Topological Methods in Data Analysis and Visualization II

We present a simple approach to the topological analysis of divergence-free 2D vector fields using discrete Morse theory. We make use of the fact that the point-wise perpendicular vector field can be interpreted as the gradient of the stream function. The topology of the divergence-free vector field is thereby encoded in the topology of a gradient vector field. We can therefore apply a formulation of computational discrete Morse theory for gradient vector fields. The inherent consistence and robustness of the resulting algorithm is demonstrated on synthetic data and an example from computational fluid dynamics.

This paper introduces a novel combinatorial algorithm to compute a hierarchy of discrete gradient vector fields for three-dimensional scalar fields. The hierarchy is defined by an importance measure and represents the combinatorial gradient flow at different levels of detail. The presented algorithm is based on Forman’s discrete Morse theory, which guarantees topological consistency and algorithmic robustness. In contrast to previous work, our algorithm combines memory and runtime efficiency. It thereby lends itself to the analysis of large data sets. A discrete gradient vector field is also a compact representation of the underlying extremal structures – the critical points, separation lines and surfaces. Given a certain level of detail, an explicit geometric representation of these structures can be extracted using simple and fast graph algorithms.

Morse-Smale complexes are gaining in popularity as a tool in scientific data analysis and visualization. The cells of the complex represent contiguous regions of uniform flow properties, and in many application domains, features can be described by carefully extracting these cells. However, existing techniques only describe how to extract ascending and descending manifolds of critical points, and their intersections; given two critical points p and q of index i and i + 1 respectively, these methods are not able to determine how many cells the intersection of ascending manifold of p and the descending manifold of q form, or distinguish between them. In this paper, we use the framework provided by discrete Morse theory to describe a combinatorial algorithm for computing all cells of the Morse-Smale complex, where the interior of each cell is simply connected, as the theory prescribes. Furthermore, we provide data structures that enable a practical implementation.

In this paper, we present two combinatorial methods to process 3-D 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 a straightforward extension of an existing 2-D technique to 3-D 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. We evaluate our method on a CFD-simulation of a gas furnace chamber. Finally, we discuss several possibilities for categorizing the invariant sets with respect to the flow.

Contours, the connected components of level sets, play an important role in understanding the global structure of a scalar field. In particular their nesting behavior and topology – often represented in form of a contour tree – have been used extensively for visualization and analysis. However, traditional contour trees only encode structural properties like number of contours or the nesting of contours, but little quantitative information such as volume or other statistics. Here we use the segmentation implied by a contour tree to compute a large number of per-contour (interval) based statistics of both the function defining the contour tree as well as other co-located functions. We introduce a new visual metaphor for contour trees, called topological cacti, that extends the traditional toporrery display of a contour tree to display additional quantitative information as width of the cactus trunk and length of its spikes. We apply the new technique to scalar fields of varying dimension and different measures to demonstrate the effectiveness of the approach.

Scalar-valued functions are ubiquitous in scientific research. Analysis and visualization of scalar functions defined on low-dimensional and simple domains is a well-understood problem, but complications arise when the domain is high-dimensional or topologically complex. Topological analysis and Morse theory provide tools that are effective in distilling useful information from such difficult scalar functions. A recently proposed topological method for understanding high-dimensional scalar functions approximates the Morse-Smale complex of a scalar function using a fast and efficient clustering technique. The resulting clusters (the so-called Morse crystals) are each approximately monotone and are amenable to geometric summarization and dimensionality reduction. However, some Morse crystals may contain loops. This shortcoming can affect the quality of the analysis performed on each crystal, as regions of the domain with potentially disparate geometry are assigned to the same cluster. We propose to use the Reeb graph of each Morse crystal to detect and resolve certain classes of problematic clustering. This provides a simple and efficient enhancement to the previous Morse crystals clustering. We provide preliminary experimental results to demonstrate that our improved topology-sensitive clustering algorithm yields a more accurate and reliable description of the topology of the underlying scalar function.

In this paper we present an efficient framework for computation of persistent homology of cubical data in arbitrary dimensions. An existing algorithm using simplicial complexes is adapted to the setting of cubical complexes. The proposed approach enables efficient application of persistent homology in domains where the data is naturally given in a cubical form. By avoiding triangulation of the data, we significantly reduce the size of the complex. We also present a data-structure designed to compactly store and quickly manipulate cubical complexes. By means of numerical experiments, we show high speed and memory efficiency of our approach. We compare our framework to other available implementations, showing its superiority. Finally, we report performance on selected 3D and 4D data-sets.

Area-preserving maps arise in the study of conservative dynamical systems describing a wide variety of physical phenomena, from the rotation of planets to the dynamics of a fluid. The visual inspection of these maps reveals a remarkable topological picture in which invariant manifolds form the fractal geometric scaffold of both quasi-periodic and chaotic regions. We discuss in this paper the visualization of such maps built upon these invariant manifolds. This approach is in stark contrast with the discrete Poincare plots that are typically used for the visual inspection of maps. We propose to that end several modified definitions of the finite-time Lyapunov exponents that we apply to reveal the underlying structure of the dynamics. We examine the impact of various parameters and the numerical aspects that pertain to the implementation of this method. We apply our technique to a standard analytical example and to a numerical simulation of magnetic confinement in a fusion reactor. In both cases our simple method is able to reveal salient structures across spatial scales and to yield expressive images across application domains.

In the study of a magnetic confinement fusion device such as a tokamak, physicists need to understand the topology of the flux (or magnetic) surfaces that form within the magnetic field. Among the two distinct topological structures, we are particularly interested in the magnetic island chains which correspond to the break up of the ideal rational surfaces. Different from our previous method [13], in this work we resort to the periodicity analysis of two distinct functions to identify and characterize flux surfaces and island chains. These two functions are derived from the computation of the fieldlines and puncture points on a Poincaré section, respectively. They are the distance measure plot and the ridgeline plot. We show that the periods of these two functions are directly related to the topology of the surface via a resonance detection (i.e., period estimation and the common denominators computation). In addition, we show that for an island chain the two functions possess resonance components which do not occur for a flux surface. Furthermore, by combining the periodicity analysis of these two functions, we are able to devise a heuristic yet robust and reliable approach for classifying and characterizing different magnetic surfaces in the toroidal magnetic fields.

We analyze characteristic curves of vector fields and report on locations where they have cusps in their spatial projection, i.e., isolated points on the curve with abruptly turning tangent direction. Cusps appear in places where a projection of the corresponding tangent curve vector field exhibits critical points. We show that such cusps are only possible for streak and path lines, whereas they cannot appear on stream and time lines. Cusps turn out to be closely related to self-intersections of characteristic curves. We utilize this information in a new algorithm to create uncluttered static visualizations of path and streak lines.

Glyphs are a widespread technique to depict local properties of different kinds of fields. In this paper we present a new glyph for singularities in non-linear vector fields. We do not simply show the properties of the derivative at the singularities as most previous methods do, but instead illustrate the behavior that goes beyond the local linear approximation. We improve the concept of linear neighborhoods to determine the size of the vicinity from which we derive the data for the glyph. To obtain data from outside this neighborhood we use integration in the vector field. The gathered information is used to depict convergence and divergence of the flow, and non-linear behavior in general. These properties are communicated by color, radius, the overall shape of the glyphs and streamlets on their surface. This way we achieve a depiction of the non-linear behavior of the flow around the singularities.

In this chapter we define the topology of two-dimensional (2D) asymmetric tensor fields in terms of two graphs corresponding to the eigenvalue and eigenvector analysis for tensor fields, respectively. Asymmetric tensor field topology can not only yield a concise representation of the field, but also provide a framework for spatial-temporal tracking of field features. Furthermore, inherent topological constraints in asymmetric tensor fields can be identified unambiguously through these graphs. We also describe efficient algorithms to compute the topology of a given 2D asymmetric tensor field. We demonstrate the utility of our graph representations for asymmetric tensor field topology with fluid simulation data sets.

Many of the recently developed methods for analysis and visualization of time-dependent flows are related to concepts, which can be subsumed under the term Lagrangian coherent structures (LCS). Thereby, no universal definition of LCS exists and different interpretations are used. Mostly, LCS are considered to be features linked to pathlines leading to the ideal conception of features building material lines. Such time-dependent features are extracted by averaging local properties of particles along their trajectories, e.g., separation, acceleration or unsteadiness. A popular realization of LCS is the finite-time Lyapunov exponent (FTLE) with its different implementations. The goal of this paper is to stimulate a discussion on the generality of the underlying assumptions and concepts. Using a few well-known datasets, the interpretation and usability of Lagrangian analysis methods are discussed.

The popularity of vector field topology in the visualization community is due mainly to the topological skeleton which captures the essential information on a vector field in a set of lines or surfaces separating regions of different flow behavior. Unfortunately, vector field topology has no straightforward extension to unsteady flow, and the concept probably most closely related to the topological skeleton are the so-called Lagrangian coherent structures (LCS). LCS are material lines or material surfaces that separate regions of different flow behavior. Ideally, such structures are material lines (or surfaces) in an exact sense and at the same time maximally attracting or repelling, but practical realizations such as height ridges of the finite-time Lyapunov exponent (FTLE) fulfill these two requirements only in an approximate sense. In this paper, we quantify the deviation from exact material lines/surfaces for several FTLE-based concepts, and we propose a numerically simpler variants of FTLE ridges that has equal or better error characteristics than classical FTLE height ridges.

We present a novel approach to obtain FTLE-like structures and visualizations for uncertain vector fields. We compute the flow map for a time interval now mapping to particle distributions rather than single positions. We use a variance-based analysis to measure the maximal stretching of the distributions thereby obtaining FTLE-like structures for uncertain vector fields. In the case of deterministic vector fields, the results are visually almost identical to FTLE. We analyze our method in the presence of errors by applying it to a number of synthetic and real world datasets.

Lagrangian coherent structures (LCS) can be extracted from time-dependent vector fields by means of ridges in the finite-time Lyapunov exponent (FTLE). While the LCS approach has proven successful in many areas and applications for the analysis of time-dependent topology, it is to some extent still an open problem how the finite time scope is appropriately chosen. One has to be aware, however, that the introduction of this finite time scope in the Lyapunov exponent, where the time scope was originally infinite, is largely responsible for the recent success of the FTLE in analysis of real-world data. Hence, there is no general upper bound for the time scope: it depends on the application and the goal of the analysis. There is, however, a clear need for a lower bound of the time scope because the FTLE converges to the eigenvalue of the rate of strain tensor as the time scope approaches zero. Although this does not represent a problem per se, it is the loss of important properties that causes ridges in such FTLE fields to lose the LCS property. LCS are time-dependent separatrices: they separate regions of different behavior over time. Thereby they behave like material constructs, advecting with the vector field and exhibiting negligible cross flow. We present a method for investigating and determining a lower bound for the FTLE time scope at isolated points of its ridges. Our approach applies the advection property to the points where attracting and repelling LCS intersect. These points are of particular interest because they are important in typical questions of Lagrangian topology. We demonstrate our approach with examples from dynamical systems theory and computational fluid dynamics.

The finite-time Lyapunov Exponent (FTLE) is useful for the visualization of time-dependent velocity fields. The ridges of this derived scalar field have been shown to correspond well to attracting or repelling material structures, so-called Lagrangian coherent structures (LCS). There are two issues involved in the computation of FTLE for this purpose. Firstly, it is often not practically possible to refine the grid for sampling the flow map until convergence of FTLE is reached. Slow conversion is mostly caused by gradient underestimation. Secondly, there is a parameter, the integration time, which has to be chosen sensibly. Both of these problems call for an examination in scale-space. We show that a scale-space approach solves the problem of gradient underestimation. We test optimal-scale ridges for their usefulness with FTLE fields, obtaining a negative result. However, we propose an optimization of the time parameter for a given scale of observation. Finally, an incremental method for computing smoothed flow maps is presented.