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1/50Department 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
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Restricted Delaunay
• Del S|M: Collection of Delaunay simplices t where Vt intersects M
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Delaunay Refinement
• Input surface M• Check
conditions• If violated,
insert• Vt∩M into S
• Output: Del S|M
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Existing Methods
• Check surface Delaunay ball size [BO05]
• Check topological disk [CDRR06]
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Limitations
• Traditional refinement maintains Delaunay triangulation in memory
• This does not scale well• Causes memory thrashing• May be aborted by OS
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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
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A Natural Solution
• Use an octree T to divide S and process points in each node v of T separately
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Two Concerns
• Termination• Mesh consistency
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Termination Trouble
• A locally furthest point in node v can be very close to a point in other nodes
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Messing Mesh Consistency
• Individual meshes do not blend consistently across boundaries
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LocDel Algorithm: Overview
• Process nodes from a queue Q• Refines nodes with parameter λ if
there are violations
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Splitting and reprocessing
• Split• Let S = ∩ S
• Split into eight children if ||S||> • Reprocess
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Splitting
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Refining node
• Augment• Assemble
R=NUS
• Compute Del R|M
• Refine• Surface Delaunay
ball larger than λ
• Fp Del R|M is not a disk
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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
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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
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Reprocessing nodes
• Needed for mesh consistency• Suppose s is
added• Enqueue each
node ' ≠ s.t. d(s, ') ≤ 2λ
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Maintaining light structures
• For each node keep:• S = S ∩
• Up ϵ S Fp
• Output: union of surface stars Up ϵ S Fp
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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
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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
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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)
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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
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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
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Topology
• For sufficiently small λ• Homeomorphism follows from [Amenta-
Choi-Dey-Leekha 02]
• Isotopy and Hausdorff distance follow from [Boissonnat-Oudot 05]
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Results
• Varying does not change the mesh qualitatively
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Results
• Optimal is platform-dependent
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Results
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Results
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Results
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Conclusions
• A simple algorithm for Delaunay refinement
• Avoids memory thrashing• Topological and geometric
guarantees• Guarantee of termination• Potentially parallelizable
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Thank You