1. Collision Detection for Deformable Models Huai-Ping Lee lhp@cs.unc.edu

2. Differences in Deformable Models Collision and self-collisions –Self collisions are often neglected for rigid bodies Preprocessing –Data structure need to be updated frequently Performance –Efficiency is very important

3. Hybrid Approach [LAM01] Goal: adapt BVHs to handle deformable models efficiently Some modification in building and updating the tree –Efficiency of updating hierarchies is more important than the tightness of BVs AABBs are preferred

4. Hybrid Approach [LAM01] For a bottom-up update strategy using AABBs [vdB97], 8-ary tree version is 10 to 20 percent faster than binary version –Fewer nodes need to be updated (if using top- down approach) –Recursion depth during collision tests is lower

5. Hybrid Approach [LAM01] Bounding volume pre-processing –8-ary AABB tree built in top-down manner –A parent AABB is split along three axis to form eight child sub-volumes –No significant difference between ways of choosing split planes Center of the box or average point of all polygons

6. Hybrid Approach [LAM01] Run-time update –Hybrid of top-down and bottom-up updates –For a tree with depth n, initially update the n/2 first levels bottom-up. –During a collision traversal, update those non- updated nodes top-down as needed

7. Hybrid Approach [LAM01] Results for hard cases –All intersecting face pairs are reported

8. Hybrid Approach [LAM01] Results for simple cases –Only the first intersecting face pair is reported

9. Hybrid Approach [LAM01] Improved bounding volume hierarchies for deformable models –More efficient update Self-collisions are not considered

10. Lazy Update [MKE03] Another improvement to BVHs –Using k-DOPs –Build the tree top-down Also reported that 4-ary and 8-ary trees are better –Lazy update Re-inserts the vertices into the leaf k-DOPs and build internal nodes bottom-up Also want to detect self-collision

11. Lazy Update [MKE03] Knowing maximum velocity of the vertices, some BVs need not be updated –Parts of the hierarchy where vertices do not travel more than a distance b can be omitted during the hierarchy update for a time t = b / v, if proximities smaller than ε close – 2b is to be detected The BVs have been fattened by ε close / 2

12. Lazy Update [MKE03] BVHs are still inappropriate when detecting self-collisions –bounding boxes will always find contacts between adjacent sub-objects Test the BVH against itself? –Need to skip some tests between adjacent sub-surfaces –Previous solutions: [VMT94] and [Pro97] –This paper uses method in [Pro97]

13. Curvature Criterion [VMT94] If: There exists a vector V for which N.V > 0 at every point of S –And: The projection of C on a plane orthogonal to V along the direction of V has no self-intersections –Then: There are no self-collisions on the surface S.

14. Curvature Criterion [VMT94] For each sub-surface –Search for V –If V exists, test the projected region for self- intersection –If both succeeded, there is no self-intersection –Otherwise, check for self intersections in the sub-surface

15. Curvature Criterion [VMT94] V can be propagated bottom-up in the tree –Divide a sphere into 14 unit vectors –In each node, keep those vectors that have positive dot products with all the normals in the BV

16. Normal Cones [Pro97] In each BV, keep a cone representing a super set of normal directions Parent cones are easily computed from child cones –α =β/2 + max(α 1, α 2 ) If α ≧ π, check for self-intersection

17. Lazy Update [MKE03] Another way to improve hierarchy update Also detects self-intersection using normal cones Results –HU=Hierarchy Update, CT=Collision Test

18. Morphing of Tree [LAM03] Accelerate the special case in which models are deformed by mesh morphing –First establish the correspondence between geometric parts in reference models, assuming all models have the same number of vertices and mesh connectivity –Interpolate between these parts –The models in each frame are formed by linear blending the n reference models

19. Morphing of Tree [LAM03] Tree building (top-down) –Add one BV per node in the tree for each reference model –Namely, each node in the tree contain n BVs BVs are updated by blending the bounding volumes of corresponding sub-models –using the same weights for linear blending

20. Morphing of Tree [LAM03] Experiment—three reference models

21. Morphing of Tree [LAM03] Compared with hybrid method [LAM01]

22. Image-Space Techniques Work with 2D or 3D discretized representation of objects –Do not perform exact collision detection due to discretization error Make use of graphics hardware –Have to worry about bandwidth to and from graphics card Too many read-backs of buffers (depth, color, stencil) will make it slower than using only CPU

23. Layered Depth Image (LDI) Decomposition [HTG03] Use discretized 3D representation to accelerate collision detection –Look like this:

24. Layered Depth Image (LDI) Decomposition [HTG03] Stage 1: Compute AABB intersection for a pair of objects (Volume-of-Intersect, VoI) Stage 2: Compute the two LDIs restricted to the VoI –like scan-line conversions

25. Layered Depth Image (LDI) Decomposition [HTG03] How to compute LDIs? –Render a 2D projection for each depth value –Like scan-conversions Need to read back the rendered image from frame buffer –For simple environment, graphics hardware version runs slower than CPU version

26. Layered Depth Image (LDI) Decomposition [HTG03] Stage 3: Perform the actual collision detection –(3a) Count the overlapping “pixels” –(3b) Check if vertices of an object are in another object’s volume

27. Layered Depth Image (LDI) Decomposition [HTG03] Results—using intersection volume (3a) –Depth complexity is the number of layers in LDI

28. Layered Depth Image (LDI) Decomposition [HTG03] Results—using vertex-in-volume –Times for LDI generation for entire objects

29. Layered Depth Image (LDI) Decomposition [HTG03] Does not need much pre-computation Can also detect self-collision –By labeling “entry” & “leaving” points explicitly Accuracy is related to the resolution of LDI Restricted to water-tight models –Otherwise the “scan-conversion” will fail Need buffer read-backs –Use graphics hardware for complex scenes!

30. CULLIDE [GRLM03] A solution to N-body problem Does not use 3D discretized representation of the models –Only use visibility queries Cull those objects that cannot be colliding –Keep a potentially colliding set (PCS) –For large environment

31. CULLIDE [GRLM03] Given an environment composed of n objects, O 1, O 2, …, O n –If O i is fully-visible with respect to all other objects, then O i cannot collide with any other object, thus is not in PCS Choose three axis to perform orthogonal projection –The second pass tests visibility of sub-objects in a similar manner Only test those still in the PCS after first pass

32. CULLIDE [GRLM03] Final step –The primitives remaining in the PCS are tested with exact collision detection methods Results 100 deforming cylinders100 cylinders * 200 polygons

33. CULLIDE [GRLM03] Visibility query done by graphics hardware –Does not need to read back buffers Accuracy governed by image resolution –Errors can be overcome by “fattened” representation [GLM04] Does not need pre-computation Suitable for any polygonal mesh, large scene Cannot be used for self-collision –Adjacent faces cannot be culled –Need decomposition of the mesh?

34. Chromatic Decomposition [Govindaraju et al. 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 To simplify the adjacency Building a corresponding graph G, and decompose it with graph coloring

35. Graph Coloring [Govindaraju et al. 05] Construct a graph G = (V, E) Each primitive p i correspond to a vertex V(p i ) in V Add an edge (V(p l ), V(p m )) to E if –Primitives p l and p m are vertex-adjacent –There exists primitive p in the mesh that is adjacent to both p l and p m Ensures each primitive in S i has only one adjacent primitive that is in S j

36. Graph Coloring [Govindaraju et al. 05] Each node is given a color that is different from its neighbors in graph G Nodes with the same color forms an independent set Each independent set has a PCS

37. Reordering [Govindaraju et al. 05] Consider each pair S i and S j, compute pairs of adjacent primitives between them –Give the adjacent primitives the same index

38. Collision Culling [Govindaraju et al. 05] Collision culling using AABB tree –Test the tree against itself –Ignore overlaps with adjacent primitives here 2.5D test: build PCS for each set –1st pass: traverse the primitives in S i from last to first Test if p i m is fully-visible against previously rendered primitives in S i and S j, namely p i >m & p j >m –2nd pass: traverse the primitives from first to last, namely test p i m against p i <m & p j <m

39. GPU Culling [Govindaraju et al. 05]

40. AABB Culling vs. GPU culling [Govindaraju et al. 05] Results of culling

41. Exact Tests [Govindaraju et al. 05] For the primitives left in the PCS, perform exact intersection tests on non-adjacent primitives –Merge the PCS of all independent sets –Use AABB tree to test these primitives first For adjacent primitives, perform elementary EE and VF tests, but do not test the shared edge or vertex

42. Benchmarks [Govindaraju et al. 05] More than 23K triangles 400-550ms during each step 13K triangles 400-500ms during each step 32,500 triangles each curtain 100ms for each curtain Path planning for a deformable object 60-90ms

43. Comparison [Govindaraju et al. 05]

44. Chromatic Decomposition [Govindaraju et al. 05] Transform self-collision detection into N- body collision detection by decomposing the mesh Use BVHs and image-space technique to do collision culling –Utilize graphics hardware

45. Conclusion BVHs are still an important tool for collision detection for deformable objects –Need to optimize update procedure Self-collision can be culled –Curvature criterion (object space) –Decompose into independent set Image-space techniques can be accelerated by graphics hardware –But accuracy is limited by discretization –Can still be powerful for culling, followed by object- space exact collision detection

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47. Reference Mezger J., Kimmerle S., Etzmuss O. 2003. Hierarchical Techniques in Collision Detection for Cloth Animation. Journal of WSCG 11, 2, 322–329. Volino P., Magnenat-Thalmann N. 1994. Efficient Self- Collision Detection on Smoothly Discretized Surface Animations using Geometrical Shape Regularity. Computer Graphics Forum 13, 3, 155–166. Provot, X. 1997. Collision and Self-Collision Handling in Cloth Model Dedicated to Design Garments. In Graphics Interface ’97 (May 1997), Canadian Information Processing Society, Canadian Human-Computer Communications Society, pp. 177–189.

48. Reference Heidelberger, B., Teschner, M., and Gross, M. 2003. Real-time volumetic intersections of deforming objects. Proc. of Vision, Modeling and Visualization. Govindaraju, N., Redon, S., Lin, M. C., and Manocha, D. 2003. CULLIDE: Interactive Collision Detection between Complex Models in Large Environments using Graphics Hardware. Proc. of Eurographics/SIGGRAPH Workshop on Graphics Hardware Govindaraju, N., Knott, D., Jain, N., Kabul, I., Tamstorf, R., Gayle, R., Lin, M. C., and Manocha, D. 2005. Interactive Collision Detection between Deformable Models using Chromatic Decomposition. Proc. of ACM SIGGRAPH.