1. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids Stefan Roettger and Thomas Ertl VIS Group University of Stuttgart

2. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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

3. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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

4. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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

5. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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  (f(x,y,z)) and  (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

6. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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 S f, S b, and l for the Volume Density Optical Model of Williams et al.

7. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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 (S f,S b,l) to the projected vertices of each tetrahedron

8. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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

9. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 First Step Goal: Accelerate pre-integration of the ray integral with graphics hardware!

10. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 Setup of 3D Texture Numerical Pre-Integration: iterate over l iterate over S b iterate over S f chromaticity C 0 =0 transparency T 0 =1 iterate over #steps (i=0...n-1) S=(1-i/(n-1))S b +i/(n-1)S f compute emission=  (S)*l/(n-1) compute absorption=exp(-  (S)*l/(n-1)) C i+1 =C i *absorption+emission T i+1 =T i *absorption Tex3D(S f,S b,l)=(C n-1,1-T n-1 )

11. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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 l=const in parallel using the graphics hardware Store transfer function in 1D texture with Tex1D(s)=(  (s)l/(n-1),exp(-  (s)l/(n-1))) Render n slices into frame buffer with the following setup for the 1D texture coordinate s:

12. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 Hardware-Accelerated Pre-Integration Setup for the 1D texture coordinate s:

13. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 Hardware-Accelerated Pre-Integration Speed Results on an SGI Octane MXI with 250 MHz MIPS R10k processor:

14. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 Hardware-Accelerated Pre-Integration Quantization artifacts due to 12/8 bits of accuray:

15. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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

16. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral Proof: The texture coordinates of the thick vertex are (S f,S b,l) but for a silhouette vertex the texture coordinates are just (S b,S b,0).

17. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral The front diagonal is the cylinder axis and the thick vertex defines the rotational parameter .

18. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral The 3D texture is resampled by a set of planes with different  To yield uniform slices each plane is projected onto the nearest base plane with  =0,90, or 180 Then the ray integral can be reconstructed for each tetrahedron from the corresponding two planes with the nearest  using two-pass multi- texturing

19. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral The resampling of the 3D texture is performed with the following mapping:

20. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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 (  0,S b )

21. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral Example: 8 slices of a 64^3 3D texture Corresponding polarized textures Transfer function

22. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral Spherical distance volume rendered with the previous 2D texture map and a bonsai tree:

23. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 2D-Reparametrization of the Ray Integral Performance comparison between 3D and 2D texture mapping:

24. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 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)

25. Stefan Roettger University of Stuttgart A Two-Step Approach for Interactive Pre-Integrated Volume Rendering of Unstructured Grids VolVis '02 Thank You!!! Questions?