From Segmented Images to Good Quality Meshes Using Delaunay Refinement

Mesh generation Applications CGAL-mesh

From Segmented Images to Good Quality Meshes using Delaunay Reﬁnement

Jean-Daniel Boissonnat

INRIA Sophia-Antipolis Geometrica Project-Team

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Grid-based methods (MC and variants)

I Produces unnecessarily large meshes

I Regularization and decimation are difﬁcult tasks

I Imposes dominant axis-aligned edges

Mesh generation Applications CGAL-mesh

A determinant step towards numerical simulations

Challenges

I Quality of the mesh : accurary, topological correctness, size and shape of the elements

I Number of elements, processing time

I Robustness issues

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

A determinant step towards numerical simulations

Challenges

I Quality of the mesh : accurary, topological correctness, size and shape of the elements

I Number of elements, processing time

I Robustness issues

Grid-based methods (MC and variants)

I Produces unnecessarily large meshes

I Regularization and decimation are difﬁcult tasks

I Imposes dominant axis-aligned edges

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

Afﬁne Voronoi diagrams and Delaunay triangulations

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

Mesh generation by Delaunay reﬁnement

A simple greedy technique that goes back to the early 80’s [Frey, Hermeline]

input : some domain D to be meshed

output : a ﬁnite set of points E ⊂ D a triangulation T ⊂ Del(E) that covers or approximate D with well-shaped simplices

main loop : REPEAT if f is a bad simplex of Del(E) take a point p so that f is not a simplex of Del(E ∪ {p}) E ← p update Del(E) UNTIL all simplices are good

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

Delaunay reﬁnement : some milestones

Polyhedral domains

1990-1995 First guaranteed method for 2D polygonal domains [Chew, Ruppert] 1998-2002 First guaranteed method for 3D polygonal domains [Shewchuk] 2000-2005 Sliver removal [Cheng et al.] 2003-2004 Sharp features [Cheng et al.]

Curved domains 1993-1997 Restricted Delaunay triangulation [Chew, Edelsbrunner & Shah] 1998 ε-sample, ﬁrst guaranteed method for surface reconstruction [Amenta & Bern] 2003 First guaranteed surface mesher [B. & Oudot] 2005 First guaranteed volume mesher [Oudot & al.] 2007 Piecewise smooth surfaces [Dey et al., Rineau & Yvinec]

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

Outline of the talk

Mesh generation Restricted Delaunay Triangulation Surface mesh generation Meshing 3D objects with curved boundaries Anisotropic mesh generation

Applications Image segmentation Manifold reconstruction Spatio-temporal scenes modeling

CGAL-mesh

Restricted Delaunay triangulation [Chew 93]

Deﬁnition The restricted Delaunay triangulation Del|S(E) is the set of facets of the Delaunay triangulation whose dual edges intersect the surface

Restricted Delaunay triangulation [Chew 93]

Deﬁnition The restricted Delaunay triangulation Del|S(E) is the set of facets of the Delaunay triangulation whose dual edges belong to Vor|S(E)

Restricted Delaunay triangulation [Chew 93]

Deﬁnition The restricted Delaunay triangulation Del|S(E) is the set of facets of the Delaunay triangulation whose dual edges belong to Vor|S(E)

Sampling conditions [Amenta & Bern 98] The medial axis of S is the set of points with at least two closest points on S

a ﬁnite point set E ⊆ S is an ε-sample of S if

∀x ∈ S, dE(x) ≤ εdMA(x)

dMA denotes the distance to the medial axis of S

I dMA is 1-Lipschitz : |dMA(x) − dMA(y)| ≤ kx − yk

I dMA(x) κ(x) ≤ 1 (κ(x) = largest absolute curvature at x)

1,1 I dMA > 0 if S is C i.e. normals exist everywhere and the normal ﬁeld is Lipschitz

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Restricted DT and sampling [B.-Oudot 05]

If all the surface Delaunay balls B(cf , rf ) of Del|S(E) are small enough (rf ≤ ε dMA(cf )), =⇒ E is an O(ε)-sample of S

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Sampling smooth surfaces

e e

c c f

Surface Delaunay balls Each facet in Del|S(E) is circumscribed by a surface Delaunay ball B(cf , rf ) centered on the surface and empty of vertices.

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Sampling smooth surfaces

e e

c c f

Surface Delaunay balls Each facet in Del|S(E) is circumscribed by a surface Delaunay ball B(cf , rf ) centered on the surface and empty of vertices.

Restricted DT and sampling [B.-Oudot 05]

If all the surface Delaunay balls B(cf , rf ) of Del|S(E) are small enough (rf ≤ ε dMA(cf )), =⇒ E is an O(ε)-sample of S

Local properties

If E is a ε-sample of S,

Normals [Amenta & Bern 1998] The angles between normals to S and to Del|S(E) are O(ε)

Curvatures [Cohen-Steiner & Morvan 2003] The curvature tensor of S can be estimated from Del|S(E)

Global properties [Amenta & Bern], [Dey et al.], [B.& Oudot]

If E is a ε-sample of S, ε ≤ 0.12

I Del|S(E) is a triangulated surface isotopic to S 2 I The Hausdorff distance between S and Del|S(E) is O(ε )

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Algorithm

INIT compute an initial (small) sample E0 ⊂ S REPEAT IF f is a bad facet insert in Del3D(cf ) update E and Del|S(E) UNTIL all facets are good

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Surface mesh generation

φ : S → R = Lipschitz function [Chew 1993, B. & Oudot 2003] ∀x ∈ S, 0 < φmin ≤ φ(x) < 0.2dMA(x)

ORACLE : For a facet f of Del|S(E), return cf , rf and φ(cf ) dΩ

A facet f is bad if rf > φ(cf )

>ε

φ : S → R = Lipschitz function [Chew 1993, B. & Oudot 2003] ∀x ∈ S, 0 < φmin ≤ φ(x) < 0.2dMA(x)

ORACLE : For a facet f of Del|S(E), return cf , rf and φ(cf ) dΩ

A facet f is bad if rf > φ(cf )

Algorithm

INIT compute an initial (small) sample E0 ⊂ S >ε REPEAT IF f is a bad facet insert in Del3D(cf ) update E and Del|S(E) UNTIL all facets are good

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes I Can be adapted to produce facets with good aspect ratio

I General (black box) model of surfaces, applicable in various contexts

? How to compute dMA(x)?

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Properties of the algorithm

I The procedure terminates and therefore

I Produces a good approximation of the surface I Produces sparse ε-samples

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes I General (black box) model of surfaces, applicable in various contexts

? How to compute dMA(x)?

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Properties of the algorithm

I The procedure terminates and therefore

I Produces a good approximation of the surface I Produces sparse ε-samples

I Can be adapted to produce facets with good aspect ratio

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes ? How to compute dMA(x)?

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Properties of the algorithm

I The procedure terminates and therefore

I Produces a good approximation of the surface I Produces sparse ε-samples

I Can be adapted to produce facets with good aspect ratio

I General (black box) model of surfaces, applicable in various contexts

I The procedure terminates and therefore

I Produces a good approximation of the surface I Produces sparse ε-samples

I Can be adapted to produce facets with good aspect ratio

I General (black box) model of surfaces, applicable in various contexts

? How to compute dMA(x)?

Collaboration with Asclepios, Ca¨ıman and Odyssee´ INRIA project-teams

Delaunay Reﬁnement Marching Cube

Delaunay reﬁnement Marching cube

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes 3-d mesh reﬁnement algorithm 1. Run the surface meshing algorithm

2. Insert points to remove the bad elements of Del|S(E) and DelO(E)

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation Meshing 3D objects with curved boundaries

[Oudot et al. 05] O def= object bounded by S The surface meshing algorithm allows to control the aspect ratio of the facets of Del|S(E) but not the aspect ratio of the tetrahedra inside O

3-d mesh reﬁnement algorithm 1. Run the surface meshing algorithm

2. Insert points to remove the bad elements of Del|S(E) and DelO(E)

bad facet f : facet whose largest circumscribing Delaunay surface ball has radius rf > αψ(cf )

bad tet. t : tetrahedron whose circumscribing ball has radius rt > ψ(ct ) or a radius-edge ratio > ρ Basic procedures

refine face(f ) : insert cf , the center of the surface Delaunay ball circumscribing f refine tet(t) : insert ct , the center of the ball circumscribing t

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Properties

I For appropriate α and ρ, the algorithm terminates

I Del|S(E) = Del|S(E ∩ S)

I DelO(E) is a 3-triangulation isotopic to O

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation

Step 2 : Apply the following rules in order

1. if f ∈ Del|S(E) has a vertex in O \ S, refine face(f )

2. if f ∈ Del|S(E) is a bad facet, refine face(f )

3. if t ∈ DelO(E) is a bad tetrahedron, 3.1 if there is no facet f ∈ Del|S(E) whose surface Delaunay ball 3 ct , refine tet(t) 3.2 else refine face(f )

Step 2 : Apply the following rules in order

1. if f ∈ Del|S(E) has a vertex in O \ S, refine face(f )

2. if f ∈ Del|S(E) is a bad facet, refine face(f )

3. if t ∈ DelO(E) is a bad tetrahedron, 3.1 if there is no facet f ∈ Del|S(E) whose surface Delaunay ball 3 ct , refine tet(t) 3.2 else refine face(f )

Properties

I For appropriate α and ρ, the algorithm terminates

I Del|S(E) = Del|S(E ∩ S)

I DelO(E) is a 3-triangulation isotopic to O

Sliver removal [Cheng et al. 2000]

The algorithm removes all tetrahedra with a small volume except slivers, i.e. tet. with almost cocircular vertices and no small edge

Slivers are removed in a postprocessing step

Surface sampling : 33.133 points, 10,07s (1.8 GHz) Sampling + volume sampling : 46.012 sommets, 20.79s

Collaboration with CERTIS

50 tissus (segmentation courtesy of France Tel´ ecom)´

6 052 vertices 37 106 cells 8, 87o smallest dihedral angle

Deﬁnition A (non uniform) anisotropic mesh is a mesh with elements whose shape depends on prescribed (variable) directions.

Motivations include

I Esthetic criteria

I Error minimization (interpolation, FEM...)

Courtesy C. Dobrzynski Various heuristic solutions certiﬁed efﬁcient solutions are missing

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Constant metric ﬁeld Any anisotropic Voronoi diagram is an afﬁne diagram

t t dMp (x, a) < dMp (x, b) ⇔ (x − a) Mp(x − a) < (x − b) Mp(x − b) t t t t ⇔ −2a Mpx + a Mpa < −2b Mpx + b Mpb

whose dual triangulation is a Delaunay anisotropic triangulation

Restricted Delaunay Triangulation Mesh generation Surface mesh generation Applications Meshing 3D objects with curved boundaries CGAL-mesh Anisotropic mesh generation The metric field A continuous metric field is given at any point p, i.e.we know a positive definite quadratic form

q t dMp (a, b) = (a − b) Mp(a − b)

Constant metric ﬁeld Any anisotropic Voronoi diagram is an afﬁne diagram

t t dMp (x, a) < dMp (x, b) ⇔ (x − a) Mp(x − a) < (x − b) Mp(x − b) t t t t ⇔ −2a Mpx + a Mpa < −2b Mpx + b Mpb

whose dual triangulation is a Delaunay anisotropic triangulation

General case The anisotropic Voronoi diagram is curved

Courtesy F. Labelle and its dual is not a triangulation in general

Algorithm [B., Wormser & Yvinec 2008]

1. Compute the star of vertex p in DelMp (E)

2. Reﬁne the mesh until all the stars are coherent

3. Merge the local triangulations

in case of 1.a high distorsion of the metric 2. quasi-cospherical conﬁgurations

Because of 2, the reﬁnement process may not terminate if we insert points at the cc of bad elements

We avoid creating quasi-cospherical conﬁgurations by inserting points in a so-called picking region [Chew, Li 00]

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Image segmentation Applications Manifold reconstruction CGAL-mesh Spatio-temporal scenes modeling

A Lagrangian approach to dynamic interfaces

B. & Pons, CVPR 2007

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Image segmentation Applications Manifold reconstruction CGAL-mesh Spatio-temporal scenes modeling

Surface reconstruction : the implicit approach

1. Compute a function over R3 whose zero-set Z either interpolates or approximates E 2. mesh Z

Various implicit functions

I natural neighbor interpolation of the sign distance functions to the tangent planes at the sample points [B. & Cazals 00]

I moving least squares (MLS) [Levin 03]

I radial basis functions [Carr et al. 01]

I Poisson based approach [Khazdan et al. 06, Alliez et al. 07]

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Image segmentation Applications Manifold reconstruction CGAL-mesh Spatio-temporal scenes modeling

Poisson Surface Reconstruction [Alliez et al. 07] Key idea: Find implicit function f such that its gradient ∆f best aligns to the principal component of the tensors.

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Image segmentation Applications Manifold reconstruction CGAL-mesh Spatio-temporal scenes modeling

Poisson Surface Reconstruction [Alliez et al. 07]

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Image segmentation Applications Manifold reconstruction CGAL-mesh Spatio-temporal scenes modeling

Spatio-temporal scenes modeling from video sequences [Aganj et al 2007] Approach

I Extend the previous algorithm to compute 4D visual hulls ⇒ 4D representation of the scene

Advantages over frame-by-frame computations

I Exploits time redundancy

I Continuous representation, allowing spatio-temporal smoothing

I Reduction of ﬂickering artefacts in synthesized views

I Handles naturally topological changes along time

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Image segmentation Applications Manifold reconstruction CGAL-mesh Spatio-temporal scenes modeling

Meshing spatio-temporal scenes

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

CGAL

The Computational Geometry Algorithm Library

A 10 years development project conducted by a european consortium ETH Zurich, MPI Saarbrucken, INRIA Sophia Antipolis Tel Aviv University...

http://www.cgal.org

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

The CGAL Library

The Computational Geometry Library

I 600.000 lines of code I 45 men-years

I 3 000 pages of doc I 50 developers for teaching, research and industry

I a tool for experimental research in Computational Geometry

I used in astrophysics, medecine, biology, geosciences, image processing, scientiﬁc computing etc.

I 10 000 downloads per year, 800 users

I startup GeometryFactory

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

An effective treatment of robustness issues

I Exact arithmetics + arithmetic ﬁlters for efﬁcient exact predicates

3D Delaunay triangulation 1.7GHz, 1GByte 106 points

ﬂoat static ﬁlters exact 48s 64s 6.000s

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

CGAL-Mesh

A tool box for mesh generation, mesh processing and surface reconstruction Available

I Delaunay and regular (weighted Delaunay) triangulations

I Constrained 2D (Delaunay) triangulations

I Mesh generation for 2D polygonal domains

I Surface mesh generation

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

CGAL-Mesh

Coming soon

I Mesh generation of curved 3D domain

I Implicit surface reconstruction

Under development

I 3D curved domains with sharp features

I Anisotropic mesh generator

I Higher dimensional meshing

I Mesh optimization

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes Mesh generation Applications CGAL-mesh

Acknowledgments

I Steve Oudot, Jean-Philippe Pons, Laurent Rineau, Camille Wormser, Mariette Yvinec

I The members of the CGAL consortium

I European projects ECG (Effective Computational Geometry for Curves and Surfaces) ACS (Algorithms for Complex Shapes)

Emerging Trends in Visual Computing From Segmented Images to Good Quality Meshes