Presented by Paul Phipps

1. Efficient collision detection using bounding volume hierarchies of k-DOPsbyJames 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 • 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

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 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 Detectionof Deformable Objects using k-DOPs