A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids

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A Two-Step Approach forInteractive
Pre-Integrated Volume Renderingof Unstructured
Grids

• Stefan Roettger and Thomas Ertl
• VIS Group
• University of Stuttgart

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Outline of the Talk

• Introduction to Unstructured Volume Rendering
• Previous 3D Texturing Approach
• Drawbacks
• First Step Hardware-Accelerated Pre-Integration
• Second Step 2D Ray Integral Reparametrization
• Results for each step
• Conclusion

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Unstructured Volume Rendering

• Given an irregular data set that consists of
  volumetric cells (typically from FEM simulation)
• How can the volume be displayed accurately?
• Numerous approaches
• Ray casting
• Ray tracing
• Sweep plane algorithms (e.g. ZSWEEP)
• PT algorithm of Shirley and Tuchman

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PT algorithm of Shirley and Tuchman

• Decompose each cell into tetrahedra
• Sort the tetrahedra in a back to front fashion
• Project each tetrahedron and render its
  decomposition into 3 or 4 triangles

Two different non-degenerate classes of the
projected tetrahedra
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Volume Density Optical Model

• For the Volume Density Optical Model of Williams
  et al. the emission and absorption along a light
  ray is defined by the transfer functions
  k(f(x,y,z)) and r(f(x,y,z)) with f(x,y,z) being
  the scalar function
• Usually the transfer functions are given as a
  linear or piecewise linear function, or as a
  lookup table

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Composing of the Tetrahedra

• For each rendered pixel the ray integral of the
  corresponding ray segment has to be computed
• Observation The ray integral depends only on Sf,
  Sb, and l for the Volume Density Optical Model of
  Williams et al.

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3D Texturing Approach

• Compute the three-dimensional ray integral by
  numerical integration and store the integrated
  chromaticity and opacity in a 3D texture
• Assign appropriate texture coords (Sf,Sb,l) to
  the projected vertices of each tetrahedron

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Pros / Cons

• Pros
• Object order method
• Hardware-accelerated approach
• Per-pixel exact rendering
• Cons
• 3D textures not available everywhere
• 3D textures are SLOW
• Numerical integration can take several minutes ?
  interactive updates not possible yet

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First Step

• Goal Accelerate pre-integration of the ray
  integral with graphics hardware!

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Setup of 3D Texture

• Numerical Pre-Integration

iterate over l iterate over Sb iterate
over Sf chromaticity C00
transparency T01 iterate over steps
(i0...n-1) S(1-i/(n-1))Sbi/(n-1)Sf
compute emissionk(S)l/(n-1)
compute absorptionexp(-r(S)l/(n-1))
Ci1Ciabsorptionemission
Ti1Tiabsorption
Tex3D(Sf,Sb,l)(Cn-1,1-Tn-1)
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Hardware-Accelerated Pre-Integration

• Hardware-Accelerated Pre-Integration means that
  the numerical pre-integration is accelerated by
  the graphics hardware
• Compute one slice of the 3D texture for lconst
  in parallel using the graphics hardware
• Store transfer function in 1D texture with
  Tex1D(s)(k(s)l/(n-1),exp(-r(s)l/(n-1)))
• Render n slices into frame buffer with the
  following setup for the 1D texture coordinate s

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Hardware-Accelerated Pre-Integration

• Setup for the 1D texture coordinate s

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Hardware-Accelerated Pre-Integration

• Speed Results on an SGI Octane MXI with 250 MHz
  MIPS R10k processor

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Hardware-Accelerated Pre-Integration

• Quantization artifacts due to 12/8 bits of
  accuray

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Second Step

• Goal Replace 3D with 2D texturing approach
• 2D-Reparametrization of the Ray Integral means
  that in cylinder coordinates the
  three-dimensional ray integral reduces to a
  two-dimensional ray integral for each
  tetrahedron.
• Claim All the rasterized pixels of a tetrahedron
  lie on a plane in texture coordinate space

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2D-Reparametrization of the Ray Integral

• Proof The texture coordinates of the thick
  vertex are (Sf,Sb,l) but for a silhouette vertex
  the texture coordinates are just (Sb,Sb,0).

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2D-Reparametrization of the Ray Integral

• The front diagonal is the cylinder axis and the
  thick vertex defines the rotational parameter a.

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2D-Reparametrization of the Ray Integral

• The 3D texture is resampled by a set of planes
  with different a
• To yield uniform slices each plane is projected
  onto the nearest base plane with a0,90, or 180
• Then the ray integral can be reconstructed for
  each tetrahedron from the corresponding two
  planes with the nearest a using two-pass
  multi-texturing

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2D-Reparametrization of the Ray Integral

• The resampling of the 3D texture is performed
  with the following mapping

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2D-Reparametrization of the Ray Integral

• For each tetrahedron the 3D texture coordinates
  of the thick vertex are transformed as follows
• For a silhouette vertex this reduces to (a,0,Sb)

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2D-Reparametrization of the Ray Integral

• Example

Transfer function
8 slices of a 643 3D texture
Corresponding polarized textures
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2D-Reparametrization of the Ray Integral

• Spherical distance volume rendered with the
  previous 2D texture map and a bonsai tree

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2D-Reparametrization of the Ray Integral

• Performance comparison between 3D and 2D texture
  mapping

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Conclusion

• Pre-Integrated Unstructured Volume Rendering
  improved for practical application
• Update rate of the transfer function was
  increased by almost a factor of hundred ?
  comfortable exploration of unstructured data sets
  by changing the transfer functions interactively
• 2D texture mapping approach allows faster
  rasterization and application on a broader range
  of graphics accelerators (e.g. on this iBook)

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Thank You!!!

• Questions?