1. Efficient Motion Updates for Delaunay Triangulations Daniel Russel Leonidas Guibas

2. 2 Motivation Delaunay of object and ambient space provides –Nicely shaped elements –Proximity information for force computation and collision detection Molecular surface area

3. 3 Delaunay Update in Physical Simulation Update Delaunay after each integrator step –All nodes move (but not very far) Delaunay changes at discrete times Can coherence be exploited?

4. 4 Measuring Coherence Motion coherence –Coherence of motion vectors Hierarchical decomposition Delaunay coherence –Fraction of preserved tetrahedra –Aim for coherence-sensitive update methods

5. 5 Some Delaunay Statistics Molecular simulation –3% fail per time step Buddha collapsing –5% to 25% failure rate Bicep –.1% to 2% Rings –.1% to 1% 700% extra tetrahedra created during reconstruction

6. 6 Update Methods KDS-like –Polynomial solvers must be used –Return extra information Point removal –Only normal predicates –Trivial when no changes necessary Hierarchic decomposition

7. 7 Kinetic Data Structures Maintain combinatorial structure –Polynomial motion of points –Polynomial certificates Discrete changes –At roots of polynomials

8. 8 KDS Delaunay Flips when the in-circle test fails

9. 9 KDS-like Methods Axis-aligned –3 passes, cubic coefficients Linear interpolation –1 pass, degree 5 Linear certificates –5 passes, degree 1 –Or n passes, degree 1 Rigid –1 pass, degree 12

10. 10 Linear Certificates In-circle predicate: Can choose any row to be all linear, or any column

11. 11 Linear Certificates In-circle predicate: Can choose any row to be all linear, or any column

12. 12 Move One Point Make (x 0 +v x t) 2 + (x 0 +v x t) 2 + (x 0 +v x t) 2 – r(t) 2 linear r(t)=2x 0 v x t+v x 2 t 2 +v y 2 t 2 +2z 0 v z t+v z 2 t 2 +qt +r 0 2 Need final pass to fix r Can expand around point location

13. 13 Move Rectilinearly Need to make (x 0 +v x t) 2 + (x 0 +v x t) 2 + (x 0 +v x t) 2 – r(t) 2 constant R will become very large –Need an initial patching phase Motion: –Grow R –Move x, then y, then z –Fix R

14. 14 Rigid Motion Rigid motions can be useful Rationalize a circle as –x=2ru –y=r(1+u 2 ) –w=(1+u 2 ) Add translation in u space –x = 2uy 0 +(1-u 2 )x 0 + t x u –y = (1-u 2 )x 0 +2uy 0 + t y u

15. 15 Rigid Optimizations Many certificates will never fail a-priori

16. 16 Point Removal point to remove mis-oriented triangle failed in-circle

17. 17 Point Removal

18. 18 Point Removal

19. 19 Point Removal

20. 20 Point Removal Details Can currently only remove isolated points Methods of picking points –Incident to a failed predicate –Incident most failed predicates 50% better

21. 21 Results Beta hairpin: 6% Bicep: Remove 10% of the points in the worst case 1% in the best Rings: 3-30% Buddha: 0-100% depending

22. 22 Redundant Work Removing a point can create transient tetrahedra –40 beta hairpin (1.1k total) –117 to 4k in bicep (22k total) –101 to 7k in the rings (6.3k total)

23. 23 Hierarchic Decomposition Objects/scenes often have natural decompositions Extract rigid transform (e.g. Horn) at each level of hierarchy

24. 24 Decomposition Now have O(n) (smaller) rigid motions to apply Can use a previous method Or directly patch Delaunay –But we don’t know how

25. 25 Decomposition and Coherence A coherent motion the magnitude of the rigid motion should decrease rapidly down the tree

26. 26 Molecular Simulations

27. 27 What should I use? Currently only point removal –Break even at 10-15% point removal which is about 2-3% tetra failure –1-2x faster for molecular frames KDS factor of 10-20 slower than rebuilding –Axis aligned motion factor of 2 slower, 50% more flips

28. 28 Exploiting the Integrator What information can the integrator provide cheaply which aids in maintaining the structure? –Objects in the scene –Decomposition of the points –Velocities (upper bounds)

29. 29 Static problems How to fill complex holes –Would like to remove more than one point –Reduce the number of transient tetrahedra How to combine rigid, non-overlapping parts –Allow hierarchy to be used directly

30. 30 Dynamic Problems KDS –How to perform filtering with KDS solvers Flip distance in 3D? Characterize stuck flips. Are there patterns in failed tetrahedra in simulation data. How to decompose moving point sets.

31. 31 Other Outcomes Software –KDS for CGAL –Various Delaunay updating methods

32. 32 Modeling Motion How do we model motion? Previous answers –Maximum distance moved –Static packing arguments –Independent rigid objects

33. 33 Properties of Motion Magnitude Coherence –How well do neighbors predict motion?

34. 34 Another Example Initial Final