1. Ray Tracing Acceleration (3)

2. Ray Tracing Acceleration Techniques
Too Slow!
Faster intersection
Fewer rays
Generalized rays N 1
Uniform grids
Spatial hierarchies
K-D
Octtree
BSP
Hierarchical grids
Hierarchical
bounding
volumes (HBV)
Tighter bounds
Faster intersector
Early ray
termination
Adaptive sampling
Beam tracing
Cone tracing
Pencil tracing

3. Roadmap (LRT book, ch 4 – Intersection Acceleration)
Approaches To Reducing Intersections
Ray-Box Intersections
Regular Grid
Hierarchical bounding volumes
BSP trees and friends
Meta-Hierarchies
Refinements to basic approaches
Hierarchical Grid Accelerator
Creation
Traversal
Kd Tree
Tree Representation
Tree construction
Further Reading

4. Spatial Data Structures
Create a data structure aware of the spatial relations
Partition space and place objects within subregions
Only consider subregions that the ray passes through
Avoid computing intersections twice if object lies inside multiple subregions

5. Spatial Decompositions:
Focus on spatial data structures that partition space into regions, or cells, of some type
Generally, cut up space with planes that separate regions
Always based on tree structures

6. Tree-structure Hierarchy: Variations
KD tree
Quadtree (2D)
Octree (3D)
BSP tree

7. Spatial Decompositions: Tree-structure Hierarchy
Kd-trees: Axis aligned planes, in alternating directions, cut space into rectilinear regions

8. Spatial Decompositions: Tree-structure Hierarchy
Octrees (Quadtrees): Axis aligned, regularly spaced planes cut space into cubes (squares)

9. Spatial Decompositions: Tree-structure Hierarchy
BSP Trees: Arbitrarily aligned planes cut space into convex regions

10. Creating Spatial Hierarchies
insert(node,prim) {
if( overlap(node->bound,prim) )
if( leaf(node) ) {
if( node->nprims > MAXPRIMS &&
node->depth < MAXDEPTH ) {
subdivide(node);
foreach child in node
insert(child,prim) }
else list_insert(node->prims,prim);
else
// Typically MAXDEPTH=16, MAXPRIMS=2-8

11. K-d trees (K-dimensional)

12. K-d trees
An axis-aligned plane is introduced to break down each volume into two sub-volumes.
The subdivision stops when the number of objects overlapping a cell falls below a certain threshold.
As in uniform grid, objects must be assigned to cells.

13. A A
Leaf nodes correspond to unique regions in space

14. A B A
Leaf nodes correspond to unique regions in space

15. A B B A
Leaf nodes correspond to unique regions in space

16. C A B B A

17. C A B C B A

18. C A D B C B A

19. C A D B C B D A

20. K-d tree (Spatial Hierarchies)B B C C D A
Leaf nodes correspond to unique regions in space

21. Kd-tree Example 4 6 1 10 8 2 3 11 3 13 2 4 5 6 7 12 8 9 10 11 12 13 9 5 1 7

22. Heuristics on where to put the partitioning plane
Median-cut: .find the plane that puts approximately equal number of objects at each side.

23. Heuristics on where to put the partitioning plane: Median-Cut
Find the plane that puts approximately equal number of objects at each side.
Build hierarchy bottom-up
Chose direction and position carefully
Surface-area heuristic

24. Heuristics on where to put the partitioning plane: Surface Area and Rays
The number of rays in a given direction that hits the object is proportional to the projected area A.

25. Crofton’s Theorem states that
Heuristics on where to put the partitioning plane: Surface Area and Rays
And rays in all directions that hits the object is proportional to the surfaces area S.
Crofton’s Theorem states that
is average projected area in all directions.

26. Crofton’s theorem: For a convex body: Example: Sphere
Heuristics on where to put the partitioning plane: Surface Area and Rays
Crofton’s theorem:
For a convex body:
Example: Sphere

27. Heuristics on where to put the partitioning plane: Surface Area and Rays
Probability of a ray hitting an object that is completely inside a cell is:

28. K-d tree (Spatial Hierarchies)B B C C D A
Point location by recursive search

29. Search for  C A D B C B D A

30. Search for 
Left or Right of partition plane A? C A D B C B D A

31. Search for  C A
Left or Right of partition plane B? D B C B D A

32. Search for  C A D B
Left or Right of partition plane C? C B D A

33. Point location by recursive search
FOUND ! C A D B C B D A
Point location by recursive search
Leaf nodes correspond to unique regions in space

34. Ray Traversal Algorithms
tmin, tmax are near and far intersection points of the ray with the current volume;
t* is the intersection of ray with partitioning plane.

35. Ray Traversal Algorithms
Recursive inorder traversal
Kaplan, Arvo, Jansen
Intersect(L,tmin,tmax)
Intersect(L,tmin,t*)
Intersect(R,t*,tmax)
Intersect(R,tmin,tmax)

36. Ray Traversal Algorithms
The previous slide applies for ray going to the right.
The dot product of ray direction with plane normal r · n tells where the ray is going with respect to the plane.

37. 3-d Trees (K-d Trees)

38. Tree-structure Hierarchy: Variations
Similarly, we can construct quadtree, octree, etc.
In BSP tree, the partitioning plane is not necessarily axis-aligned
As in uniform grid, objects must be assigned to cells
KD tree
Quadtree (2D)
Octree (3D)
BSP tree

39. Other Partitioning Structures
Octree
Ray can parse through large empty areas
Requires less space than grid
Subdivision takes time
Binary Space Partition (BSP) Tree
Planes can divide models nearly in half
Trees better balanced, shallower
Added ray-plane intersections

40. octree

41. Octrees

42. Octrees

44. BSP Example 1 1 2 2 4 A 3
out 5 7 3 A
out B 6 8
out 6 5 C B C
out D
out
Notes:
Splitting planes end when they intersect their parent node’s planes
Internal node labeled with planes, leaf nodes with regions D 4 8 7

45. BSP Example

47. BSP Trees

48. Ray Tracing BSP Trees