2. Contraction kernels and Combinatorial maps Luc Brun L.E.R.I. University of Reims -France luc.brun@univ-reims.fr and Walter Kropatsch P.R.I.P Vienna Univ. of Technology-Austria krw@prip.tuwien.ac.at

3. Content of the talk  Combinatorial maps  Expected Advantages  Irregular Pyramids  Contraction Kernels  Conclusion

4. Combinatorial Maps Definition  G=(V,E)  G=(D, ,  )  decompose each edge into two half- edges(darts) : -  : edge encoding D ={-6,…,-1,1,…,6} 1 -5 2 -34 -4 5 -2 6 -6 3

5. Combinatorial Maps Definition  G=(D, ,  )   : vertex encoding  * (1)=(1,  * (1)=(1,3  * (1)=(1,3,2) 12 3 -3 4 -4 5 -5 -2 6 -6

6. Combinatorial Maps Properties  Computation of the dual graph :  * (-1)=(-1,3,4  * (-1)=(-1,3,4,6) 1 5 -5 -4 -3 -6 6 2 -2 4 3 12 3 -3 4 -4 5 -5 -2 6 -6  * (-1)=(-1,3  * (-1)=(-1 G=(D, ,  )G=(D,  =   ,  )

7. Reduction operations  Removal operation: not allowed for bridges  Contraction operation: not defined for self- loops 3 -3 4 -4 5 -5 -2 6 -6 12 d =  *(3) 1 5-5-4 -3 -6 6 2 -2 4 3 1 5 -5 -4-4 -66 2 -2 4 d =  *(3) 4 -4 5 -5 -2 6 -6 12

8. Reduction operation Property  Removal and Contraction preserve the orientation 1 2 3 4 d c b a 1 2 3 4 d c b a

9. Expected Advantages  May encode many topological features (multiple boundaries, surrounding relationships...)  Encode explicitely the orientation of edges around each vertex  Efficient encoding of the dual (may be implicitely encoded)  May be extended to higher dimensions

10. Irregular Pyramids  Definition: Stack of successively reduced graphs  Advantages  Efficient computation of global features through local computations  Describe several level of details of a same image  Construction scheme  Contraction parameter: Defines which edges must be contracted  Contraction operations

11. Contraction Kernel  G=(D, ,  ), K  D  K is a contraction Kernel iff  K defines a forest of G,  K preserves the image boundary  SD=D-K is called the set of surviving darts.

12. Example of Contraction : K Selection of K 1 Contraction of K 1 Selection of redundant double edges Contraction of K 2 Removal of redundant double edges Selection of K 2

13. Equivalent Contraction kernels K1K1 K2K2 K3K3

14. Reduction operation  Example 1 23 4 56 7 89 10 1112 13141516 17181920 21222324 K=

15. Reduction operation  Example 1 23 4 56 7 89 10 1112 131415 16 171819 20 212223 24 K= G=(D, ,  )  G’=(D-K,  ’,  ) ?

16. Reduction operation  How to compute the contracted combinatorial map ?  What is the value of  ’(-2) ? 12 4 1314 15 -2 2 4 13 14 15 -2 13

17. Reduction operation  How to compute the contracted combinatorial map ?  What is the value of  ’(-2) ? 12 4 1314 15 -2 -13 17 7 2 4 14 15 -2 17 7

18. Connecting Walk 1 2 3 4 56 7 89 10 1112 1314 1516 1718 1920 2122 2324 -2 CW(-2)=-2.13.17.21. 10 For each d  SD

19. Construction of the contracted map 1 2 3 4 56 7 89 10 11 12 1314 1516 1718 1920 2122 2324 -2 For each d in SD=D-K d’=  (d) While( d’  K) d’=  (d)  ’(d)=d’  Sequential Algorithm

20. Construction of the contracted map 1 2 3 4 56 7 89 10 1112 1314 1516 1718 1920 2122 2324 -2 2 3 56 89 11 12 1516 1920 2324 -2 18 14 22 7 4 -11

21. Parallel computation of the contracted map  Each dart traverses in parallel its connecting walk 12 1011 13 17 21 -2 -1 13 17 21 10 11 13 17 21 10 11 11 17 21 10 11 11 11 21 10 11 11 11 11 10 11 11 11 11 11 11 11 11 11 11 11 Survive[d]=d While(Survive[d]  K) Survive[d]=Survive[  (d)]

22. Conclusion  Construction of the pyramid by contraction of Combinatorial maps.  Sequential/Parallel algorithms based on Contraction Kernels  Equivalent Contraction Kernels  increase/decrease the decimation ratio

23. Perspectives  Explicit / Implicit encoding  General/optimised contraction kernel