Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat INRIA – Sophia Antipolis, Francefirstname.lastname@sophia.inria.fr

1. Initialization

Use the weighted points corresponding to the protecting ballsAt least 3 points per material

Build the current mesh which is a restricted Delaunay triangulation: DelVol = set of tetrahedra whose circumcentres are inside any material DelSurf = set of boundary facets between two tetrahedra in different materials

3. Sliver exudation

Algorithm:(1) If there is a bad boundary facet f in DelSurf then refine-facet(f)(2) If there is a bad tetrahedron t in DelVol then

compute the weighted circumcentre cif c is included in a weighted surface Delaunay ball of some boundary facet f then refine-facet(f)else refine-tet(f)

2. Refinement

III. Mesh generation

Conclusion

References

Results

Restricted Delaunay triangulationMethod

Detection and extraction of multi-material junctions where 3 or more materials meet.

The algorithm uses the digital subdivision of the input 3D image: surface patches - connected sets of surfels : intersection of 2 materials edges - connected sets of linels : intersection of 3 or more materials corners - pointels : intersection of 4 or more materials

After a single scan of the image, the algorithm outputs a 1D cellular complex composed of digital edges and corners. A digital edge may be either a closed curve with no endpoint or an edge with two endpoints.

I. Junction extraction

II. Junction protection

Protection is required:→ constraining the edges and the corners in the final mesh prevents

the Delaunay refinement to terminate when surface patches make small angles.

Delaunay triangulation restricted to the blue curve:→ set of edges whose dual Voronoi edges intersect the curve

Delaunay triangulation restricted to the yellow region:→ set of triangles whose circumcentres are in the region

A set of points in 2D and its Voronoi diagram

Delaunay triangulation

Feature preserving Delaunay mesh generation from 3D multi-material images

Introduction

Segmented 3D images impose particular challenges for meshing algorithms since they contain up to three types of multi-material junctions: surface patches, edges and corners.

Multi-material junction features are expected to be properly represented in the output mesh.

We present a Feature Preserving Delaunay Refinement (FPDR) algorithm which builds high-quality tetrahedral meshes from 3D multi-material images, with a user-specified resolution.

The basic idea is to constrain the Delaunay refinement algorithm to respect not only the surface patches but also the edges and the corners.The method extends the refinement algorithm in [1] and uses the protecting balls scheme [2] to preserve the corners and to reconstruct accurately the edges.

Balls properties: Every 1-junction is completely covered by the protecting balls of its samples Any 2 adjacent balls on a given 1-junction overlap without containing each other's centre Any 2 balls on different 1-junctions do not intersect No 3 or 4 balls have a common intersection

The algorithm inserts points into the current mesh until mesh simplices meet the refinement criteria.

The method is related to the concept of the restricted Delaunay triangulation [3].

We introduced an efficient and robust Feature Preserving Delaunay Refinement meshing strategy from a segmented 3D image.

The algorithm outputs high-quality tetrahedral meshes where all multi-material junctions are consistently represented:

corners are preserved and unique in the final mesh, edges are accurately approximated by a set of consecutive Delaunay edges, without zigzagging, surface patches are smoothly represented by a set of restricted Delaunay facets.

[1] Pons et al. High-quality consistent meshing of multi-label datasets. In IPMI 2007.[2] Dey et al. Delaunay refinement for piecewise smooth complexes. In SODA 2007.[3] Boissonnat & Yvinec. Algorithmic geometry. Cambridge University Press 1998.[4] Cheng et al. Sliver exudation. In SCG 1999.

Acknowledgments

The method has been implemented using primitives provided by CGAL - Computational Geometry Algorithms Library, http://www.cgal.org. Segmented medical images are courtesy of IRCAD, http://www.ircad.frThis work was supported by FOCUS K3D Coordination Action, EU Contract ICT-2007.4.2.

Our meshing strategy is composed of three main steps:

The Delaunay refinement algorithm is tuned to use a weighted Delaunay triangulation.

The weighted distance from a point x in R3 to a weighted point (p,r) is ||x-p||² - r².

Criteria for boundary facets: Topology Size Shape Approximation

Criteria for tetrahedra: Size Shape

A bad element is a simplex which does not fulfil the refinement criteria:

Refinement of boundary facets and tetrahedra: refine-facet(f)→ insertion of weighted surface Delaunay ball centre refine-tet(f)→ insertion of weighted circumcentre

surface Delaunay ball

After the refinement, the mesh is optimized using a variant of the sliver exudation algorithm [4].

1. Simultaneous meshing of multiple domains

2. Topology and geometry approximation guarantees

3. Control of elements' size and shape: → possibly non-uniform sizing field

4. If the resulting sampling is dense enough, every material is represented by a submesh of tetrahedra

→ boundary facets provide a good and watertight approximation of the surface,

free of self intersections→ tetrahedra form a good mesh of the volume

5. Initial points are kept in the final mesh→ corners are preserved→ any 2 consecutive points sampled on a digital edge

remain connected with a restricted Delaunay edge

6. Refinement points are inserted outside the protecting balls:→ refinement terminates

Given a 3D multi-material image, our algorithm meshes simultaneously the set of regions with different labels. It builds a unique tetrahedral mesh where every anatomical structure is represented by a submesh.

with feature preservationwithout feature preservation

pointel

linel

surfel

voxel

Algorithm:(1) Sample points on the junctions

- all corners - points on edges with user-given density d.

(2) Cover the features with protecting balls centered on the sampled points.

.

.

.

3D digital subdivision

digital edges and corners

sampled points

protecting balls

50 loops

100 loops

end

4° 170°

148°60°15°dihedral angle distribution

without feature preservation

with feature preservation

168°9°

with feature preservation

without feature preservation