Visualizing the Behavior of Higher Dimensional Dynamical Systems

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Visualizing the Behavior of Higher Dimensional
Dynamical Systems

• Rainer Wegenkittl, Helwig Loffelmann, and Eduard
  Groller
• IEEE Visualization 97
• http//www.cg.tuwien.ac.at/research/
  vis/dynsys/ndim/ndim_crc.pdf
• presented by John T. Bell
• CS 526 - Spring 2004

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Motivation

• A lot of very interesting ( scientifically )
  phenomenon are inherently high-dimensional.
  This is increasing as simulations increase the
  number of properties calculated at each point.
• Dynamic systems in particular employ continuous
  time derivatives at each point in n-space.
• However humans generally view the world in 3-D.
• So how do we display m-D data defined in n-D
  space in a cognitively effective manner?

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Previous Methods of Visualizing Multi-Dimensional
Data

• Attribute Mapping ( e.g. Color Coding )
• Geometric Coding ( e.g. Glyphs, Icons, Chernoff
  Faces )
• Sonification
• Reduction of Dimension - Focusing or Linking
• Parallel Coordinates

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Attribute Mapping ( e.g. Color ) I
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Attribute Mapping ( e.g. Color ) II
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Geometric Coding - Glyphs I

• Sphere Boids
• Arrow Dart Boids
• Flow Ribbons
• Test Particles
• Ellipsoid Boids for Tensor Fields
• Probes for Fluid Flow Visualization

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Geometric Coding - Glyphs II

• G. David Kerlick, Moving Iconic Objects in
  Scientific Visualization, IEEE 1990.
• Dart Boids and Flow Ribbons

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Geometric Coding - Glyphs III

• Willem C. de Leeuw and Wijk, A Probe for Local
  Field Visualization

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Geometric Coding - Color Icons

• Haim Levkowitz, Color Icons Merging Color and
  Texture Perception for Integrated Visualization
  of Multiple Parameters, IEEE 1991.

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Geometric Coding - Chernoff Faces

• Herman Chernoff, as reproduced in Edward Tufte,
  The Visual Display of Quantitative Information,
  p. 142.

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Sonification ( Color Icons )
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Reduction of Dimensioning I - Focusing

• Focusing on a smaller dimensional view of the
  original data
• Subsetting
• Panning
• Zooming
• Slicing
• Projection
• Specialty Fisheye views, and rooms

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Reduction of Dimensioning II - Linking

• Collections of focused subsets, linked.
• E.g. Tuftes Small Multiples

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Parallel Coordinates

• Alfred Inselberg Bernard Dimsdale, Parallel
  Coordinates A Tool for Visualizing
  Multi-Dimensional Geometry

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Properties of High D Dynamic Systems
Visualizations

• Many phenomena described by systems of N
  differential equations in N state variables.
• gt N-dimensional domain with N-dimensional
  slope vector defined continuously over domain.
• Visualization needs to account for the flow the
  topology, not just the independent data values.
• Approach 1 Show derivatives, shear, curvature,
  vorticity, etc. at each point using icons.
• Approach 2 Flow trajectories.

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Three New Approaches to High-D Visualization

• Extruded Parallel Coordinates - Employ the Z
  direction, so lines become surfaces.
• Linking With Wings - Plot trajectories in 3-D,
  based on 3 of N variables Then add coordinate
  axes along trajectories for more D.
• 3-D Parallel Coordinates - Instead of linear
  axes, use 2-D planes instead Either coincident,
  parallel, or otherwise arranged.

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Extruded Parallel Coordinates
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Linking With Wings
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3-D Parallel Coordinates - I
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3-D Parallel Coordinates - II
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Four-Dimensional Hedgehog
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References - I

• Rainer Wegenkittl, Helwig Loffelmann, and Eduard
  Groller, Visualizing the Behavior of Higher
  Dimensional Dynamical Systems, Proceedings of
  the 8th IEEE Visualization 97 Conference,
  http//www.cg.tuwien.ac.at/research/
  vis/dynsys/ndim/ndim_crc.pdf
• Scientific Visualization Web Site at Technische
  Universitat Wien, http//www.cg.tuwien.ac.at/resea
  rch/vis-dyn-syst/
• G. D. Kerlick, Moving Iconic Objects in
  Scientific Visualization, IEEE Visualiztion 90
  Proceedings, pp. 124-129, 1990.
• W.C. de Leeuw, J.J. van Wijk, A Probe for Local
  Flow Field Visualization, IEEE Visualization 93
  Proceedings, pp. 39-45, 1993.
• A. Inselberg, B. Dimsdale, Parallel Coordinates
  A Tool for Visualizing Multidimensional
  Geometry, Visualization 90 Proceedings, pp.
  361-378, 1990.
• H. Levkowitz, Color Icons Merging Color and
  Texture Perception for Integrated Visualization
  of Multiple Parameters, IEEE Visualization 91
  Proceedings, pp. 164-170, 1991.

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References - II

• Edward Tufte, The Visual Display of Quantitative
  Information, Graphics Press, Cheshire, CT, 1983
• K.W. Brodie et. al. ( ed.s ) Scientific
  Visualization, Techniques and Applications.
• ( H. Chernoff, The Use of Faces to Represent
  Points in K-Dimensional Space Graphically,
  Journal of the American Statistical Association,
  68, pp. 361-368, 1993. - As reported in Tufte
  above. )
• T. Mihalisin, J. Timlin, J. Schwegler,
  Visualization and Analysis of Multi-variate
  Data A Technique for all Fields, EIII
  Visualization 91 Proceedigns, pp. 171-178, 1991
• Delft Technical University Scientific
  Visualization Web Site, http//visualization.tudel
  ft.nl/index.html
• Daniel Harms, Extending the McCabe-Thiele Method
  to Multicomponent Distillation Using Virtual
  Reality, M.S. Project Report, University of
  Illinois Chicago, 2001.

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How Can We ExtendMcCabe-Thiele ?
?
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First Consider a Single Stage
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Then Stack Multiple Stages
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Augment With Color and Detail
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Close Ups of Feedand Top Stages
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Frames Add T, P, X, Y Info
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McCabe-Thiele SpacingShows (In)Efficient Stages
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Two Related Towers ...
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Can Now Be Combined
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Of Course Too Much DataCan Still Be Overwhelming