1/50 Department of Computer Science and Engineering Localized Delaunay Refinement for Sampling and...

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