While 2D and 3D vector fields are ubiquitous in computational sciences, their use in graphics is often limited to regular grids, where computations are easily handled through finite-difference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vector fields on arbitrary tetrahedral grids. We introduce a variational, multiscale decomposition of vector fields into three intuitive components: a divergence-free part, a curl-free part, and a harmonic part. We show how our discrete approach matches its well-known smooth analog, called the HelmotzHodge decomposition, and that the resulting computational tools have very intuitive geometric interpretation. We demonstrate the versatility of these tools in a series of applications, ranging from data visualization to fluid and deformable object simulation.

We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2-manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simulating differentiability. We then simplify Morse complexes by cancelling pairs of critical points in order of increasing persistence. Keywords Computational topology, PL manifolds, Morse theory, topological persistence, hierarchy, algorithms, implementation, terrains 1. INTRODUCTION In this paper, we define the Morse complex decomposing a piecewise linear 2-manifold and present algorithms for constructing and simplifying this complex. 1.1 Motivation Physical simulation problems often start with a space and measurements over this space. If the measurements are scalar values, we talk about a height function of that space. We use this name throughout the paper, although the functions can ...

Abstract. Topology-based visualization of planar turbulent flows results in visual clutter due to the presence of numerous features of very small scale. In this paper, we attack this problem with a topology simplification method for vector and tensor fields defined on irregular grids. This is the generalization of previous work dealing with structured grids. The method works for all interpolation schemes. 1

One of the reasons that topological methods have a limited popularity for the visualization of complex 3D flow fields is the fact that such topological structures contain a number of separating stream surfaces. Since these stream surfaces tend to hide each other as well as other topological features, for complex 3D topologies the visualizations become cluttered and hardly interpretable. This paper proposes to use particular stream lines called saddle connectors instead of separating stream surfaces and to depict single surfaces only on user demand. We discuss properties and computational issues of saddle connectors and apply these methods to complex flow data. We show that the use of saddle connectors makes topological skeletons available as a valuable visualization tool even for topologically complex 3D flow data.

We introduce a scheme of control polygons to design topological skeletons for vector fields of arbitrary topology. Based on this we construct piecewise linear vector fields of exactly the topology specified by the control polygons. This way a controlled construction of vector fields of any topology is possible. Finally we apply this method for topology-preserving compression of vector fields consisting of a simple topology.

We present a new method for topological segmentation in steady three-dimensional vector fields. Depending on desired properties, the algorithm replaces the original vector field by a derived segmented data set, which is utilized to produce separating surfaces in the vector field. We define the concept of a segmented data set, develop methods that produce the segmented data by sampling the vector field with streamlines, and describe algorithms that generate the separating surfaces. This method is applied to generate local separatrices in the field, defined by a movable boundary region placed in the field. The resulting partitions can be visualized using standard techniques for a visualization of a vector field at a higher level of abstraction. 1.

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One of the reasons that topological methods have a limited popularity for the visualization of complex 3D flow fields is the fact that their topological structures contain a number of separating stream surfaces. Since these stream surfaces tend to hide each other as well as other topological features, for complex 3D topologies the visualizations become cluttered and hardly interpretable. One solution of this problem is the recently introduced concept of saddle connectors which treats separation surfaces emanating from critical points. In this paper we extend this concept to separation surfaces starting from boundary switch curves. This way we obtain a number of particular stream lines called boundary switch connectors. They connect either two boundary switch curves or a boundary switch curve with a saddle. We discuss properties and computational issues of boundary switch connectors and apply them to topologically complex flow data.

Abstract — This paper describes approaches to topologically segmenting 2D time-dependent vector fields. For this class of vector fields, two important classes of lines exist: stream lines and path lines. Because of this, two segmentations are possible: either concerning the behavior of stream lines, or of path lines. While topological features based on stream lines are well established, we introduce path line oriented topology as a new visualization approach in this paper. As a contribution to stream line oriented topology we introduce new methods to detect global bifurcations like saddle connections and cyclic fold bifurcations as well as a method to tracking all isolated closed stream lines. To get the path line oriented topology we segment the vector field into areas of attracting, repelling and saddle-like behavior of the path lines. We compare both kinds of topologies and apply them to a number of test data sets. Index Terms — flow visualization, vector field topology, bifurcations, stream lines, path lines I.

(a) 184 first order critical points. The box around the molecule represents the chosen area for topological simplification. (b) Topologically simplified representation with one higher order critical point elucidates the far field behavior of the benzene. Figure 1: Topological representations of the electrostatic field of the benzene molecule. This paper presents an approach to extracting and classifying higher order critical points of 3D vector fields. To do so, we place a closed convex surface s around the area of interest. Then we show that the complete 3D classification of a critical point into areas of different flow behavior is equivalent to extracting the topological skeleton of an appropriate 2D vector field on s, if each critical point is equipped with an additional Bit of information. Out of this skeleton, we create an icon which replaces the complete topological structure inside s for the visualization. We apply our method to find a simplified visual representation of clusters of critical points, leading to expressive visualizations of topologically complex 3D vector fields.