Collision Detection II. These slides are mainly from Ming Lin s course notes at UNC Chapel Hill

Transcription

1 Collision Detection II These slides are mainly from Ming Lin s course notes at UNC Chapel Hill

2 Some Possible Approaches Geometric methods Algebraic techniques Hierarchical Bounding Volumes Spatial Partitioning Others (e.g. optimization) ILE5030 Computer Animation and Special Effects 2

3 BVH vs. Spatial Partitioning BVH: SP: - Object centric - Space centric - Spatial redundancy - Object redundancy (Recall from Ming Lin s Notes) ILE5030 Computer Animation and Special Effects 3

4 BVH-Based Collision Detection ILE5030 Computer Animation and Special Effects 4

5 Type of Bounding Volumes Spheres Ellipsoids Axis-Aligned Bounding Boxes (AABB) Oriented Bounding Boxes (OBBs) Convex Hulls k-discrete Orientation Polytopes (k-dop) Spherical Shells Swept-Sphere Volumes (SSVs) ILE5030 Computer Animation and Special Effects 5

6 Trade-off in Choosing BV s Sphere AABB OBB 6-dop Convex Hull increasing complexity & tightness of fit decreasing cost of (overlap tests + BV update) ILE5030 Computer Animation and Special Effects 6

7 Evaluating Bounding Volume Hierarchies Cost Function: F = N u x C u + N bv x C bv + N p x C p F: total cost function for interference detection N u : no. of bounding volumes updated C u : cost of updating a bounding volume, N bv : no. of bounding volume pair overlap tests C bv : cost of overlap test between 2 BVs N p : no. of primitive pairs tested for interference C p : cost of testing 2 primitives for interference ILE5030 Computer Animation and Special Effects 7

8 Designing Bounding Volume Hierarchies The choice governed by these constraints: It should fit the original model as tightly as possible (to lower N bv and N p ) Testing two such volumes for overlap should be as fast as possible (to lower C bv ) It should require the BV updates as infrequently as possible (to lower N u ) ILE5030 Computer Animation and Special Effects 8

9 Observations Simple primitives (spheres, AABBs, etc.) do very well with respect to the second constraint. But they cannot fit some long skinny primitives tightly. More complex primitives (minimal ellipsoids, OBBs, etc.) provide tight fits, but checking for overlap between them is relatively expensive. Cost of BV updates needs to be considered. ILE5030 Computer Animation and Special Effects 9

10 Building Hierarchies Choices of Bounding Volumes cost function & constraints Top-Down vs. Bottum-up speed vs. fitting Depth vs. breadth branching factors Splitting factors where & how ILE5030 Computer Animation and Special Effects 10

11 BVH-based Methods Sphere-trees K discrete oriented polytopes (K-DOPs) OBB-trees ILE5030 Computer Animation and Special Effects 11

12 Sphere-Trees A sphere-tree is a hierarchy of sets of spheres, used to approximate an object Advantages: Simplicity in checking overlaps between two bounding spheres Invariant to rotations and can apply the same transformation to the centers, if objects are rigid Shortcomings: Not always the best approximation (esp bad for long, skinny objects) Lack of good methods on building sphere-trees ILE5030 Computer Animation and Special Effects 12

13 k-dop s k-dop: k-discrete orientation polytope a convex polytope whose facets are determined by half-spaces whose outward normals come from a small fixed set of k orientations For example: In 2D, an 8-dop is determined by the orientation at +/- {45,90,135,180} degrees In 3D, an AABB is a 6-dop with orientation vectors determined by the +/-coordinate axes. ILE5030 Computer Animation and Special Effects 13

14 Choices of k-dops in 3D 6-dop: defined by coordinate axes 14-dop: defined by the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,1), (1,-1,1), (1,1,-1) and (1,-1,-1) 18-dop: defined by the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,0), (1,0,1), (0,1,1), (1,-1,0), (1,0,-1) and (0,1,-1) 26-dop: defined by the vectors (1,0,0), (0,1,0), (0,0,1), (1,1,1), (1,-1,1), (1,1,-1), (1,-1,-1), (1,1,0), (1,0,1), (0,1,1), (1,-1,0), (1,0,-1) and (0,1,-1) ILE5030 Computer Animation and Special Effects 14

15 Building an OBBTree Recursive top-down construction: partition and refit ILE5030 Computer Animation and Special Effects 15

16 Building an OBB Tree Given some polygons, consider their vertices... ILE5030 Computer Animation and Special Effects 16

17 Building an OBB Tree and an arbitrary line ILE5030 Computer Animation and Special Effects 17

18 Building an OBB Tree and an arbitrary line ILE5030 Computer Animation and Special Effects 18

19 Building an OBB Tree Project onto the line Consider variance of distribution on the line ILE5030 Computer Animation and Special Effects 19

20 Building an OBB Tree Different line, different variance ILE5030 Computer Animation and Special Effects 20

21 Building an OBB Tree Maximum Variance ILE5030 Computer Animation and Special Effects 21

22 Building an OBB Tree Minimal Variance ILE5030 Computer Animation and Special Effects 22

23 Building an OBB Tree Given by eigenvectors of covariance matrix of coordinates of original points ILE5030 Computer Animation and Special Effects 23

24 Building an OBB Tree Choose bounding box oriented this way ILE5030 Computer Animation and Special Effects 24

25 Building an OBB Tree Good Box ILE5030 Computer Animation and Special Effects 25

26 Building an OBB Tree Add points: worse Box ILE5030 Computer Animation and Special Effects 26

27 Building an OBB Tree More points: terrible box ILE5030 Computer Animation and Special Effects 27

28 Building an OBB Tree Compute with extremal points only ILE5030 Computer Animation and Special Effects 28

29 Building an OBB Tree Even distribution: good box ILE5030 Computer Animation and Special Effects 29

30 Building an OBB Tree Uneven distribution: bad box ILE5030 Computer Animation and Special Effects 30

31 Building an OBB Tree Fix: Compute facets of convex hull... ILE5030 Computer Animation and Special Effects 31

32 Building an OBB Tree Better: Integrate over facets ILE5030 Computer Animation and Special Effects 32

33 Building an OBB Tree and sample them uniformly ILE5030 Computer Animation and Special Effects 33

34 OBB Overlap test L h a s h b L is a separating axis iff: s > h + h ILE5030 Computer Animation and Special Effects 34 a b

35 Separating Axis Theorem Given two convex shapes, if we can find an axis along which the projection of the two shapes does not overlap, then the shapes don't overlap ILE5030 Computer Animation and Special Effects 35

36 OBB Overlap Test Project boxes onto axis. If intervals don t overlap, it is a separating axis. A separating axis exists if and only if boxes are disjoint. ILE5030 Computer Animation and Special Effects 36

37 Building an OBB Tree: Summary OBB Fitting algorithm: covariance-based use of convex hull not foiled by extreme distributions O(n log n) fitting time for single BV O(n log 2 n) fitting time for entire tree ILE5030 Computer Animation and Special Effects 37

38 Hybrid Hierarchy of Swept Sphere Volumes E. Larsen, S. Gottschalk, M.C. Lin, D. Manocha, Fast Proximity Queries with Swept Sphere Volumes, ICRA 2000 Point Swept Spheres (PSS) Line Swept Spheres (LSS) Rectangle Swept Spheres (RSS) ILE5030 Computer Animation and Special Effects 38

39 Swept Sphere Volumes PSS LSS RSS ILE5030 Computer Animation and Special Effects 39

40 SSV Fitting Use OBB s code based upon Principle Component Analysis For PSS, use the largest dimension as the radius For LSS, use the two largest dimensions as the length and radius For RSS, use all three dimensions ILE5030 Computer Animation and Special Effects 40

41 Collision Detection in Large Environments Need to overcome the bottleneck of O(n 2 ) pairwise tests Two phases Broad phase: identify potential collisions (roughly) Spatial partitioning Sweep and prune Narrow phase: check each pair for exact collision Convex objects Spatial partitioning BVH ILE5030 Computer Animation and Special Effects 41

42 N-body Collision Detection Architecture Transform Sweep & Prune Overlap Simulation Exact Collision Detection Parameters Analysis & Response Collision ILE5030 Computer Animation and Special Effects 42

43 Sweep and Prune Compute the axis-aligned bounding box (fixed vs. dynamic) for each object Dimension Reduction by projecting boxes onto each x, y, z- axis Sort the endpoints and find overlapping intervals Possible collision -- only if projected intervals overlap in all 3 dimensions ILE5030 Computer Animation and Special Effects 43

44 Sweep & Prune T = 1 e 3 e 2 b e 3 1 b 2 b 1 b 2 e 1 e 2 b 3 e 3 b 1 T = 2 e 2 e 1 b 2 e 3 b 1 b 1 b 2 e 1 e 2 b 3 e 3 b 3 ILE5030 Computer Animation and Special Effects 44

45 Acceleration Using Graphics Hardware Govindaraju et al., CULLIDE, EWGH 03 Visibility-based pruning Compute image visibility to check for overlaps An object O does not collide with a set of objects S if O is fully-visible with respect to S Cull those objects that cannot be colliding Accuracy governed by image resolution Suitable for any polygon mesh, large scene Multiple deformable, breakable objects Cannot handle self-collision ILE5030 Computer Animation and Special Effects 45

46 Chromatic Decomposition Govindaraju et al., SIGGRAPH 05 Modify CULLIDE to handle self-collision transforms self-collision detection into pair-wise N-body CD between non-adjacent primitives Decompose the mesh into k independent sets S 1,,S k For every pair of independent set, (S i, S j ), ensure each primitive in S i has only one adjacent primitive that is in S j Building a corresponding graph G, and decompose it with graph coloring ILE5030 Computer Animation and Special Effects 46

47 Collision Detection Packages Various packages by UNC Chapel Hill QuickCD general-purpose CD library based on K-DOPs SOLID 3D polygonal objects undergo rigid motion Interactive 3D graphics OPCODE Similar to SOLID and RAPID, but more memory friendly and faster ILE5030 Computer Animation and Special Effects 47

48 Videos Govindaraju, Knott, Jain, Kabul, Tamstorf, Gayle, Lin, and Manocha, SIGGRAPH ILE5030 Computer Animation and Special Effects 48

49 Videos ILE5030 Computer Animation and Special Effects 49

50 Videos ILE5030 Computer Animation and Special Effects 50