In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration.

In this thesis, we introduce several tools, each coming from a different branch of mathematics, for analyzing real vector fields and their associated flows. Beginning with a discussion about generalized vector field decompositions, that mainly have been derived from the classical Helmholtz-Hodge-decomposition, we decompose a field into a kernel and a rest respectively to an arbitrary vector-valued linear differential operator that allows us to construct decompositions of either toroidal flows or flows obeying differential equations of second (or even fractional) order and a rest. The algorithm is based on the fast Fourier transform and guarantees a rapid processing and an implementation that can be directly derived from the spectral simplifications concerning differentiation used in mathematics. Moreover, we present two combinatorial methods to process 3D steady vector fields, which both use graph algorithms to extract features from the underlying vector field. Combinatorial approaches are known to be less sensitive to noise than extracting individual trajectories. Both of the methods are extensions of an existing 2D technique to 3D fields. We observed that the first technique can generate overly coarse results and therefore we present a second method that works using the same concepts but produces more detailed results. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow. Existing methods for analyzing separation of streamlines are often restricted to a finite time or a local area. In the frame of this work, we introduce a new method that complements them by allowing an infinite-time-evaluation of steady planar vector fields. Our algorithm unifies combinatorial and probabilistic methods and introduces the concept of separation in time-discrete Markov chains. We compute particle distributions instead of the streamlines of single particles. We encode the flow into a map and then into a transition matrix for each time direction. Finally, we compare the results of our grid-independent algorithm to the popular Finite-Time-Lyapunov-Exponents and discuss the discrepancies. Gauss\'' theorem, which relates the flow through a surface to the vector field inside the surface, is an important tool in flow visualization. We are exploiting the fact that the theorem can be further refined on polygonal cells and construct a process that encodes the particle movement through the boundary facets of these cells using transition matrices. By pure power iteration of transition matrices, various topological features, such as separation and invariant sets, can be extracted without having to rely on the classical techniques, e.g., interpolation, differentiation and numerical streamline integration.

In vielen Forschungsbereichen wie Medizin, Natur- oder Ingenieurwissenschaften spielt die wissenschaftliche Visualisierung eine wichtige Rolle und hilft Wissenschaftlern neue Erkenntnisse zu gewinnen. Der Hauptgrund hierfür ist, dass Visualisierungen das Unsichtbare sichtbar machen können. So können Visualisierungen beispielsweise den Verlauf von Nervenfasern im Gehirn von Probanden oder den Luftstrom um Hindernisse herum darstellen. Diese Arbeit trägt insbesondere zum Teilgebiet der Strömungsvisualisierung bei, welche sich mit der Untersuchung von Prozessen in Flüssigkeiten und Gasen beschäftigt. Eine beliebte Methode, um Einblicke in komplexe Datensätze zu erhalten, besteht darin, einfache und bekannte Strukturen innerhalb eines Datensatzes aufzuspüren. In der Strömungsvisualisierung führt dies zum Konzept der lokalen Linearisierung und Linearität im Allgemeinen. Dies liegt daran, dass lineare Vektorfelder die einfachste Form von nicht-trivialen Feldern darstellen und diese sehr gut verstanden sind. In der Regel werden simulierte Datensätze in einzelne Zellen diskretisiert, welche auf linearer Interpolation basieren. Beispielsweise können auch stationäre Punkte in der Vektorfeldtopologie mittels linearen Strömungsverhaltens charakterisiert werden. Daher ist Linearität allgegenwärtig. Durch das Verständnis von lokalen linearen Strömungsverhalten in Vektorfeldern konnten verschiedene Visualisierungsmethoden erheblich verbessert werden. Ähnliche Erfolge sind auch für andere Methoden zu erwarten. In dieser Arbeit wird das Konzept der Linearität in der Visualisierung weiterentwickelt. Zunächst wird eine bestehende Definition von linearen Nachbarschaften hin zu affin-linearen Nachbarschaften erweitert. Affin-lineare Nachbarschaften sind Regionen mit einem überwiegend linearem Strömungsverhalten. Es wird eine detaillierte Diskussion über die Definition sowie die gewählten Fehlermaße durchgeführt. Weiterhin wird ein Region Growing-Verfahren vorgestellt, welches affin-lineare Nachbarschaften um beliebige Positionen bis zu einem bestimmten, benutzerdefinierten Fehlerschwellwert extrahiert. Um die lokale Linearität in Vektorfeldern zu messen, wird ein komplementärer Ansatz, welcher die Qualität der bestmöglichen linearen Näherung für eine gegebene n-Ring-Nachbarschaft berechnet, diskutiert. In einer ersten Anwendung werden affin-lineare Nachbarschaften an stationären Punkten verwendet, um deren Einflussbereich sowie ihre Wechselwirkung mit der sie umgebenden, nichtlinearen Strömung, aber auch mit sehr nah benachbarten stationären Punkten zu visualisieren. Insbesondere bei sehr großen Datensätzen kann die analytische Beschreibung der Strömung innerhalb eines linearisierten Bereichs verwendet werden, um Vektorfelder zu komprimieren und vorhandene Visualisierungsansätze zu beschleunigen. Insbesondere sollen eine Reihe von Komprimierungsalgorithmen für gitterbasierte Vektorfelder verbessert werden, welche auf der sukzessiven Entfernung einzelner Gitterkanten basieren. Im Gegensatz zu vorherigen Arbeiten sollen affin-lineare Nachbarschaften als Grundlage für eine Segmentierung verwendet werden, um eine obere Fehlergrenze bereitzustellen und somit eine hohe Qualität der Komprimierungsergebnisse zu gewährleisten. Um verschiedene Komprimierungsansätze zu bewerten, werden die Auswirkungen ihrer jeweiligen Approximationsfehler auf die Stromlinienintegration sowie auf integrationsbasierte Visualisierungsmethoden am Beispiel der numerischen Berechnung von Lyapunov-Exponenten diskutiert. Zum Abschluss dieser Arbeit wird eine mögliche Erweiterung des Linearitätbegriffs für Vektorfelder auf zweidimensionalen Mannigfaltigkeiten vorgestellt, welche auf einer adaptiven, atlasbasierten Vektorfeldzerlegung basiert. / In many research areas, such as medicine, natural sciences or engineering, scientific visualization plays an important role and helps scientists to gain new insights. This is because visualizations can make the invisible visible. For example, visualizations can reveal the course of nerve fibers in the brain of test persons or the air flow around obstacles. This thesis in particular contributes to the subfield of flow visualization, which targets the investigation of processes in fluids and gases. A popular way to gain insights into complex datasets is to identify simple and known structures within a dataset. In case of flow visualization, this leads to the concept of local linearizations and linearity in general. This is because linear vector fields represent the most simple class of non-trivial fields and they are extremely well understood. Typically, simulated datasets are discretized into individual cells that are based on linear interpolation. Also, in vector field topology, stationary points can be characterized by considering the local linear flow behavior in their vicinity. Therefore, linearity is ubiquitous. Through the understanding of local linear flow behavior in vector fields by applying the concept of local linearity, some visualization methods have been improved significantly. Similar successes can be expected for other methods. In this thesis, the use of linearity in visualization is investigated. First, an existing definition of linear neighborhoods is extended towards the affine linear neighborhoods. Affine linear neighborhoods are regions of mostly linear flow behavior. A detailed discussion of the definition and of the chosen error measures is provided. Also a region growing algorithm that extracts affine linear neighborhoods around arbitrary positions up to a certain user-defined approximation error threshold is introduced. To measure the local linearity in vector fields, a complementary approach that computes the quality of the best possible linear approximation for a given n-ring neighborhood is discussed. As a first application, the affine linear neighborhoods around stationary points are used to visualize their region of influence, their interaction with the non-linear flow around them as well as their interaction with closely neighbored stationary points. The analytic description of the flow within a linearized region can be used to compress vector fields and accelerate existing visualization approaches, especially in case of very large datasets. In particular, the presented method aims at improving over a series of compression algorithms for grid-based vector fields that are based on edge collapse. In contrast to previous approaches, affine linear neighborhoods serve as the basis for a segmentation in order to provide an upper error bound and also to ensure a high quality of the compression results. To evaluate different compression approaches, the impact of their particular approximation errors on streamline integration as well as on integration-based visualization methods is discussed on the example of Finite-Time Lyapunov Exponent computations. To conclude the thesis, a first possible extension of linearity to fields on two-dimensional manifolds, based on an adaptive atlas-based vector field decomposition, is given.