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 Winter School 4 Mesh generation Sophia Antipolis 1 / 23

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 d Ω o O P dO 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] Chew [93] Ruppert [95] B. & Oudot [03,04] Shewchuk [02] Cheng et al. [04] Winter School 4 Mesh generation Sophia Antipolis 3 / 23

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 ? Winter School 4 Mesh generation Sophia Antipolis 4 / 23

Restricted Delaunay triangulation [Chew 93] Definition The restricted Delaunay triangulation Del X ( P ) to X ⊂ R d is the nerve of Vor ( P ) ∩ X If P is an ε -sample, any ball centered on X that circumscribes a facet of Del X ( P ) has a radius ≤ ε rch ( M ) Winter School 4 Mesh generation Sophia Antipolis 5 / 23

Restricted Delaunay triangulation [Chew 93] Definition The restricted Delaunay triangulation Del X ( P ) to X ⊂ R d is the nerve of Vor ( P ) ∩ X If P is an ε -sample, any ball centered on X that circumscribes a facet of Del X ( P ) has a radius ≤ ε rch ( M ) Winter School 4 Mesh generation Sophia Antipolis 5 / 23

Restricted Delaunay triangulation [Chew 93] Definition The restricted Delaunay triangulation Del X ( P ) to X ⊂ R d is the nerve of Vor ( P ) ∩ X If P is an ε -sample, any ball centered on X that circumscribes a facet of Del X ( P ) has a radius ≤ ε rch ( M ) Winter School 4 Mesh generation Sophia Antipolis 5 / 23

Restricted Delaunay triangulation [Chew 93] Definition The restricted Delaunay triangulation Del X ( P ) to X ⊂ R d is the nerve of Vor ( P ) ∩ X Winter School 4 Mesh generation Sophia Antipolis 6 / 23

A variant of the nerve theorem [Edelsbrunner & Shah 1997] Let M be a compact manifold without boundary. If, for any face f ∈ Vor ( P ) f ∩ M � = ∅ , s.t. f intersects M transversally 1 f ∩ M = ∅ or is a topological ball 2 then Del M ( P ) ≈ M Winter School 4 Mesh generation Sophia Antipolis 7 / 23

A variant of the nerve theorem [Edelsbrunner & Shah 1997] Let M be a compact manifold without boundary. If, for any face f ∈ Vor ( P ) f ∩ M � = ∅ , s.t. f intersects M transversally 1 f ∩ M = ∅ or is a topological ball 2 then Del M ( P ) ≈ M Winter School 4 Mesh generation Sophia Antipolis 7 / 23

Proof of the closed ball property Barycentric subdivision of Vor ( P ) ∩ M of Del M ( P ) Winter School 4 Mesh generation Sophia Antipolis 8 / 23

Good sampling, scale and dimension Winter School 4 Mesh generation Sophia Antipolis 9 / 23

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 ) = inf x ∈ 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 )) Winter School 4 Mesh generation Sophia Antipolis 10 / 23

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 ) = inf x ∈ 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 )) Winter School 4 Mesh generation Sophia Antipolis 10 / 23

Restricted Delaunay triang. of ( ε, η ) -nets [Amenta et al. 1998-], [B. & Oudot 2005] If S ⊂ R 3 is a compact surface of positive reach without boundary P is an ( ε, η ) -net with ε/η ≤ ξ 0 and ε small enough then Del | S ( S ) provides good estimates of normals There exists a homeomorphism φ : Del | S ( P ) → S sup x ( � φ ( x ) − x � ) = O ( ε 2 ) Winter School 4 Mesh generation Sophia Antipolis 11 / 23

Surface mesh generation by Delaunay refinement [Chew 1993, B. & Oudot 2003] φ : S → R = Lipschitz function ∀ x ∈ S , 0 < φ min ≤ φ ( x ) < ε rch ( x ) ORACLE : For a facet f of Del | S ( P ) , return c f , r f and φ ( c f ) A facet f is bad if r f > φ ( c f ) Algorithm INIT compute an initial (small) sample P 0 ⊂ S REPEAT IF f is a bad facet insert in Del3D( c f ) , update P and Del | S ( P ) UNTIL all facets are good Winter School 4 Mesh generation Sophia Antipolis 12 / 23

Surface mesh generation by Delaunay refinement [Chew 1993, B. & Oudot 2003] φ : S → R = Lipschitz function ∀ x ∈ S , 0 < φ min ≤ φ ( x ) < ε rch ( x ) ORACLE : For a facet f of Del | S ( P ) , return c f , r f and φ ( c f ) A facet f is bad if r f > φ ( c f ) Algorithm INIT compute an initial (small) sample P 0 ⊂ S REPEAT IF f is a bad facet insert in Del3D( c f ) , update P and Del | S ( P ) UNTIL all facets are good Winter School 4 Mesh generation Sophia Antipolis 12 / 23

The meshing algorithm in action Winter School 4 Mesh generation Sophia Antipolis 13 / 23

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 φ 0 > 0 the algorithm terminates P is a loose ε -sample of S ◮ Each facet has a S -radius r f ≤ φ ( c f ) < ε rch ( S ) ◮ Del | S ( P ) has a vertex on each cc of S (INIT) Winter School 4 Mesh generation Sophia Antipolis 14 / 23

�� � dx Size of the sample = Θ φ 2 ( x ) S Upper bound B p = B ( p , φ ( p ) 2 ) , p ∈ P dx dx � φ 2 ( x ) ≥ � � (the B p are disjoint) φ 2 ( x ) S p ( B p ∩ S ) area ( B p ∩ S ) ≥ 4 � φ ( x ) ≤ φ ( p ) + � p − x � 9 p φ 2 ( p ) ≤ 3 2 φ ( p ) ) ≥ � p C = C |P| reach( S ) p φ ( p ) 2 Lower bound Use a covering instead of a packing Winter School 4 Mesh generation Sophia Antipolis 15 / 23

�� � dx Size of the sample = Θ φ 2 ( x ) S Upper bound B p = B ( p , φ ( p ) 2 ) , p ∈ P dx dx � φ 2 ( x ) ≥ � � (the B p are disjoint) φ 2 ( x ) S p ( B p ∩ S ) area ( B p ∩ S ) ≥ 4 � φ ( x ) ≤ φ ( p ) + � p − x � 9 p φ 2 ( p ) ≤ 3 2 φ ( p ) ) ≥ � p C = C |P| reach( S ) p φ ( p ) 2 Lower bound Use a covering instead of a packing Winter School 4 Mesh generation Sophia Antipolis 15 / 23

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) Winter School 4 Mesh generation Sophia Antipolis 16 / 23

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 Winter School 4 Mesh generation Sophia Antipolis 17 / 23

Results on smooth implicit surfaces Winter School 4 Mesh generation Sophia Antipolis 18 / 23

Meshing 3D domains Input from segmented 3D medical images [INSERM] [SIEMENS] Winter School 4 Mesh generation Sophia Antipolis 19 / 23

Comparison with the Marching Cube algorithm Delaunay refinement Marching cube Winter School 4 Mesh generation Sophia Antipolis 20 / 23

Meshing with sharp features A polyhedral example Winter School 4 Mesh generation Sophia Antipolis 21 / 23

Meshing 3D multi-domains Input from segmented 3D medical images [IRCAD] Winter School 4 Mesh generation Sophia Antipolis 22 / 23

Surface reconstruction from unorganized point sets Courtesy of P . Alliez Winter School 4 Mesh generation Sophia Antipolis 23 / 23