Time-Dependent 2-D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures

This paper presents an approach to a time-dependent variant of the concept of vector field topology for 2-D vector fields. Vector field topology is defined for steady vector fields and aims at discriminating the domain of a vector field into regions of qualitatively different behaviour. The presented approach represents a generalization for saddle-type critical points and their separatrices to unsteady vector fields based on generalized streak lines, with the classical vector field topology as its special case for steady vector fields. The concept is closely related to that of Lagrangian coherent structures obtained as ridges in the finite-time Lyapunov exponent field. The proposed approach is evaluated on both 2-D time-dependent synthetic and vector fields from computational fluid dynamics

Extraction of topological structures in 2D and 3D vector fields

feature extraction, feature tracking, vector field visualizationMagdeburg, Univ., Fak. für Informatik, Diss., 2008von Tino WeinkaufZsfassung in dt. Sprach

The State of the Art in Flow Visualization: Partition-Based Techniques

Flow visualization has been a very active subfield of scientific visualization in recent years. From the resulting large variety of methods this paper discusses partition-based techniques. The aim of these approaches is to partition the flow in areas of common structure. Based on this partitioning, subsequent visualization techniques can be applied. A classification is suggested and advantages/disadvantages of the different techniques are discussed as well

Spectral, Combinatorial, and Probabilistic Methods in Analyzing and Visualizing Vector Fields and Their Associated Flows

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

Topological analysis, visualization, and design of vector fields on surfaces

Analysis, visualization, and design of vector fields on surfaces have a wide variety of major applications in both scientific visualization and computer graphics. On the one hand, analysis and visualization of vector fields provide critical insights to the flow data produced from simulation or experiments of various engineering processes. On the other hand, many graphics applications require vector fields as input to drive certain graphical processes. This thesis addresses vector field analysis and design for both visualization and graphics applications. Topological analysis of vector fields provides the qualitative (or structural) information of the underlying dynamics of the given vector data, which helps the domain experts identify the critical features and behaviors efficiently. In this dissertaion, I introduce a more complete vector field topology called Entity Connection Graph (ECG) by including periodic orbits, an essential component in vector field topology. An efficient technique for periodic orbit extraction is introduced and incorporated into the algorithm for ECG construction. The analysis results are visualized using the improved evenly-spaced streamline placement with all separation features being highlighted. This is the first time that periodic orbits have been extracted from surface flows. Through applications to engine simulation datasets, I demonstrate how the extracted topology helps engineers interpret the flow data that contains certain desirable behaviors which indicate the ideal engineering process. Accuracy is typically of paramount importance for visualization and analysis tasks. However, the trajectory-based vector field topology approaches are sensitive to small perturbations such as error and noise which are contained in the given data and introduced during data acquisition and processing. This makes rigorous interpretation of vector field topology and flow dynamics difficult. To overcome that, I advocate the use of Morse decomposition to define a more reliable vector field topology called Morse Connection Graph (MCG). In particular, I present the pipeline of Morse decomposition of an input vector field. A technique based on the idea of [tau]-map is introduced to produce desirably fine Morse decompositions of vector fields. To address the issue of slow performance of the global [tau]-map framework, I describe a hierarchical MCG refinement framework. It enables the [tau]-map approach to be conducted within a Morse set of interest which greatly reduces the computation cost and leads to faster analysis. It is my hope that the work on Morse decomposition will invoke the investigation of other similar data analysis problems such as scalar field and tensor field analysis. The techniques of time-independent vector field design have been well-studied. However, there is little attention on the systematic design of time-varying vector fields on surfaces. This dissertation addresses this by developing a design system that allows the creation and modification of time-varying vector fields on surfaces. More specifically, I present a number of novel techniques to enable efficient design over important characteristics in the vector field such as singularity paths, pathlines, and bifurcations. These vector field features are used to generate a vector field by either blending basis vector fields or performing a constrained optimization process. Unwanted singularities and bifurcations can lead to visual artifacts, and I address them through singularity and bifurcation editing. I demonstrate the capabilities of the design system by applying it to the design of two types of vector fields: the orientation field and the advection field for the application of texture synthesis and animation

Localized flow, particle tracing, and topological separation analysis for flow visualization

Since the very beginning of the development of computers they have been used to accelerate the knowledge gain in science and research. Today they are a core part of most research facilities. Especially in natural and technical sciences they are used to simulate processes that would be hard to observe in real world experiments. Together with measurements from such experiments, simulations produce huge amounts of data that have to be analyzed by researchers to gain new insights and develop their field of science