1
Probabilistic SurfacesPoint Based Primitives
to Show Surface Uncertainty
Authors Gevorg Grigoryan and Penny
Rheingans Presented by Allan Spale, Spring 2004
2
Uncertainty Is Inevitable

• Technical fields have measurements with an amount
  of uncertainty
• Example 25 mm /- 2 mm
• Visualizations require a merging of the main data
  being displayed in addition to the uncertainty
  values
• Especially crucial when decision-making must be
  done by users of the visualization

3
Traditional Uncertainty Visualization

• Examples
• Vertical bars on bar graphs
• Probability density curves and surfaces
• Adjacent displays of data and its uncertainty
• Problem
• Does not scale well to multidimensional data

4
Uncertainty VisualizationDiscrete Points

• Examples
• Glyphs 12,16 sonification (sound used to
  convey data) 7, 9, 11 discrete point
  distribution 14 procedural annotation 2
• Advantage
• Useful sampling
• Disadvantage
• Using discrete points limits its application to
  continuous domains

5
Uncertainty VisualizationPseudo-coloring

• Example
• Coloring a mountain range to show the likelihood
  of an avalanche or mud slide
• Advantage
• Uncertainty displayed continuously
• Disadvantages
• Uncertainty relates to a variable, not to a
  single point or group of points
• Disconnect between uncertainty and geometry
• Further reading in 17

6
Uncertainty VisualizationGeometry Modification

• Examples
• Fat surfaces (surfaces with thickness) 1, 12
• Displaced/perturbed geometry 10, 12
• Animation 4,
• IFS (iterated function system) fractal
  interpolation 18
• Advantages
• Fat surfaces illustrate potential range of data
  point locations
• Animation easy to detect uncertainty
• Fractals good for introducing geometric
  variability
• Disadvantages
• Animation not printable to paper, motion
  distracts from other variables

7
Uncertainty VisualizationProbabilistic Surfaces

• Application
• MRI of brain showing probabilities of cancerous
  tumor boundaries
• Requirements to be successful
• Information about both surface geometry and
  regional uncertainty
• Intuitive and not distracting
• Display of other variables in addition to
  uncertainty

8
Tumor Growth ModelGeneral Information

• Gompertz model 15
• V(t) V(0) exp( A/B ( 1 exp( -Bt ) ) )
• Parameters
• V(0) initial volume of tumor at time t 0
• A, B tumor growth parameters
• Trends
• t ? (as t approaches infinity)
• A/B final tumor size
• B rate of initial growth
• Used to build a large tumor from small tumors
  dispersed in a region

9
Tumor Growth ModelUncertainty

• Uncertainty equation
• d (V(t))/d t V(0) exp( A/B ( 1 exp( -Bt ) )
  ) A exp( -Bt )
• V(t) A exp( -Bt )
• Equation information
• Growth rate proportional to current tumor volume
  and an exponentially decaying term
• Inverse proportionality between cell density and
  uncertainty
• High cell density ? low uncertainty

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Tumor Growth ModelUncertainty

• Uncertainty equation
• d (V(t))/d t V(0) exp( A/B ( 1 exp( -Bt ) )
  ) A exp( -Bt )
• V(t) A exp( -Bt )
• Equation information
• t ? (as t approaches infinity)
• Uncertainty ? 0
• Growth rate ? 0
• Tumor reaches a constant volume

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Tumor Growth ModelMetastasis

• Description
• As cancer develops, cells separate from tumor
• Cells travel in blood vessels and settle in a new
  place
• Parameter in the growth model
• At an increasing time interval d t, tumor size
  increases
• At some size threshold, metastasis occurs
• New tumor center created at a random distance
  from the origin in the direction of the blood
  vessel flow
• Output as age along with an uncertainty value

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Geometry Alone vs.Geometry with Uncertainty
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Approach toUncertainty Points

• For each surface point, displace it along the
  surface normal of the point
• Point displacement is proportional to the points
  uncertainty value
• Reasons to use points instead of polygons
• Levoy Whitted (1985) pioneered the use of
  points as display primitives (paper reference
  8)
• Increasing scene complexity reduces algorithm
  design complexity
• Do not have to consider surfaces at polygons

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Improved Approach toUncertainty Points

• Transparency is appropriate for displaying
  uncertainty
• High uncertainty ? high transparency
• Result
• Highly uncertain regions will contain points with
  blurriness and large displacements

15
ImplementationInitialization

• Read in surface and uncertainty information
  (polygon mesh with uncertainty data at vertices
  uncertainty values between 0 and 1)
• Note The text for all implementation steps are
  directly quoted from the paper.

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ImplementationCreating Random Points

1. Create N random points inside each triangle (user
   is able to control the density of points)
2. Interpolate uncertainty values and normals from
   the vertices of the triangle onto each point in
   each triangle

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ImplementationCalculating Displacement

• For each point P do
• Calculate the displacement
• disp rand( )
• ( uncertainty at P )a
• ( scale factor )
• rand() returns a random number between the values
  of 0 and 1
• a and scale factor are controlled by the user

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Implementation Point Displacement Transparency

• For each point P do
• Displace P in the direction of the normal at P
• Calculate the transparency, alpha
• alpha 1
• ( uncertainty at P )b
• b is controlled by the user

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ImplementationOptional Pseudo-coloring

• For each point P do
• Pseudo-coloring
• If pseudo-coloring is enabled, assign the color
  of P by mapping the uncertainty at P through the
  current color map
• Otherwise, assign the default color
• End loop

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ImplementationDisplaying the Primitives

• Display all points
• Polygons
• If polygonal model usage is enabled, display
  all the polygons

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Implementation Results

• Simplest version
• Certain and uncertain areas are clearly displayed
• The uncertainty area of the tumor boundary is
  clearly defined
• Size of region proportional to level of
  uncertainty
• Best suited for interactive display
• Disadvantage
• Artifacts difficulty making dense areas smooth

Certain Region
Uncertain Region
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EnhancementAdding Polygonal Rendering

• Improving computational efficiency
• Use point rendering for uncertain areas
• Use polygonal rendering for certain areas
  regardless of its point density
• Advantages
• Well-defined low uncertainty areas
• More pleasing to view and more informative

Certain Region
Uncertain Region
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EnhancementAdjusting Transparency

• Adjust transparency according to uncertainty
  level using the equation
• a 1.0 err c
• err uncertainty value scaled from 0 to 1
• c rate that transparency increases with
  uncertainty constant value

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Reviewing Uncertainty Visualization Requirements

• Informative
• Polygonal (certainty) and point (uncertainty)
  rendering
• Intuitive/not distracting
• Transparency value uncertainty value
• Additional information
• Inclusion of tumor age
• Correlation between tumor age and uncertainty
• Seemingly contradictory items (blue, green
  arrows) remain valid for Equation 2

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Application to Diagnostic Data

• Datasets of CAT scan of human kidney tumors
• Creation of the visualization
• Input
• Each point is in 3-D with a density value
• Output
• Geometry used isovalue of tumor density to
  create the isosurface volume
• Certainty at the tumor surface, used inverse
  density gradient
• Results
• High density gradient ? sharp border
• Low density gradient ? fuzzy border
• Certainty data coexists with geometry data
• Scalable to larger datasets

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Images of Diagnostic Data

• Red arrows ? high uncertainty blue arrows ? low
  uncertainty

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Performance Tests

• System
• Red Hat Linux 6.1, Intel 1 GHz, 256 MB RAM,
  NVIDIA GeForce3 graphics card
• Metrics
• Polygons Polygon count in the model
• Density Points per single polygon in point-based
  model
• Points Point count in the model
• Display Time for displaying model after finished
  calculations
• Build Time for generating all primitives,
  calculating, etc.
• Miscellaneous information
• Figure 3 Basic point-based model
• Figure 4 Hybrid polygonal-point model
• Use of transparency does not affect running time

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Performance Data (from Table 1)
FFigure, POLYPolygons, DENDensity, PTPoints,
DISPDisplay, BBuild
F1 POLY18,532 DN/A PTN/A DISP0.034
B0.034
F5 POLY18,532 D20 PT370,640 DISP0.16
B1.33
F2 POLY18,532 DN/A PTN/A DISP0.034,
B0.033
F6 POLY28,088 D20 PT561,176 DISP0.25
B2.02
F3 POLY18,532 D100 PT1,853,200 DISP0.61
B5.9
F7 POLY26,486 D20 PT529,720 DISP0.32
B2.78
F4 POLY18,532 D20 PT370,640 DISP0.16
B1.33
F8 POLY75,248 D30 PT2,257,440 DISP0.89
B7.2
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Summary Future Work

• Point-based model for uncertainty visualization
  is successful
• Geometry and uncertainty data can coexist
• Adding polygonal mesh increases run-time
  efficiency
• Use of transparency creates an intuitive model
• Future improvements
• More parameters to display uncertainty
• Specular coefficient
• Refractive index

30
Thanks for listening
31
Paper References

• 1 R.E. Barnhill, K. Opitz, and H. Pottmann. Fat
  surfaces A trivariate approach to triangle-based
  interpolation on surfaces. Computer Aided
  Geometric Design, 9(5)365378, 1992.
• 2 Andrej Cedilnik and Penny Rheingans.
  Procedural annotation of uncertain information.
  In Proceedings of IEEE Visualization 00, pages
  7784. IEEE, 2000.
• 3 Baoquan Chen and Minh Xuan Nguyen. POP A
  hybrid point and polygon rendering system for
  large data. In Proceedings of Visualization 2001,
  pages 4552. IEEE, 2001.
• 4 C.R. Ehlschlaeger, A.M. Shortridge, and M.F.
  Goodchild. Visualizing spatial data uncertainty
  using animation. Computers in GeoSciences,
  23(4)387395, 1997.
• 5 Markus Gross. Are points the better graphics
  primitives? Computer Graphics Forum, 20(3),
  2001.
• 6 A.R. Kansal, S. Torquato, G.R.IV Harsh, E.A.
  Chiocca, and T.S. Deisboeck. Simulated brain
  tumor growth dynamics using a three-dimensional
  cellular automaton. Journal of Theoretical
  Biology, 203367382, 2000.
• 7 Gregory Kramer. Auditory Display,
  Sonification, Audification, and Auditory
  Interfaces, pages 178. Addison-Wesley, 1994.
• 8 Marc Levoy and Turner Whitted. The use of
  points as a display primitive. Technical Report
  85-022, University of North Carolina at Chapel
  Hill, January 1985.
• 9 S. K. Lodha, C. M. Wilson, and R. E. Sheehan.
  LISTEN sounding uncertainty visualization. In
  Proceedings of Visualization 96, pages 189195.
  IEEE, 1996.

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Paper References

• 10 Suresh Lodha, Robert Sheehan, Alex Pang, and
  Craig Wittenbrink. Visualizing geometric
  uncertainty of surface interpolants. In
  Proceedings of Graphics Interface, pages 238
  245, May 1996.
• 11 R. Minghim and A.R. Forrest. An illustrated
  analysis of sonification for scientific
  visualization. In Proceedings of Visualization
  95, pages 110117. IEEE, 1995.
• 12 A. T. Pang, C. M. Wittenbrink, and S. K.
  Lodha. Approaches to uncertainty visualization.
  The Visual Computer, 13(8)370 390, 1997.
• 13 Hanspeter Pfister and Jeroen van Baar.
  Surfels Surface elements as rendering
  primitives. In Kurt Akele, editor, Proceedings of
  SIGGRAPH 2000, pages 335342. ACM SIGGRAPH,
  Addison Wesley Longman, July 2000.
• 14 P. Rheingans and S. Joshi. Visualization of
  molecules with positional uncertainty. In E.
  Groller, H. Loffelmann, and W. Ribarsky,
  editors, Data Visualization 99, pages 299306.
  Springer-Verlag Wien, 1999.
• 15 G.G. Steel. Growth Kinetics of Tumors.
  Oxford Clarendon Press, 1977.
• 16 C. M. Wittenbrink, A. T. Pang, and S.K.
  Lodha. Glyphs for visualizing uncertainty in
  vector fields. IEEE Transactions on Visualization
  and Computer Graphics, 2(3)226279, 1996.
• 17 C.M. Wittenbrink, A.T. Pang, and S. Lodha.
  Verity visualization Visual mappings. Technical
  Report UCSC-CRL-95-48, University of California
  Santa Cruz, 1995.
• 18 Craig M. Wittenbrink. IFS fractal
  interpolation for 2D and 3D visualization. In
  Proceedings to Visualization 95, pages 7784.
  IEEE, October November 1995.

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Credits and References

• Non-paper sources
• Microsoft Excel 2000, Help Error bars in
  charts
• Microsoft Excel 2000, Help Examples of Chart
  Types XY Scatter
• Paper sources
• All images labeled figure
• Internet
• IFS http//www.cse.ucsc.edu/research/slvg/ifs.htm
  l
• Fat Surfaces http//www.ticam.utexas.edu/reports/
  1999/9908.pdf

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Questions and Comments