Localized Delaunay Refinement for Sampling and Meshing

1. Tamal K. Dey Joshua A. Levine Andrew G. Slatton The Ohio State University Localized Delaunay Refinement for Sampling and Meshing

2. Restricted Delaunay • Del S|M: Collection of Delaunay simplices t where Vt intersects M

3. Delaunay Refinement • Input surface M • Check conditions • If violated, insert • Vt∩M into S • Output: Del S|M

4. Existing Methods Check surface Delaunay ball size [BO05] Check topological disk [CDRR06]

5. Limitations Traditional refinement maintains Delaunay triangulation in memory This does not scale well Causes memory thrashing May be aborted by OS

6. Our Contribution A simple algorithm that avoids the scaling issues of the Delaunay triangulation Avoids memory thrashing Topological and geometric guarantees Guarantee of termination Potentially parallelizable

7. A Natural Solution Use an octree T to divide S and process points in each node v of T separately

8. Two Concerns • Termination • Mesh consistency

9. Termination Trouble A locally furthest point in node v can be very close to a point in other nodes

10. Messing Mesh Consistency Individual meshes do not blend consistently across boundaries

11. LocDel Algorithm: Overview Process nodes from a queue Q Refines nodes with parameter λ if there are violations

12. Splitting and reprocessing  Split Let S = ∩ S Split  into eight children if ||S||> Reprocess 

13. Splitting 

14. Refining node  Augment Assemble R=NUS Compute Del R|M Refine Surface Delaunay ball larger than λ Fp  Del R|M is not a disk

15. Returned points for violations Checking Violations Large triangle t incident to p ϵ S Radius of surface ball > λ Return (p,p*) where p* is furthest dual(t) ∩ M Non-disk surface star Fp Return (p,p*) where p* is the furthest dual(t) ∩ M among all triangles

16. Point Insertions Modified insertion strategy If nearest point s ϵ S to p* is within λ/8 and s ≠ p, then add s to R Else add p* to R p* augments S, but s does not

17. Reprocessing nodes • Needed for mesh consistency • Suppose s is added • Enqueue each node ' ≠ s.t. d(s, ') ≤ 2λ

18. Maintaining light structures • For each node  keep: • S = S ∩ • Up ϵ S Fp • Output: union of surface stars Up ϵ S Fp

19. Termination If insertions are finite, so are enqueues and splits Augmenting R by an existing point does not grow S Consider inserting a new point s Nearest point ≠ p → at least λ/8 from S Insertion due to triangle size → at least λ from S Else → at least εM from S by Proposition 1

20. Termination Proposition 1 [Cheng-Dey-Ramos-Ray 2007]: εM>0 s.t. if intersections of all edges of Vp with M lie within εM of p then Fp forms a topological disk

21. Guarantees The underlying space of the output mesh is a 2-manifold without boundary Each point in the output is within distance λ of M λ*>0 s.t. if λ<λ* the output is isotopic to M with Hausdorff distance of O(λ2)

22. Manifoldness • We require surface stars to fit together globally • Consistency condition: In the output complex UpFp, a triangle abc is in Fa if and only if it is also in Fb and Fc

23. Manifoldness Theorem: At termination UFp  Del S|M Consider the last time  is processed; t in  Size condition → t in Del S|M when  is done If t  Del S|M afterward, there is a point s in Delaunay ball. But, s causes  to be reprocessed

24. Topology For sufficiently small λ Homeomorphism follows from [Amenta-Choi-Dey-Leekha 02] Isotopy and Hausdorff distance follow from [Boissonnat-Oudot 05]

25. Results • Varying  does not change the mesh qualitatively

26. Results • Optimal is platform-dependent

27. Results

28. Results

29. Results

30. Conclusions A simple algorithm for Delaunay refinement Avoids memory thrashing Topological and geometric guarantees Guarantee of termination Potentially parallelizable

31. Thank You