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This work is concerned with adaptive screen-space sampling for volume ray-casting. The goal is to reduce the
number of rays being cast into the scene and, thus, the overall number of sampling points. We guarantee reliable
images through explicit error control using an error estimator that is founded in the field of finite element methods
(FEM). FEM theory further provides a well-founded theory to prove the efficiency of the presented algorithm via
convergence analysis. We, therefore, compare the convergence behavior of our method against uniform subdivisions
and a refinement scheme that was presented in the context of CPU volume ray-casting. Minimizing
the number of sampling points is of interest for rendering large datasets where each evaluation might need an expensive
decompression. Furthermore, with increasing screen resolutions high-resolution images are created more
efficiently with our method.

We propose a combinatorial algorithm to track critical points of 2D
time-dependent scalar fields. Existing tracking algorithms such as Feature
Flow Fields apply numerical schemes utilizing derivatives of the data,
which makes them prone to noise and involve a large number of computational
parameters. In contrast, our method is robust against noise
since it does not require derivatives, interpolation, and numerical integration.
Furthermore, we propose an importance measure that combines the
spatial persistence of a critical point with its temporal evolution. This
leads to a time-aware feature hierarchy, which allows us to discriminate
important from spurious features. Our method requires only a single,
easy-to-tune computational parameter and is naturally formulated in an
out-of-core fashion, which enables the analysis of large data sets. We apply
our method to a number of data sets and compare it to the stabilized
continuous Feature Flow Field tracking algorithm.

Memory-Efficient Computation of Persistent Homology for 3D Images using Discrete Morse Theory
(2011)

A framework is proposed for extracting features in 2D transient flows, based on the acceleration field to ensure Galilean invariance. The minima of the acceleration magnitude, i.e. a superset of the acceleration zeros, are extracted and discriminated into vortices and saddle points --- based on the spectral properties of the velocity Jacobian. The extraction of topological features is performed with purely combinatorial algorithms from discrete computational topology. The feature points are prioritized with persistence, as a physically meaningful importance measure. These features are tracked in time with a robust algorithm for tracking features. Thus a space-time hierarchy of the minima is built and vortex merging events are detected. The acceleration feature extraction strategy is applied to three two-dimensional shear flows: (1) an incompressible periodic cylinder wake, (2) an incompressible planar mixing layer and (3) a weakly compressible planar jet. The vortex-like acceleration feature points are shown to be well aligned with acceleration zeros, maxima of the vorticity magnitude, minima of pressure field and minima of λ2.