1/50 Department of Computer Science and Engineering Localized Delaunay Refinement for Sampling and Meshing Tamal K. Dey Joshua A. Levine Andrew G. Slatton The Ohio State University

2/50 Department of Computer Science and Engineering Restricted Delaunay Del S| M : Collection of Delaunay simplices t where V t intersects M

3/50 Department of Computer Science and Engineering Delaunay Refinement Input surface M Check conditions If violated, insert V t ∩ M into S Output: Del S| M

4/50 Department of Computer Science and Engineering Existing Methods Check surface Delaunay ball size [BO05] Check topological disk [CDRR06]

5/50 Department of Computer Science and Engineering Limitations Traditional refinement maintains Delaunay triangulation in memory This does not scale well Causes memory thrashing May be aborted by OS

6/50 Department of Computer Science and Engineering 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

7/50 Department of Computer Science and Engineering A Natural Solution Use an octree T to divide S and process points in each node v of T separately

8/50 Department of Computer Science and Engineering Two Concerns Termination Mesh consistency

9/50 Department of Computer Science and Engineering Termination Trouble A locally furthest point in node v can be very close to a point in other nodes

10/50 Department of Computer Science and Engineering Messing Mesh Consistency Individual meshes do not blend consistently across boundaries

11/50 Department of Computer Science and Engineering LocDel Algorithm: Overview Process nodes from a queue Q Refines nodes with parameter λ if there are violations

12/50 Department of Computer Science and Engineering Splitting and reprocessing Split Let S = ∩ S Split into eight children if ||S ||>  Reprocess

13/50 Department of Computer Science and Engineering Splitting

14/50 Department of Computer Science and Engineering Refining node Augment Assemble R =N US Compute Del R | M Refine Surface Delaunay ball larger than λ F p  Del R | M is not a disk

15/50 Department of Computer Science and Engineering Returned points for violations Checking Violations Large triangle t incident to p S Radius of surface ball > λ Return (p,p*) where p* is furthest dual(t) ∩ M Non-disk surface star F p Return (p,p*) where p* is the furthest dual(t) ∩ M among all triangles

16/50 Department of Computer Science and Engineering Point Insertions Modified insertion strategy If nearest point s 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

17/50 Department of Computer Science and Engineering Point insertions

18/50 Department of Computer Science and Engineering Reprocessing nodes Needed for mesh consistency Suppose s is added Enqueue each node ' ≠ s.t. d(s, ') ≤ 2λ

19/50 Department of Computer Science and Engineering Maintaining light structures For each node keep: S = S ∩ U p S F p Output: union of surface stars U p S F p

20/50 Department of Computer Science and Engineering 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 ε M from S by Proposition 1

21/50 Department of Computer Science and Engineering Termination Proposition 1 [Cheng-Dey-Ramos-Ray 2007]:  ε M >0 s.t. if intersections of all edges of V p with M lie within ε M of p then F p forms a topological disk

22/50 Department of Computer Science and Engineering 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(λ 2 )

23/50 Department of Computer Science and Engineering Manifoldness We require surface stars to fit together globally Consistency condition: In the output complex U p F p, a triangle abc is in F a if and only if it is also in F b and F c

24/50 Department of Computer Science and Engineering Manifoldness Theorem: At termination UF p  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

25/50 Department of Computer Science and Engineering Topology For sufficiently small λ Homeomorphism follows from [Amenta-Choi- Dey-Leekha 02] Isotopy and Hausdorff distance follow from [Boissonnat-Oudot 05]

26/50 Department of Computer Science and Engineering Results Varying  does not change the mesh qualitatively

27/50 Department of Computer Science and Engineering Results Optimal  is platform- dependent

28/50 Department of Computer Science and Engineering Results

29/50 Department of Computer Science and Engineering Results

30/50 Department of Computer Science and Engineering Results

31/50 Department of Computer Science and Engineering Conclusions A simple algorithm for Delaunay refinement Avoids memory thrashing Topological and geometric guarantees Guarantee of termination Potentially parallelizable

32/50 Department of Computer Science and Engineering Thank You