Probing for Surface Mesh Generation through Delaunay Refinement · 2018. 11. 29. · De nition of...

Definition of the problem Delaunay refinement Limitations Proposal Experimental results

Probing for Surface Mesh Generationthrough Delaunay Refinement

Hugo Feree and Pierre Alliez

INRIA

July 31, 2009

Hugo Feree and Pierre Alliez Surface mesh generation


Outline

Definition of the problem

Delaunay refinement

Limitations

Proposal

Experimental results



Goal: mesh a smooth closed 2-manifold surface given:

I a convex bounding domain Ω

I an intersection oracle

I a lower bound ε > 0 on the local feature size

I approximation and shape criteria

I nothing more! (especially no initial point set)

with guarantees of:

I topology

I approximation

I shape

I parsimony



Delaunay refinement

Algorithm [Chew, 1993]:I Compute the Voronoi diagram and the dual 3D Delaunay

triangulation of the initial point setI Compute the Voronoi diagram restricted to the surfaceI Check the facets by probing their dual Voronoi edgeI While there are bad facets, refine them

Many good properties [Boissonnat, Oudot, 2005, 2007]:I no self intersectionI same topologyI bound and convergence of the

I Hausdorff distanceI areaI normals (the Voronoi edges become orthogonal to the surface)

I shape and approximation quality

But the initialization is left to the user!



Delaunay refinement

Algorithm [Chew, 1993]:I Compute the Voronoi diagram and the dual 3D Delaunay

triangulation of the initial point setI Compute the Voronoi diagram restricted to the surfaceI Check the facets by probing their dual Voronoi edgeI While there are bad facets, refine them

Many good properties [Boissonnat, Oudot, 2005, 2007]:I no self intersectionI same topologyI bound and convergence of the

I Hausdorff distanceI areaI normals (the Voronoi edges become orthogonal to the surface)

I shape and approximation quality

But the initialization is left to the user!Hugo Feree and Pierre Alliez Surface mesh generation


Delaunay refinement versus marching cubes

Marching cubes:

I careful research

I axis dependent (not isotropic)I surface independent, which implies:

I heavy cost: ∼ Vε3 (lots of short probes)I oversampling

Delaunay refinement:

I adapts itself to the surface

I axis-independent

I parsimonious

I uses long probes (Delaunay edges) to both refine and discoverthe surface

I only lacks a careful seeding



Delaunay refinement versus marching cubes

Figure: Left: Delaunay refinement (7040 vertices) Right: marching cubes(10440 vertices)



Results with connected surfaces

Figure: Meshes of connected surfaces



Delaunay refinement’s probing ability

Figure: All the toruses are meshed after refinement, with only four initialpoints on the central sphere



TheoremIf S0 is a connected component of diameter d including no othercomponent, at distance at least d of any other connectedcomponent and containing at most two points of the triangulation,then S0 will not be meshed by Delaunay refinement.

Figure: Illustration of the theorem in dimension 2



Application of the theorem

Figure: The ’worst’ case



I Probe along the edges of a regular grid: marching cubes

I Probe randomly (done in the surface mesher of CGAL)

Figure: Statistics on random spheres with random initial points



I Probe along the edges of a regular grid: marching cubes

I Probe randomly (done in the surface mesher of CGAL)

Figure: Cluster of vertices



Dealing with isolated points

Figure: An isolated point




Figure: A persistent facet is made




Figure: The bad facet is refined




Figure: All the component is meshed




Figure: Iteration on a grid of spheres




Figure: Iteration on a grid of spheresHugo Feree and Pierre Alliez Surface mesh generation


Probing the space

Idea: the Voronoi edges are use to refine and discover the surface⇒ the (largest) Voronoi cells define where the space was notprobed

Figure: Illustration in dimnsion 2: probing a cell

Figure: Probing a Voronoi cell: illustration in dimension 2



Algorithm

Q ← ∅P0 = probe(Ω) // Find a first pointinitialize component(P0)while Q 6= ∅ dorefineif ∃P isolated theninitialize component(P)// Deal with isolated points

elserepeat

C =popQprobe(C ) // Probe a cell

until intersection found or Q = ∅end if

end while



Examples

Figure: Simple Delaunay refinement (blue) and after isolated pointstreatment (red)



Statistics with random spheres

Figure: Statistics for mesh generation with isolated points treatment



An adaptive method

This algorithm can be accelerated knowing (optional) informations:

I some initial pointsI the number of connected componentsI a maximum inside radius R > ε

Ω

ε R

RR

Figure: R can be arbitrarily large compared with ε and its knowledge canavoid many probing steps: illustration in dimension 2



Conclusion

We have provided:I a generic adaptative meshing algorithm based on Delaunay

refinementI with the same quality and parsimonyI but without asking for a user’s initialization.

I a partial implementation with concluding results.

Rest to do:

I an implementation of the whole algorithm

I an accurate study of complexity in time and in number ofvertices



Thanks!

Questions?


Definition of the problemDelaunay refinementLimitationsProposalExperimental results

Probing for Surface Mesh Generation through Delaunay Refinement · 2018. 11. 29. · De nition of...

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