Mesh Generation Jean-Daniel Boissonnat Geometrica, INRIA - - PowerPoint PPT Presentation

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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Meshing surfaces and 3D domains

visualization and graphics applications CAD and reverse engineering geometric modelling in medecine, geology, biology etc. autonomous exploration and mapping (SLAM) scientific computing : meshes for FEM

P d dO Ω O

• Winter School 4

Mesh generation Sophia Antipolis 2 / 23

Mesh generation : from art to science

Grid methods

Lorensen & Cline [87] : marching cube Lopez & Brodlie [03] : topological consistency Plantiga & Vegter [04] : certified topology using interval arithmetic

Morse theory

Stander & Hart [97] B., Cohen-Steiner & Vegter [04] : certified topology

Delaunay refinement

Hermeline [84] Ruppert [95] Shewchuk [02] Chew [93]

• B. & Oudot [03,04]

Cheng et al. [04]

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Main issues

Sampling

How do we choose points in the domain ? What information do we need to know/measure about the domain ?

Meshing

How do we connect the points ? Under what sampling conditions can we compute a good approximation of the domain ?

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Restricted Delaunay triangulation

[Chew 93]

Definition

The restricted Delaunay triangulation DelX(P) to X ⊂ Rd is the nerve of Vor(P) ∩ X

If P is an ε-sample, any ball centered on X that circumscribes a facet of DelX(P) has a radius ≤ ε rch(M)

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Restricted Delaunay triangulation

[Chew 93]

Definition

The restricted Delaunay triangulation DelX(P) to X ⊂ Rd is the nerve of Vor(P) ∩ X

If P is an ε-sample, any ball centered on X that circumscribes a facet of DelX(P) has a radius ≤ ε rch(M)

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Restricted Delaunay triangulation

[Chew 93]

Definition

The restricted Delaunay triangulation DelX(P) to X ⊂ Rd is the nerve of Vor(P) ∩ X

If P is an ε-sample, any ball centered on X that circumscribes a facet of DelX(P) has a radius ≤ ε rch(M)

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Restricted Delaunay triangulation

[Chew 93]

Definition

The restricted Delaunay triangulation DelX(P) to X ⊂ Rd is the nerve of Vor(P) ∩ X

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A variant of the nerve theorem

[Edelsbrunner & Shah 1997]

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

1

f intersects M transversally

2

f ∩ M = ∅ or is a topological ball then DelM(P) ≈ M

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A variant of the nerve theorem

[Edelsbrunner & Shah 1997]

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

1

f intersects M transversally

2

f ∩ M = ∅ or is a topological ball then DelM(P) ≈ M

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Proof of the closed ball property

Barycentric subdivision

• f Vor(P) ∩ M
• f DelM(P)

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Good sampling, scale and dimension

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Sampling conditions

[Federer 1958], [Amenta & Bern 1998]

Medial axis of M : axis(M)

set of points with at least two closest points on M

Reach

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

(ǫ, η)-net of M

• 1. P ⊂ M, ∀x ∈ M :

d(x, P) ≤ ǫ rch(x)

• 2. ∀p, q ∈ P,

p − q ≥ η min(rch(p), rch(q))

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Sampling conditions

[Federer 1958], [Amenta & Bern 1998]

Medial axis of M : axis(M)

set of points with at least two closest points on M

Reach

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

(ǫ, η)-net of M

• 1. P ⊂ M, ∀x ∈ M :

d(x, P) ≤ ǫ rch(x)

• 2. ∀p, q ∈ P,

p − q ≥ η min(rch(p), rch(q))

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

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

If S ⊂ R3 is a compact surface of positive reach without boundary P is an (ε, η)-net with ε/η ≤ ξ0 and ε small enough then Del|S(S) provides good estimates of normals There exists a homeomorphism φ : Del|S(P) → S supx(φ(x) − x) = 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) < ε rch(x) ORACLE : For a facet f of Del|S(P), return cf , rf and φ(cf ) A facet f is bad if rf > φ(cf )

Algorithm

INIT compute an initial (small) sample P0 ⊂ S REPEAT IF f is a bad facet insert in Del3D(cf ) , update P and Del|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) < ε rch(x) ORACLE : For a facet f of Del|S(P), return cf , rf and φ(cf ) A facet f is bad if rf > φ(cf )

Algorithm

INIT compute an initial (small) sample P0 ⊂ S REPEAT IF f is a bad facet insert in Del3D(cf ) , update P and Del|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 sample P is sparse ∀p ∈ P, d(p, P \ {p}) = p − q ≥ min(φ(p), φ(q)) ≥ φ(p) − p − q ⇒ p − q ≥ 1

2 φ(p) ≥ 1 2 φ0 > 0

the algorithm terminates

P is a loose ε-sample of S

◮ Each facet has a S-radius rf ≤ φ(cf ) < ε rch(S) ◮ Del|S(P) has a vertex on each cc of S

(INIT)

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Size of the sample = Θ

• S

dx φ2(x)

• Upper bound

Bp = B(p, φ(p)

2 ), p ∈ P

• S

dx φ2(x) ≥ p

• (Bp∩S)

dx φ2(x)

(the Bp are disjoint)

≥ 4

9

• p

area(Bp∩S)

φ2(p)

φ(x) ≤ φ(p) + p − x ≤ 3

2 φ(p))

≥

p C = 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

• S

dx φ2(x)

• Upper bound

Bp = B(p, φ(p)

2 ), p ∈ P

• S

dx φ2(x) ≥ p

• (Bp∩S)

dx φ2(x)

(the Bp are disjoint)

≥ 4

9

• p

area(Bp∩S)

φ2(p)

φ(x) ≤ φ(p) + p − x ≤ 3

2 φ(p))

≥

p C = 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 surface ˆ S

◮ homeomorphic to S ◮ close to S (Hausdorff/Fr´

echet distance, approximation of normals)

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Applications

Implicit surfaces f(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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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

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