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Flow, vector field, and tensor field visualization

Introduction

Many phenomena in science cannot be represented adequately by a single value (or scalar). As soon as directional information is of importance, vectors or tensors are used. An example for this is the simulation of air in a closed room. While single values are enough to represent the temperature at every position in the room, vectors are needed to represent the flow of the air in the room. This example also shows the strong connection between vector field and flow visualization. Vector fields are usually the basis for flow visualization applications.

In more complex cases, where not only a single direction but also a directional distribution is given, even a vector is not sufficient. In this case, tensors are used. Tensors are the generalization of scalars and vectors. A scalar is a tensor of zeroth order and a vector is a tensor of first order.

Arrow glyphs visualizing the local flow direction of simulated air in a closed room.
Pathlines visualizing the path of massless particles in the air flow.

Vector field and flow visualization

A typical source for vector fields are computational fluid dynamics applications. In this case, the vector field describes the flow direction of the fluid. A standard technique for visualizing them is the usage of arrow glyphs showing the local flow direction. Another common method is the visualization of tangential curves, .e.g, path lines for time-dependent vector fields.

Both approaches have drawbacks in that they visualize only instantaneous local properties (glyphs) or are prone to visual clutter (path lines). For time-dependent vector fields, a new approach became popular, the visualization of Lagrangian coherent structures by using the finite-time Lyapunov (FTLE) exponent. The FTLE is a measure for the separation of particles after a given time. Low FTLE values indicate low separation and therefore coherent regions in the flow. High FTLE values indicate separation and occur between different coherent regions in the flow. Hence, the FTLE field is separating the flow field in regions of coherent behavior.

Computing the FTLE field is computationally expensive, because an integral curve has to be computed for every element of resulting field. We developed a hierarchical scheme to accelerate this computation. With our method, the computational complexity is reduced from linear to logarithmic. As a result, the computation of a time series of 3D FTLE fields can be done in a few minutes instead of several hours.

2D forward FTLE field of simulated air in a closed room.
3D forward FTLE field of the Arnold-Beltrami-Childress flow.

Tensor field visualization

Second order tensors are used to describe the directional distribution of properties. They are used, e.g., to describe diffusion or mechanical stress. Such data are generated e.g., with diffusion tensor magnetic resonance tomographie.

One approach to visualize these tensors is the usage of special glyphs, similar to arrow glyphs for vectors. However, only local information is directly visible in this case, relations between neighboring positions are only indirectly recognizable. Another common approach is the display of lines following the major eigenvector of the tensor. In this case, not only local properties are visualized. However, this approach is only feasible for low number of lines. Visualizing the complete dataset with this approach results in massive visual clutter.

A possible solution for both problems, visualizing not only local properties and avoiding visual clutter, is the extraction and visualization of structural information from the data. We therefore developed a method to visualize coherent structures in second-order tensor fields.

Visualization of coherent structures in a human brain.
Visualization of nerve tracts in the brain. Coherent regions are consistent with bundles of nerve fibers.

Publications

2011

Finding and Classifying Critical Points of 2D Vector Fields: A Cell-Oriented Approach Using Group Theory
Effenberger, Felix; Weiskopf, Daniel: Finding and Classifying Critical Points of 2D Vector Fields: A Cell-Oriented Approach Using Group Theory. In: IEEE Computing and Visualization in Science: No. 8 (2011), pp. 377-396.
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Flow Radar Glyphs - Static Visualization of Unsteady Flow with Uncertainty
Hlawatsch, Marcel; Leube, Philipp; Nowak, Wolfgang; Weiskopf, Daniel: Flow Radar Glyphs - Static Visualization of Unsteady Flow with Uncertainty. In: IEEE Transactions on Visualization and Computer Graphics: No. 12 (2011), pp. 1949-1958.
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Hierarchical Line Integration
Hlawatsch, Marcel; Sadlo, Filip; Weiskopf, Daniel: Hierarchical Line Integration. In: IEEE Transactions on Visualization and Computer Graphics: No. 8 (2011), pp. 1148-1163.
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Coherent Structures of Characteristic Curves in Symmetric Second Order Tensor Fields
Hlawatsch, Marcel; Vollrath, Joachim E.; Sadlo, Filip; Weiskopf, Daniel: Coherent Structures of Characteristic Curves in Symmetric Second Order Tensor Fields. In: IEEE Transactions on Visualization and Computer Graphics: No. 6 (2011), pp. 781-794.
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Pagot, Christian; Osmari, Daniel; Sadlo, Filip; Weiskopf, Daniel; Ertl, Thomas; Comba, João L. D.: Efficient Parallel Vectors Feature Extraction from Higher-Order Data. In: Computer Graphics Forum (EuroVis 2011): No. 3 (2011), pp. 84-91.
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Visualization of Cell-Based Higher-Order Fields
Sadlo, Filip; Üffinger, Markus; Pagot, Christian; Osmari, Daniel; Comba, João L. D.; Ertl, Thomas; Munz, Claus-Dieter; Weiskopf, Daniel: Visualization of Cell-Based Higher-Order Fields. In: Computing in Science & Engineering: No. 3 (2011), pp. 84-91.
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GPU-based 2D Flow Simulation Steering using Coherent Structures
Ament, Marco; Frey, Steffen; Sadlo, Filip; Ertl, Thomas; Weiskopf, Daniel: GPU-based 2D Flow Simulation Steering using Coherent Structures. In: Proceedings of the 2nd International Conference on Parallel, Distributed, Grid and Cloud Computing for Engineering, p. paper18, 2011.
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Pagot, Christian; Vollrath, Joachim E.; Sadlo, Filip; Weiskopf, Daniel; Ertl, Thomas; Comba, João L. D.: Interactive Isocontouring of High-Order Surfaces. In: Dagstuhl Follow-Ups, p. vol.2, 2011.
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2010

Fuchs, Raphael; Kemmler, Jan; Schindler, Benjamin; Waser, Jürgen; Sadlo, Filip; Hauser, Helwig; Peikert, Ronald: Toward a Lagrangian Vector Field Topology. In: Computer Graphics Forum: No. 3 (2010), pp. 1163-1172.
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Time-Dependent 2D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures
Sadlo, Filip; Weiskopf, Daniel: Time-Dependent 2D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures. In: Computer Graphics Forum: No. 1 (2010), pp. 88-100.
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Interactive Isocontouring of High-Order Surfaces
Pagot, Christian; Vollrath, Joachim E.; Sadlo, Filip; Weiskopf, Daniel; Ertl, Thomas; Comba, João L. D.: Interactive Isocontouring of High-Order Surfaces. In: Dagstuhl Follow-Ups. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2010.
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Trajectory-Augmented Visualization of Lagrangian Coherent Structures in Unsteady Flow
Falk, Martin; Seizinger, Alexander; Sadlo, Filip; Üffinger, Markus; Weiskopf, Daniel: Trajectory-Augmented Visualization of Lagrangian Coherent Structures in Unsteady Flow. In: International Symposium on Flow Visualization (ISFV14), 2010.
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Interactive High-Quality Visualization of Higher-Order Finite Elements
Üffinger, Markus; Frey, Steffen; Ertl, Thomas: Interactive High-Quality Visualization of Higher-Order Finite Elements. In: Computer Graphics Forum (CGF) Volume 29(2) (Euro Graphics 2010) Pages 115-136, pp. 337-346, 2010.
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2009

Peikert, Ronald; Sadlo, Filip: A Robust Stream Surface Method for the Visualization of Vector Field Singularities. In: Computer Graphics and Geometry: No. 2 (2009), pp. 2-13.
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Baysal, Kudret; Schafhitzel, Tobias; Ertl, Thomas; Rist, Ulrich: Extraction and Visualization of Flow Features. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, Springer Verlag, 2009.
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Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection
Sadlo, Filip; Rigazzi, Alessandro; Peikert, Ronald: Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection. In: Topology-Based Methods in Visualization III (Proceedings of TopoInVis 2009). 2009.
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Dynamic Grid Refinement for Fluid Simulations on Parallel Graphics Architectures
Ament, Marco; Straßer, Wolfgang: Dynamic Grid Refinement for Fluid Simulations on Parallel Graphics Architectures. In: Proceedings of the Eurographics Symposium on Parallel Graphics and Visualization, pp. 9-15, 2009.
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Topologically Relevant Stream Surfaces for Flow Visualization
Peikert, Ronald; Sadlo, Filip: Topologically Relevant Stream Surfaces for Flow Visualization. In: Proceedings of Spring Conference on Computer Graphics, pp. 43-50, 2009.
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2008

Animation of Orthogonal Texture Patterns for Vector Field Visualization
Bachthaler, Sven; Weiskopf, Daniel: Animation of Orthogonal Texture Patterns for Vector Field Visualization. In: IEEE Transactions on Visualization and Computer Graphics: No. 4 (2008), pp. 741-755.
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Output-Sensitive 3D Line Integral Convolution
Falk, Martin; Weiskopf, Daniel: Output-Sensitive 3D Line Integral Convolution. In: IEEE Transactions on Visualization and Computer Graphics: No. 4 (2008), pp. 820-834.
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Laramee, Robert S.; Erlebacher, Gordon; Garth, Christoph; Schafhitzel, Tobias; Theisel, Holger; Tricoche, Xavier; Weinkauf, Tino; Weiskopf, Daniel: Applications of Texture-Based Flow Visualization. In: Engineering Applications of Computational Fluid Mechanics (2008), pp. 264-274.
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Li, Guo-Shi; Tricoche, Xavier; Weiskopf, Daniel; Hansen, Charles D.: Flow Charts: Visualization of Vector Fields on Arbitrary Surfaces. In: IEEE Transactions on Visualization and Computer Graphics: No. 5 (2008), pp. 1067-1080.
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Flow Feature Visualization Using Logical Operators on Multivariate Fields
Botchen, Ralf P.; Lauser, Andreas; Weiskopf, Daniel; Ertl, Thomas: Flow Feature Visualization Using Logical Operators on Multivariate Fields. In: Electronic Proceedings International Symposium on Flow Visualization '08, 2008.
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Particle-based vortex core line tracking taking into account vortex dynamics
Schafhitzel, Tobias; Baysal, Kudret; Rist, Ulrich; Weiskopf, Daniel; Ertl, Thomas: Particle-based vortex core line tracking taking into account vortex dynamics. In: Proceedings International Symposium on Flow Visualization '08 (to appear ), 2008.
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Topology-Preserving lambda2-based Vortex Core Line Detection for Flow Visualization
Schafhitzel, Tobias; Vollrath, Joachim E.; Gois, Joao P.; Weiskopf, Daniel; Castelo, Antonio; Ertl, Thomas: Topology-Preserving lambda2-based Vortex Core Line Detection for Flow Visualization. In: Proceedings of EG/IEEE TCVG Symposium on Visualization Eurovis '08 , pp. 1023-1030, 2008.
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GPU-Based Streamlines for Surface-Guided 3D Flow Visualization
Üffinger, Markus; Klein, Thomas; Strengert, Magnus; Ertl, Thomas: GPU-Based Streamlines for Surface-Guided 3D Flow Visualization. In: Proceedings of Vision Modeling and Visualization 2008 (VMV 2008), pp. 91-100, 2008.
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