1. Graph Based Representations in different application Domains Luc Brun and Myriam Mokhtari L.E.R.I. Reims University (France)

2. Content of the talk n 2D Segmentation n Shape Representation n 3D Segmentation n Links

3. 2D Segmentation n Encoding the 2D partition

4. 2D Segmentation n Encoding the 2D partition Simple graph [Piroddi 01]

5. 2D Segmentation n Encoding the 2D partition Simple graph

6. 2D Segmentation n Encoding the 2D partition Simple graph

7. 2D Segmentation n Encoding the 2D partition Simple graph

8. 2D Segmentation n Encoding the 2D partition Simple graph Dual graph

9. 2D Segmentation n Encoding the 2D partition Simple graph Dual graph 1 2 3 -34 5 -4-5 6 -6 -2

10. 2D Segmentation n Simple Graph n Efficient encoding n Dual Graph n Encodes topological features: multiple boundaries, surrounding relationships. May be encoded by usual graph libraries. n 2D Combinatorial maps n Encodes topological features. Allows an implicit encoding of the dual graph. May be extended to higher dimensions

11. Shape Representation n Encoding the shape of an object: Pattern Recognition, Matching –Skeleton –Medial Axis

12. Shape Representation n Medial graph Illustration from [Salotti 01]

13. Shape Representation n Crest graph/River Graph –Crest Graph n Vertex: Hill altitude n edge: lowest altitude of the paths between two hills –River Graph n Dual vertex: Basin altitude n dual edge: higher altitude of the paths between two basins.

14. Shape Representation n Crest Graph 25 10 5 55 55 555 255 5105 555 5 5 5 55 55 5 5 5 5 555 255 5105 55 5 5 55 55 5 5 25 5 5

15. Shape Representation n Crest Graph Original ImageSobel operatorCrest Graph Illustration from [Glantz 01]

16. Shape Representation n Shock Graph 3-002 1-002 1-003 3-003 1-008 1-007 1-009 3-002 3-004 1-004 1-005 1-001 3-001 1-0071-0081-009 3-001 1-001 3-003 1-0021-003 3-004 1-0041-005 1234 Illustration from [Siddiqi98]

17. n Medial graph n Exact reconstruction n Crest graph n Not restricted to distance map, support a parallel implementation, does not encode narrowing dead ends. n Shock Graph n Exact reconstruction, Highly structurated graph. Shape Representation

18. 3D Segmentation n Encoding a 3D scene CUBE CUBE, HOLE 3HBAR, HOLE BOLT HEAD [Bauckhage 01]

19. 3D Segmentation 2-2 -4 1 -3 3   -22 1  4 -4-3 3  -4 n Encoding 3D partition : 3D combi. maps n G=(D, ,  )  G=(D, , ,  ) Example from [Braquelaire98]

20. 3D Segmentation n Two approaches: – Splitting scheme: Braquelaire, Desbarats, Domenger, Wüthrich n Tracking of surface boundaries –Bottom-up scheme: Bertrand, Damiand, Fiorio n Successive simplifications of the 3D map encoding the partition

21. Links n 3D dual graph - 3D combinatorial maps n Medial graph, Crest Graph, Shock Graph: which choice for which application ? n 3D Irregular Pyramids based on 3D combinatorial maps n 3D Split & Merge algorithms based on a combination of [Braquelaire] and [Bertrand]