Contraction Kernels and Combinatorial Maps: A Comprehensive Overview

1. 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

2. Content of the talk • Combinatorial maps • Expected Advantages • Irregular Pyramids • Contraction Kernels • Conclusion

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

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

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

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

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

8. 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

9. 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

10. 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.

11. Example of Contraction : K Contraction of K1 Selection of K2 Selection of K1 Removal of redundant double edges Selection of redundant double edges Contraction of K2

12. Equivalent Contraction kernels K1 K2 K3

13. Reduction operation • Example K= 1 2 3 13 14 15 16 4 5 6 17 18 19 20 7 8 9 21 22 23 24 10 11 12

14. Reduction operation • Example K= 1 2 3 16 13 14 15 4 5 6 20 17 18 19 G=(D,,) G’=(D-K,’,) ? 7 8 9 24 21 22 23 10 11 12

15. 2 -2 13 15 14 4 Reduction operation • How to compute the contracted combinatorial map ? • What is the value of ’(-2) ? 1 -1 2 -2 13 13 14 15 4

16. 17 2 -2 15 4 14 7 Reduction operation • How to compute the contracted combinatorial map ? • What is the value of ’(-2) ? 1 -1 2 -2 13 14 15 -13 4 17 7

17. Connecting Walk For each d SD 1 -1 2 -2 3 13 14 15 16 4 5 6 17 18 19 20 7 8 9 CW(-2)=-2. -1. 13. 17. 21. 10 21 22 23 24 10 11 12

18. Construction of the contracted map • Sequential Algorithm For each d in SD=D-K d’=(d) 1 -1 2 -2 3 While( d’  K) d’=(d) 13 14 15 16 4 5 6 ’(d)=d’ 17 18 19 20 7 8 9 21 22 23 24 11 10 12

19. Construction of the contracted map 2 -2 1 3 -1 2 -2 3 14 15 16 13 14 15 16 4 4 5 6 5 6 7 11 19 20 18 17 18 19 20 7 8 9 8 9 23 24 21 22 23 24 22 10 12 11 -11 12 -11

20. -1 13 17 21 10 11 -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 Parallel computation of the contracted map • Each dart traverses in parallel its connecting walk Survive[d]=d While(Survive[d]  K) Survive[d]=Survive[ (d)] 1 2 -2 -1 13 17 21 10 11

21. 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

22. Perspectives • Explicit / Implicit encoding • General/optimised contraction kernel