. A principal
component analysis of all space-time positions of all critical points has been used to
determine their principal spatial direction as well as their common center of move-
ment. This dimensionality reduction effectively reduced the amount of information
and the resulting visualizations aid in understanding the most dominating trends of
the data set.
4.2.3 Filtering of Features
A common method for dealing with a large number of mainly noise-induced features
is to filter them according to one or more criteria. The goal is to quantify each feature
point in terms of a certain “feature strength” and to keep only the most dominant
(parts of the) features.
In the following, we will detail this concept using the example of extremal
structures—lines and surfaces at which the scalar function value becomes mini-
mal or maximal with respect to the local neighborhood. These features are important
 
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