### Topology-Based Methods in Visualization II: v. 2 (Mathematics and Visualization)

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. 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.

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.

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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 Methods in Data Analysis and Visualization II

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.

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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.

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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.

## Publications of Tino Weinkauf

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 , in this work we resort to the periodicity analysis of two distinct functions to identify and characterize flux surfaces and island chains.

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. 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.

Vector fields, represented as vector values sampled on the vertices of a triangulation, are commonly used to model physical phenomena. To analyze and understand vector fields, practitioners use derived properties such as the paths of massless particles advected by the flow, called streamlines.

### Theory, Algorithms, and Applications

However, currently available numerical methods for computing streamlines do not guarantee preservation of fundamental invariants such as the fact that streamlines cannot cross. The resulting inconsistencies can cause errors in the analysis, e.

We propose an alternate representation for triangulated vector fields that exchanges vector values with an encoding of the transversal flow behavior of each triangle. We call this representation edge maps. This work focuses on the mathematical properties of edge maps; a companion paper discusses some of their applications.

Edge maps allow for a multi-resolution approximation of flow by merging adjacent streamlines into an interval based mapping. Consistency is enforced at any resolution if the merged sets maintain an order-preserving property. At the coarsest resolution, we define a notion of equivalency between edge maps, and show that there exist 23 equivalence classes describing all possible behaviors of piecewise linear flow within a triangle.

We analyze characteristic curves of vector fields and report on locations where they have cusps in their spatial projection, i.

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