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Feature preserving Delaunay mesh generation from 3D multi-material images
1. Symposium on Geometry Processing
17th July 2009 - Berlin
Feature preserving
Delaunay mesh generation
from 3D multi-material images
Dobrina Boltcheva, Mariette Yvinec, Jean-Daniel Boissonnat
2. From medical images to meshed models
Segmentation Meshing
CT-scan image Multi-material image 3D models
3. Motivations and Applications
● Visualization 3D
● Analysis
● Preoperative planning
● Biomedical simulations
● Augmented reality
● ....
6. Previous work
Volume mesh generation
Grid/Octree based methods [Nielson97, Hartman98, Bajaj07, …]
Delaunay refinement based methods [Oudot05, Rineau06,
Yvinec07, Pons07…]
7. Delaunay refinement strategy
Advantages
1. Simultaneous meshing of multiple domains
2. Topology and geometry approximation guarantees
3. Control of elements' size and shape:
possibly non-uniform sizing field
8. Delaunay refinement strategy
Drawbacks
1. The 1-junctions are usually zigzagging
2. The 0-junctions are not preserved and may be multiple
9. Feature preserving Delaunay refinement algorithm
Overview
Step 1: Detect 0- and 1-junctions in the input 3D image
Step 2: Sample points on the 0- and 1-junctions and
protect the junctions with balls centred on these points
Step 3: Run the Delaunay refinement algorithm
with the protecting balls as initial set of vertices
10. Restricted Delaunay triangulation
Background 2D
Voronoi diagram Delaunay triangulation
Delaunay triangulation Delaunay triangulation
restricted to the blue curve: restricted to the yellow region:
→ set of edges whose dual Voronoi → set of triangles whose
edges intersect the curve circumcentres are in the region
11. Restricted Delaunay triangulation
Background 3D
Delaunay triangulation Delaunay triangulation
restricted to the surfaces: restricted to the volumes:
→ set of triangles whose dual Voronoi → set of tetrahedra whose
edges intersect any surface circumcentres are inside any volume
12. Delaunay refinement algorithm
The basic
1. Initialization – at least 3 points per material
2. Refinement – inserting new vertices,
maintaining the Delaunay triangulation,
its restriction to volumes and boundary facets
until there is no bad element left
13. Delaunay refinement algorithm
Criteria
Criteria for boundary facets:
➔ Topology
➔ Size
➔ Shape
➔ Approximation surface Delaunay ball
Criteria for tetrahedra:
➔ Size
➔ Shape
A bad element does not fulfil all the criteria.
Refinement boundary facets and tetrahedra:
refine_facet(f) → insertion of its surface Delaunay ball centre
refine_tet(t) → insertion of its circumcentre
14. Delaunay refinement algorithm
The algorithm
1. Initialization – at least 3 points per material
2. Refinement
a) If there is a bad boundary facet f
then refine_facet(f)
b) If there is a bad tetrahedron t
- compute the circumcentre c
- if c is included in a surface Delaunay ball
of some boundary facet f
- then refine_facet(f)
- else refine_tet(t)
3. Sliver exudation
15. Delaunay refinement algorithm
Results
● The algorithm terminates
● If the resulting sampling is dense enough,
every image 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 approximation
of the volume
17. Feature preserving extension
Step 1: Multi-material junction extraction
Step 2: Junction protection
Step 3: Adaptation of the algorithm
18. Feature preserving extension
Step 1. Junction extraction
We use the digital subdivision of the input 3D image.
2-junctions: surface patches
→ surfels between 2 materials
1-junctions: digital edges
→ linels between 3 or more materials
0-junctions: corner vertices
→ pointels between 4 or more materials
19. Feature preserving extension
Step 1. Junction extraction
Digital 3D subdivision 1D cellular complex:
Five 1-junctions and two 0-junctions
20. Feature preserving extension
Step 2. Junction protection
1. Sample all 0-junctions
2. Sample points on 1-junctions with
a user-given density d.
3. Cover junctions with
protecting balls b(p,r) with r=2/3*d
centred on sampled points
21. Feature preserving extension
Step 2. Junction protection
Ball properties:
● Every 1-junction is completely covered
by the protecting balls of its samples
Fig.1
● 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,
exception Fig.1 Fig.2
● No 3 balls have a common intersection,
exception Fig.1 and Fig.2
● No 4 balls have a common intersection,
Fig.3
exception Fig.3
22. Feature preserving extension
Step 3. Algorithm adaptation
The algorithm is tuned to use a weighted Delaunay triangulation.
1. Initialization – with the weighted points corresponding
to the protecting balls
2. Refinement
a) If there is a bad boundary facet f
then refine_facet(f)
b) If there is a bad tetrahedron t
- compute the weighted circumcentre c
- if c is included in a weighted surface Delaunay ball
of some boundary facet f
- then refine_facet(f)
- else refine_tet(t)
3. Sliver exudation
23. Feature preserving extension
Step 3. Criteria adaptation
Boundary facet f with initial vertices whose protecting balls intersect
● 3 → f is never refined
● 2 or 1 → topology and sizing criteria
Tetrahedron t with initial vertices whose protecting balls intersect
● 4 → t is never refined
● 3 → sizing
(check surface Delaunay ball of the constrained facet)
● 2 or 1 → sizing
24. Feature preserving extension
Why does it work?
1. Refinement points inserted only outside the protecting balls:
→ meshing independent 2-junctions
2. Relaxed quality criteria in proximity to 0- and 1-junctions:
→ algorithm termination
Result: any 2 consecutive points on a 1-junction
remain connected with a restricted Delaunay edge
Note: the difference between digital and trilinear
junction definitions is hidden by
the protecting balls