Symposium on Geometry Processing
                   17th July 2009 - Berlin


      Feature preserving
  Delaunay mesh gen...
From medical images to meshed models



                Segmentation                    Meshing




CT-scan image         ...
Motivations and Applications

●   Visualization 3D
●   Analysis
●   Preoperative planning
●   Biomedical simulations
●   A...
Multi-material meshing challenges

                                          virtual endoscopy


1. Conforming meshes for ...
Previous work
                     Surface mesh generation

 Marching Cubes extensions [Hege97,
  Wu03, …]




 Dual Mar...
Previous work
                       Volume mesh generation

 Grid/Octree based methods [Nielson97, Hartman98, Bajaj07, …...
Delaunay refinement strategy
                Advantages


1. Simultaneous meshing of multiple domains

2. Topology and geo...
Delaunay refinement strategy
               Drawbacks


1. The 1-junctions are usually zigzagging

2. The 0-junctions are ...
Feature preserving Delaunay refinement algorithm
                          Overview

Step 1: Detect 0- and 1-junctions in ...
Restricted Delaunay triangulation
                         Background 2D




     Voronoi diagram                Delaunay ...
Restricted Delaunay triangulation
                             Background 3D




Delaunay triangulation                  D...
Delaunay refinement algorithm
                           The basic


1. Initialization – at least 3 points per material

2...
Delaunay refinement algorithm
                                Criteria

Criteria for boundary facets:
➔ Topology

➔ Size

...
Delaunay refinement algorithm
                      The algorithm


1. Initialization – at least 3 points per material

2....
Delaunay refinement algorithm
                         Results


●   The algorithm terminates

●   If the resulting sampli...
Delaunay refinement algorithm
                        Results

BUT: 0 and 1-junctions are poorly represented...




      ...
Feature preserving extension




Step 1: Multi-material junction extraction

Step 2: Junction protection

Step 3: Adaptati...
Feature preserving extension
                    Step 1. Junction extraction



We use the digital subdivision of the inpu...
Feature preserving extension
                Step 1. Junction extraction




Digital 3D subdivision            1D cellular...
Feature preserving extension
     Step 2. Junction protection


1. Sample all 0-junctions

2. Sample points on 1-junctions...
Feature preserving extension
                      Step 2. Junction protection

Ball properties:
●   Every 1-junction is c...
Feature preserving extension
                 Step 3. Algorithm adaptation

The algorithm is tuned to use a weighted Delau...
Feature preserving extension
                   Step 3. Criteria adaptation

Boundary facet f with initial vertices whose ...
Feature preserving extension
                         Why does it work?

1. Refinement points inserted only outside the pr...
Results

Parameters: (30°, 12mm, 2mm, 4, 14mm)




                                   9°           168°
#vertices = 6 142,...
Results




 Digital 3D subdivision
                                    1D cellular complex




Protecting balls
Results

Parameters: (25°, 14mm, 4mm, 4, 18mm)




                                                           4°          ...
Acknowledgements




Thank you for your attention !
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Feature preserving Delaunay mesh generation from 3D multi-material images

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Presentation given by Dobrina Boltcheva (Postdoc at INRIA) at the Symposium on Geometry Processing, Berlin, Germany, 17th July 2009.

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

  1. 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. 2. From medical images to meshed models Segmentation Meshing CT-scan image Multi-material image 3D models
  3. 3. Motivations and Applications ● Visualization 3D ● Analysis ● Preoperative planning ● Biomedical simulations ● Augmented reality ● ....
  4. 4. Multi-material meshing challenges virtual endoscopy 1. Conforming meshes for all materials → simultaneous mesh generation right kidney 2. Preserving multi-material junctions: 2-junctions: surface patches artery 1-junctions: edges 0-junctions: corner vertices
  5. 5. Previous work Surface mesh generation  Marching Cubes extensions [Hege97, Wu03, …]  Dual Marching Cubes extensions [Gibson98, Nielson04, Bertram05, Reitinger05, Kobbelt06…]  Delaunay based methods [Amenta01, Boissonnat03, Meyer08…]
  6. 6. Previous work Volume mesh generation  Grid/Octree based methods [Nielson97, Hartman98, Bajaj07, …]  Delaunay refinement based methods [Oudot05, Rineau06, Yvinec07, Pons07…]
  7. 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. 8. Delaunay refinement strategy Drawbacks 1. The 1-junctions are usually zigzagging 2. The 0-junctions are not preserved and may be multiple
  9. 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. 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. 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. 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. 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. 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. 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
  16. 16. Delaunay refinement algorithm Results BUT: 0 and 1-junctions are poorly represented... 2 792 vertices, 5 681 triangles 20mm edge length 36 208 vertices 74 287 triangles 10mm edge length
  17. 17. Feature preserving extension Step 1: Multi-material junction extraction Step 2: Junction protection Step 3: Adaptation of the algorithm
  18. 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. 19. Feature preserving extension Step 1. Junction extraction Digital 3D subdivision 1D cellular complex: Five 1-junctions and two 0-junctions
  20. 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. 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. 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. 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. 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
  25. 25. Results Parameters: (30°, 12mm, 2mm, 4, 14mm) 9° 168° #vertices = 6 142, #boundary facets = 5 439, #tetrahedra = 31 043, Time = 24 sec
  26. 26. Results Digital 3D subdivision 1D cellular complex Protecting balls
  27. 27. Results Parameters: (25°, 14mm, 4mm, 4, 18mm) 4° 170° #vertices = 12 381, #boundary facets = 9 646, #tetrahedra = 64 485, Time = 98 sec
  28. 28. Acknowledgements Thank you for your attention !

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