1. Presented by Paul Phipps
Efficient collision detection using bounding volume hierarchies of k-DOPs by James T. Klosowski, Martin Held, Joseph S.B. Mitchell, Henry Sowizral, and Karel Zikan
Presented by Paul Phipps

2. Overview Background Collision Detection Perspective k-DOP
Cost Function
Design Choices for BV Tree
Tumbling the k-DOPs
Experimental Results
Future Work

3. Background Collision Detection Do fewer comparisons Approaches
Pure detection
Detect and report
Do fewer comparisons
Between pairs of objects
Between pairs of primitives
Approaches
Spatial Decomposition
Octrees, k-d trees, BSP-trees, brep-indices, tetrahedral meshes, and (regular) grids
Bounding Volumes Hierarchy
Spheres, axis-aligned bounding boxes (AABBs), oriented bounding box (OBB) (“RAPID” uses OBBTrees)
Miscellaneous

4. Collision Detection Perspective
Assumptions
Rigid bodies
Discrete points in time
Typical input:
Static object (the environment)
Moving object (the flying hierarchy)
Goals:
Accuracy
Real-time rates
Haptics can require over 1000 collision queries per second

6. Contributions… “k-DOP” (Discrete Orientation Polytope)
convex polytope whose facets are determined by halfspaces whose outward normals come from a small fixed set of k orientations
Axis Aligned Bounding Box == 6-DOP
((using axes +x, -x, +y, -y, +z, and -z)
k-DOPs used in experiments:
6-DOP, 14-DOP, 18-DOP, 26-DOP

7. …Contributions
Compare ways to construct a Bounding Volume Hierarchy (“BV-tree”) of k-DOPs
Algorithms
Maintain k-DOP BV-tree for moving objects
Translation
Rotation
Fast collision detection
Using BV-trees of moving object and of environment
Results with real and simulated data

10. Cost Function N = number of occurrences C = cost per occurrence
Bounding Volume Overlap Tests
Primitive Overlap Tests
Updates of Hierarchy nodes
N = number of occurrences
C = cost per occurrence

11. k-DOP Advantages over other BV
Tighter fit than AABB or Sphere
The higher the k  the lower Nv, Np, and Nu
Only k values to remember for a BV (using opposite-pointing orientations)
Simpler overlap tests than OBB
Just do (k / 2) interval overlap tests
The parameter k can be chosen to get a good balance between tight fit and quick overlap test

12. Design Choices for BV Tree
Branching degree 2 (binary tree) is good
Simple to implement
Simple to traverse tree
Splitting rule for pre-computing the tree structure
Pick either x, y, or z axis (using various tests)
Sort along that axis, then use
Median
Mean
Recur

14. Tumbling the k-DOPs “hill climbing” method “approximation method”
For Root Node
“hill climbing” method
Use pre-computed convex hulls
Are extreme vertices still extreme?
If not, “climb” to more extreme neighbors
Advantage: tight
“approximation method”
Rotate vertices of k-DOP
Get new k-DOP
Don’t accumulate error: Rotate from pre-computed vertices in Model-Space
Advantage: fast
For non-Root Nodes

15. Algorithm 1

22. Future Work Use of temporal coherence Multiple flying objects
Dynamic environments
Deformable objects
Numerically Controlled Verification

23. Overview Background Collision Detection Perspective k-DOP
Cost Function
Design Choices for BV Tree
Tumbling the k-DOPs
Experimental Results
Future Work

24. Questions?

25. Continuous Collision Detection of Deformable Objects using k-DOPs