Mesh Generation

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Mesh Generation

Jean-Daniel Boissonnat Geometrica, INRIA

http://www-sop.inria.fr/geometrica

Winter School, University of Nice Sophia Antipolis January 26-30, 2015

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Restricted Delaunay triangulation

^{[Chew 93]}

Definition

Therestricted Delaunay triangulationDelX(P)to X⊂R^d is the nerve of Vor(P)∩X

If P is anε-sample, any ball centered onXthat circumscribes a facet ofDelX(P) has a radius≤εrch(M)

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Restricted Delaunay triangulation

^{[Chew 93]}

Definition

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Restricted Delaunay triangulation

^{[Chew 93]}

Definition

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Restricted Delaunay triangulation

^{[Chew 93]}

Definition

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A variant of the nerve theorem

[Edelsbrunner & Shah 1997]

LetMbe a compact manifold without boundary. If, for any face f ∈Vor(P) s.t. f ∩M6=∅,

1 f intersectsMtransversally

2 f ∩M=∅or is a topological ball thenDel_M(P)≈M

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A variant of the nerve theorem

[Edelsbrunner & Shah 1997]

LetMbe a compact manifold without boundary. If, for any face f ∈Vor(P) s.t. f ∩M6=∅,

1 f intersectsMtransversally

2 f ∩M=∅or is a topological ball thenDel_M(P)≈M

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Proof of the closed ball property

Barycentric subdivision

ofVor(P)∩M ofDel_M(P)

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Good sampling, scale and dimension

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Sampling conditions

[Federer 1958], [Amenta & Bern 1998]

Medial axis ofM: axis(M)

set of points with at least two closest points onM

Reach

∀x∈M, rch(x)=infimum of the radii of the medial balls tangent toMatx rch(M)=infx∈Mrch(x)

(, η)-sample ofM

1.P ⊂M,∀x∈M: d(x,P)≤rch(M) 2.∀p,q∈ P, kp−qk ≥ηrch(M)

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Sampling conditions

[Federer 1958], [Amenta & Bern 1998]

Medial axis ofM: axis(M)

set of points with at least two closest points onM

Reach

∀x∈M, rch(x)=infimum of the radii of the medial balls tangent toMatx rch(M)=infx∈Mrch(x)

(, η)-sample ofM

1.P ⊂M,∀x∈M: d(x,P)≤rch(M) 2.∀p,q∈ P, kp−qk ≥ηrch(M)

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Restricted Delaunay triang. of ( ε, η )-samples

[Amenta et al. 1998-], [B. & Oudot 2005]

If

S ⊂R³ is a compact surface of positive reach without boundary

P is an(ε, η)-sample withε/η≤ξ₀andε small enough

then

Del|S(S)provides good estimates of normals

There exists ahomeomorphism φ:Del_|S(P)→ S

sup_x(kφ(x)−xk) =O(ε²)

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Surface mesh generation by Delaunay refinement

[Chew 1993, B. & Oudot 2003]

φ:S→R= Lipschitz function

∀x∈S, 0< φmin≤φ(x)< εrch(x)

ORACLE: For a facetf ofDel|S(P), returncf,rf andφ(cf) A facetf isbadifrf > φ(cf)

Context and Motivation

Delaunay refinement meshing engine

The algorithms refines:

Bad facets: f ∈Del_|S(P) – oversized (sizing field)

– badly shaped (min angle bound) – inaccurate (distance bound) Bad Tetrahedra : t∈Del_|O(P) – oversized (sizing field)

– badly-shaped (radius-egde ratio) Required oracle on domain to be meshed

• point location in domain and subdomains

• intersection detection/computation between boundary surfaces and segments (Delaunay edges)

PA-MY (INRIA Geometrica) CGALmesh ARC-ADT-AE-2010 9 / 36

Algorithm

INIT compute an initial (small) sampleP0⊂S REPEAT IFf is a bad facet

insert in Del3D(cf), updatePandDel|S(P) UNTIL all facets are good

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Surface mesh generation by Delaunay refinement

[Chew 1993, B. & Oudot 2003]

φ:S→R= Lipschitz function

∀x∈S, 0< φmin≤φ(x)< εrch(x)

ORACLE: For a facetf ofDel|S(P), returncf,rf andφ(cf) A facetf isbadifrf > φ(cf)

Context and Motivation

Delaunay refinement meshing engine

The algorithms refines:

Bad facets: f ∈Del_|S(P) – oversized (sizing field)

– badly shaped (min angle bound) – inaccurate (distance bound) Bad Tetrahedra : t∈Del_|O(P) – oversized (sizing field)

– badly-shaped (radius-egde ratio) Required oracle on domain to be meshed

• point location in domain and subdomains

• intersection detection/computation between boundary surfaces and segments (Delaunay edges)

Algorithm

INIT compute an initial (small) sampleP0⊂S REPEAT IFf is a bad facet

insert in Del3D(cf), updatePandDel|S(P) UNTIL all facets are good

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The meshing algorithm in action

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The algorithm outputs a good sample

The output sampleP is sparse

∀p∈ P,d(p,P \ {p}) =kp−qk

≥min(φ(p), φ(q))

≥φ(p)− kp−qk

⇒ kp−qk ≥ ¹₂φ(p)≥ ¹₂φ₀ >0 the algorithm terminates

P is a looseε-sample ofS

I Each facet has aS-radiusr_f ≤φ(cf)< εrch(S)

I Del_|_S(P)has a vertex on each cc ofS (INIT)

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Size of the sample = Θ R

S dx φ²(x)

Upper bound

Bp=B(p,^φ(p)₂ ),p∈ P R

S dx φ²(x) ≥P

p

R

(Bp∩S) dx

φ²(x) (theBpare disjoint)

≥⁴₉P

p

area^(Bp∩S)

φ²(p) φ(x)≤φ(p) +kp−xk

≤ ³₂ φ(p))

≥P

pC = C|P|

reach(S)

p φ(p)2

Lower bound

Use a covering instead of a packing

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Size of the sample = Θ R

S dx φ²(x)

Upper bound

Bp=B(p,^φ(p)₂ ),p∈ P R

S dx φ²(x) ≥P

p

R

(Bp∩S) dx

φ²(x) (theBpare disjoint)

≥⁴₉P

p

area^(Bp∩S)

φ²(p) φ(x)≤φ(p) +kp−xk

≤ ³₂ φ(p))

≥P

pC = C|P|

reach(S)

p φ(p)2

Lower bound

Use a covering instead of a packing

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The full result

The Delaunay refinement algorithm produces a good (dense and sparse) sample a triangulated surfaceˆS

I homeomorphic toS

I close toS (Hausdorff/Fr ´echet distance, approximation of normals)

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Applications

Implicit surfacesf(x,y,z) =0

Isosurfaces in a 3d image (Medical images) Triangulated surfaces (Remeshing)

Point sets (Surface reconstruction)

see cgal.org, CGALmesh project

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Results on smooth implicit surfaces

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CGALmesh Achievements

Meshing 3D domains

Input from segmented 3D medical images

[INSERM] [SIEMENS]

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Comparison with the Marching Cube algorithm

Delaunay refinement Marching cube

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Meshing with sharp features

A polyhedral example

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Meshing 3D multi-domains

Input from segmented 3D medical images [IRCAD]

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Surface reconstruction from unorganized point sets

Courtesy of P. Alliez