# 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.

## 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.

## 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.

# Publications

## 2014

 Hlawatsch, Marcel; Sadlo, Filip; Jang, Hajun; Weiskopf, Daniel: Pathline Glyphs. In: Computer Graphics Forum: Vol. 33, No. 2 (2014), pp. 497-506. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Karch, Grzegorz K.; Sadlo, Filip; Weiskopf, Daniel; Ertl, Thomas: Streamline-Based Concepts for Space-Time Analysis of 2D Time-Dependent Flow. In: Proceedings of International Symposium on Flow Visualization (ISFV16), 2014. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Sadlo, Filip: Lyapunov Time for 2D Lagrangian Visualization. In: Topological and Statistical Methods for Complex Data (to appear). [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Peikert, Ronald; Pobitzer, Armin; Sadlo, Filip; Schindler, Benjamin: A Comparison of Finite-Time and Finite-Size Lyapunov Exponents. In: Topological Methods in Data Analysis and Visualization III, pp. 187-200, 2014. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Sadlo, Filip; Karch, Grzegorz K.; Ertl, Thomas: Topological Features in Time-Dependent Advection-Diffusion Flow. In: Topological Methods in Data Analysis and Visualization III, pp. 217-231, 2014. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Bolemann, Thomas; Üffinger, Markus; Sadlo, Filip; Ertl, Thomas; Munz, Claus-Dieter: Direct Visualization of Piecewise Polynomial Data. In: (to appear). [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2013

 Hlawatsch, Marcel; Sadlo, Filip; Weiskopf, Daniel: Predictability-Based Adaptive Mouse Interaction and Zooming for Visual Flow Exploration.. In: International Journal for Uncertainty Quantification: Vol. 3, No. 3 (2013), pp. 225-240. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Üffinger, Markus; Sadlo, Filip; Ertl, Thomas: A Time-Dependent Vector Field Topology Based on Streak Surfaces. In: IEEE Transactions on Visualization and Computer Graphics: Vol. 19, No. 3 (2013), pp. 379-392. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Sadlo, Filip; Bachthaler, Sven; Dachsbacher, Carsten; Weiskopf, Daniel: Space-Time Flow Visualization of Dynamics in 2D Lagrangian Coherent Structures. In: Gabriela Csurka, Martin Kraus, Robert S. Laramee, Paul Richard, José Braz : Communications in Computer and Information Science. Heidelberg: Springer, 2013. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Karch, Grzegorz K.; Sadlo, Filip; Meister, Christian; Rauschenberger, Philipp; Schulte, Kathrin; Weigand, Bernhard; Ertl, Thomas: Visualization of Piecewise Linear Interface Calculation. In: Proceedings of Pacific Visualization Symposium (PacificVis) 2013, pp. 121-128, 2013. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Üffinger, Markus; Sadlo, Filip; Munz, Claus-Dieter; Ertl, Thomas: Toward Wall Function Consistent Interpolation of Flow Fields . In: EuroVis (Short Papers), pp. 85-89, 2013. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Karch, Grzegorz K.; Sadlo, Filip; Songoro, Harald; Gjonaj, Erion; Weiland, Thomas; Ertl, Thomas: Visualizing edge-conforming discrete field quantities in electromagnetic field problems with interfaces. In: ILASS–Europe, 25th European Conference on Liquid Atomization and Spray Systems, Chania, Greece, 2013, 2013. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2012

 Bachthaler, Sven; Sadlo, Filip; Weeber, Rudolf; Kantorovich, Sofia S.; Holm, Christian; Weiskopf, Daniel: Magnetic Flux Topology of 2D Point Dipoles. In: Computer Graphics Forum: Vol. 31, No. 3 (2012), pp. 955-964. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Karch, Grzegorz K.; Sadlo, Filip; Weiskopf, Daniel; Hansen, Charles D.; Li, Guo-Shi; Ertl, Thomas: Dye-Based Flow Visualization. In: IEEE Computing in Science and Engineering: Vol. 14, No. 6 (2012), pp. 80-86. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Karch, Grzegorz K.; Sadlo, Filip; Weiskopf, Daniel; Munz, Claus-Dieter; Ertl, Thomas: Visualization of Advection-Diffusion in Unsteady Fluid Flow. In: Computer Graphics Forum: Vol. 31, No. 3 (2012), pp. 1105-1114. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 Bachthaler, Sven; Sadlo, Filip; Dachsbacher, Carsten; Weiskopf, Daniel: Space-Time Visualization of Dynamics in Lagrangian Coherent Structures of Time-Dependent 2D Vector Fields. In: International Conference on Information Visualization Theory and Applications, pp. 573-583, 2012. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Mello Machado, Gustavo; Sadlo, Filip; Müller, Thomas; Müller, Daniel; Ertl, Thomas: Visualizing Solar Dynamics Data. In: Workshop on Vision, Modeling and Visualization (VMV), pp. 95-102, 2012. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Netzel, Rudolf; Ament, Marco; Burch, Michael; Weiskopf, Daniel: Spectral Analysis of Higher-Order and BFECC Texture Advection. In: Workshop on Vision, Modeling and Visualization (VMV), pp. 87-94, 2012. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Sadlo, Filip; Üffinger, Markus; Ertl, Thomas; Weiskopf, Daniel: On the Finite-Time Scope for Computing Lagrangian Coherent Structures from Lyapunov Exponents. In: Topological Methods in Data Analysis and Visualization II (TopoInVis 2011), pp. 269-281, 2012. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Üffinger, Markus; Sadlo, Filip; Kirby, Mike; Hansen, Charles D.; Ertl, Thomas: FTLE Computation Beyond First Order Approximation. In: Eurographics 2012 (Short Papers) (to appear). [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2011

 Effenberger, Felix; Weiskopf, Daniel: Finding and Classifying Critical Points of 2D Vector Fields: A Cell-Oriented Approach Using Group Theory. In: Computing and Visualization in Science: Vol. 13, No. 8 (2011), pp. 377-396. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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: Vol. 17, No. 12 (2011), pp. 1949-1958. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Hlawatsch, Marcel; Sadlo, Filip; Weiskopf, Daniel: Hierarchical Line Integration. In: IEEE Transactions on Visualization and Computer Graphics: Vol. 17, No. 8 (2011), pp. 1148-1163. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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: Vol. 17, No. 6 (2011), pp. 781-794. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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: Vol. 30, No. 3 (2011), pp. 751-760. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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: Vol. 13, No. 3 (2011), pp. 84-91. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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, vol. 2, pp. 276-291, 2011. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Sadlo, Filip; Rigazzi, Alessandro; Peikert, Ronald: Time-Dependent Visualization of Lagrangian Coherent Structures by Grid Advection. In: Topological Methods in Data Analysis and Visualization, pp. 151-165, 2011. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 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: Vol. 29, No. 3 (2010), pp. 1163-1172. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Sadlo, Filip; Weiskopf, Daniel: Time-Dependent 2D Vector Field Topology: An Approach Inspired by Lagrangian Coherent Structures. In: Computer Graphics Forum: Vol. 29, No. 1 (2010), pp. 88-100. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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 (2010), pp. 337-346. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2009

 Peikert, Ronald; Sadlo, Filip: A Robust Stream Surface Method for the Visualization of Vector Field Singularities. In: Computer Graphics and Geometry: Vol. 11, No. 2 (2009), pp. 2-13. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Peikert, Ronald; Sadlo, Filip: Topologically Relevant Stream Surfaces for Flow Visualization. In: Proceedings of Spring Conference on Computer Graphics, pp. 43-50, 2009. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]

## 2008

 Bachthaler, Sven; Weiskopf, Daniel: Animation of Orthogonal Texture Patterns for Vector Field Visualization. In: IEEE Transactions on Visualization and Computer Graphics: Vol. 14, No. 4 (2008), pp. 741-755. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] Falk, Martin; Weiskopf, Daniel: Output-Sensitive 3D Line Integral Convolution. In: IEEE Transactions on Visualization and Computer Graphics: Vol. 14, No. 4 (2008), pp. 820-834. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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: Vol. 14, No. 5 (2008), pp. 1067-1080. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]
 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details] 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. [XPS] [PDF] [DOI] [OpenXML] [BibTeX] [Slides] [Details]