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46
Discrete Multiscale Vector Field Decomposition
, 2003
"... 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 finitedifference methods. In this paper, we propose a set of simple and accurate tools for the analysis of 3D discrete vecto ..."
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Cited by 68 (8 self)
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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 finitedifference 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 divergencefree part, a curlfree part, and a harmonic part. We show how our discrete approach matches its wellknown 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.
Saddle connectors  an approach to visualizing the topological skeleton of complex 3D vector fields
 IN PROC. IEEE VISUALIZATION
, 2003
"... 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 feature ..."
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Cited by 52 (17 self)
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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 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.
Hierarchical Morse Complexes for Piecewise Linear 2Manifolds
, 2001
"... We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. While Morse complexes are defined only in the smooth category, we extend the construction to the piecewise linear category by ensuring structural integrity and simu ..."
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Cited by 47 (5 self)
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We present algorithms for constructing a hierarchy of increasingly coarse Morse complexes that decompose a piecewise linear 2manifold. 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 2manifold 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 ...
Vector and Tensor Field Topology Simplification, Tracking, and Visualization
 PhD. thesis, Schriftenreihe Fachbereich Informatik (3), Universität
, 2002
"... Abstract. Topologybased 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 ge ..."
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Cited by 40 (3 self)
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Abstract. Topologybased 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
Designing 2D Vector Fields of Arbitrary Topology
 Computer Graphics Forum (Eurographics
, 2002
"... 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 ..."
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Cited by 30 (16 self)
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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 topologypreserving compression of vector fields consisting of a simple topology.
Topological segmentation in threedimensional vector fields
 IEEE Transactions on Visualization and Computer Graphics
, 2004
"... We present a new method for topological segmentation in steady threedimensional 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 conce ..."
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Cited by 27 (5 self)
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We present a new method for topological segmentation in steady threedimensional 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.
1 TopologyBased Flow Visualization, The State of the Art
"... Flow visualization research has made rapid advances in recent years, especially in the area of topologybased flow visualization. The ever increasing size of scientific data sets favors algorithms that are capable of extracting important subsets of the data, leaving the scientist with a more managea ..."
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Cited by 25 (5 self)
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Flow visualization research has made rapid advances in recent years, especially in the area of topologybased flow visualization. The ever increasing size of scientific data sets favors algorithms that are capable of extracting important subsets of the data, leaving the scientist with a more manageable representation that may be visualized interactively. Extracting the topology of a flow achieves the goal of obtaining a compact representation of a vector or tensor field while simultaneously retaining its most important features. We present the state of the art in topologybased flow visualization techniques. We outline numerous topologybased algorithms categorized according to the type and dimensionality of data on which they operate and according to the goaloriented nature of each method. Topology tracking algorithms are also discussed. The result serves as a useful introduction and overview to research literature concerned with the study of topologybased flow visualization.
Gridindependent detection of closed stream lines in 2D vector fields
 in Proc. Vision, Modeling and Visualization
, 2004
"... We present a new approach to detecting isolated closed stream lines in 2D vector fields. This approach is based on the idea of transforming the 2D vector field into an appropriate 3D vector field such that detecting closed stream lines in 2D is equivalent to intersecting certain stream surfaces in 3 ..."
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Cited by 21 (3 self)
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We present a new approach to detecting isolated closed stream lines in 2D vector fields. This approach is based on the idea of transforming the 2D vector field into an appropriate 3D vector field such that detecting closed stream lines in 2D is equivalent to intersecting certain stream surfaces in 3D. Contrary to preexisting methods, our approach does not rely on any underlying grid structure of the vector field. We demonstrate the applicability and stability by applying it to a test data set. 1
Boundary switch connectors for topological visualization of complex 3D vector fields
 In Proc. VisSym 04
, 2004
"... 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 feature ..."
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Cited by 20 (16 self)
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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.