1. Unstructured Volume Rendering Jian Huang, CS 594, Spring 2002 This set of slides reference slides developed by Prof. Torsten Moeller, SFU, Canada

2. Grid Types uniformrectilinearregularcurvilinear Structured Grids: regularirregularhybridcurved Unstructured Grids:

3. Unstructured Grids still assume scalar data translation - as before translation matrix still the same classification - as before because it is still a scalar data type shading - same shading model(s) interpolation - problematic compositing - still BTF or FTB problem - how to sort?

4. Ray-Casting Garrity 1990 needs to figure out, where the ray is going then needs to figure out how to integrate along that ray assumes arbitrary polyhedra (as opposed to just tetrahedra as many other algorithm) image order FTB

5. Ray-Casting - basic algorithm For each ray –find intersection with first cell –interpolate a color, opacity value –while not outside the volume find exit point of ray in cell interpolate a color, opacity value integrate opacity/color along ray within cell

6. Ray-Casting - intersection point just look at all faces compute the intersection with all faces of the current cell and take the nearest requires cell organized data structure first cell - only consider faces that belong to only one cell A D C B

7. Ray-Casting - intersection point Intersection of ray defined by point d and direction vector e: d a c b

8. Ray-Casting - rendering integral classify scalar values at nodes into color/opacity it then linearly interpolates these values within a plane for the intersection points integrate along the cell by multiplying the average with the distance (another linear interpolation at work) needs the gradient for shading A D C B

9. Ray-Casting - gradients Vertices are not nicely aligned in x/y/z direction hence, we need to solve an equation system using any three pairs of neighboring points

10. Ray-Casting - conclusion Bottleneck - size of image (# of pixels) results for 512 2 image: 960 cells - 81.6 seconds 1490 cells - 147.6 seconds 16,896 cells - 149.2 seconds 381,548 cells - 591.6 seconds may miss small polyhedra!!

11. Cell Projection Max, Hanrahan, Crawfis 1990 Shirley, Tuchman 1990 object order (cannot miss small structures) makes sense if one cell projects to many pixels based on tetrahedral decomposition

12. Cell Projection - basic algorithm Decompose volume into tetrahedral cells classify each tetrahedron according to its projected profile relative to a viewpoint find the positions of the tetrahedra vertices after the perspective transformation decompose projections of the tetrahedra into triangles find color and opacity values for the triangle vertices using ray integration in the original world coordinates scan convert the triangles on a graphics workstation

13. Cell Projection - tetrahedra Decompose volume into tetrahedral cells 5 or 6 tetrahedra are possible want least amount of computation for 5 - need to make sure that there will be no “cracks”

14. Cell Projection - tetrahedra for 5 - need to make sure that there will be no “cracks” I.e. edges need to line up properly

15. Cell Projection - projection classify each tetrahedron according to its projected profile relative to a viewpoint find the positions of the tetrahedra vertices after the perspective transformation

16. Projected Tetrahedra - decompose decompose projections of the tetrahedra into triangles

17. Cell Integration find color and opacity values for the triangle vertices using ray integration in the original world coordinates Sabella assumes, that volume density  (x,y,z) glows with an energy of C  per unit length and absorbs with an optical density of  (C and  are constants for a fixed material

18. Cell Integration How can we evaluate the rendering integral:

19. Cell Integration How can we evaluate D? linear interpolation (A=P(a) - P is the ray equation): and now:

20. Cell Integration How can we find S?  (u) is basically the transfer function it’s integral S(t) over a linear function could be pre- computed there is no restriction to the transfer function really scan convert the triangles on a graphics workstation

21. Projected Tetrahedra Shirley et al: find color and opacity values for the triangle vertices using ray integration in the original world coordinates at the thickest point scan convert the triangles on a graphics workstation then linearly interpolate this integral on the scanline can be implemented in hardware 0 I

22. Cell Integration Max et al: the integral is computed for each pixel/object scanline along the front and back face to determine front and back values of S(u). then we evaluate I with our formula

23. Depth Sorting But we need to repeat all these steps for all cells how to sort those cells? Max et al -> topological sort first: create a directed graph –nodes - cells –edges - faces –direction of edge means “in front of”

24. Depth Sorting do a topological sort of this graph (in linear time - linear in the order of the number of edges) only works if there are no cycles!

25. Rendering by Slicing Yagel et al 1996 uses the texture mapping hardware only implemented for tetrahedral grids cannot use the same algorithm as for regular grids where the slices are aligned along major axis we have to intersect the underlying geometry

26. Slicing - basic algorithm For any change in the viewing parameters –transform grid points into image space –determine z min and z max –dz = (z min -z max )/NumSlices –for i:=1 to NumSlices do slice the transformed cells with plane z slice render 2D polymesh and compose (hardware) z slice := z slice +dz for any change in the viewing parameters re-render polylist

27. Slicing - adaptive slicing Problem - might miss small detail idea: just keep (adaptively) subdividing until we sliced every polyhedra in that particular interval the compositing step must take care of different distance of slices

28. Slicing - progressive slicing Problem: How many slices do we need for a good image? Idea: time-quality tradeoff start with very few slices -> interactive display as long as the user isn’t changing the viewing parameters, keep refining -> hence progressively better images!

29. Splatting Mao, Hong, Kaufman 1995/96 extends the regular splatting ideas to irregular domain based on a resampling step based on Poisson distribution of new sample points

30. Splatting - basic algorithm For each cell –compute a random point –compute its extent (sphere/ellipse) by interpolating the gradients from the vertices –see whether it fits in the cell sort each ellpsoid/sphere splat each ellipsoid/sphere

31. Splatting - Poisson distribution Poisson distribution

32. Splatting - basic algorithm Pre-processing takes very long (days!!!) number of points increases dramatically alternative - just compute approximation of “bean- shapes” at the vertices (ellipsoids)

33. Splatting - sorting Mao doesn’t report anything specific can be done by quicksort better - topological sort