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

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

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

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

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

…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

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

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

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

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

Algorithm 1

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

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

Questions?

Continuous Collision Detection of Deformable Objects using k-DOPs