1. SA2014.SIGGRAPH.ORG SPONSORED BY

2. SA2014.SIGGRAPH.ORG SPONSORED BY Approximate Pyramidal Shape Decomposition Ruizhen Hu Honghua Li Hao Zhang Daniel Cohen-Or Simon Fraser UniversityZhejiang UniversityTel Aviv University

3. SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal shape 3 Terrain 2.5D Heightfield

4. SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal shape 4

5. SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 5

6. SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 6 3D layered printing

7. SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 7 3D layered printing Molding/casting

8. SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 8 3D layered printing Molding/casting Point inside/outside testing

9. SA2014.SIGGRAPH.ORG SPONSORED BY Various applications 9 3D layered printing Molding/casting Point inside/outside testing

10. SA2014.SIGGRAPH.ORG SPONSORED BY However, 10 Many common objects are not pyramidal… Pyramidal decomposition

11. SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal decomposition Exact pyramidal decomposition ◦Minimum number of pyramidal parts 11 NP-hard NP-hard [Fekete and Mitchell 2001]

12. SA2014.SIGGRAPH.ORG SPONSORED BY Pyramidal decomposition Approximate pyramidal decomposition ◦Parts being only approximately pyramidal ◦Still aiming to produce as few parts as possible 12

13. SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 13 Exact Cover Problem {1234} …

14. SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 14 {1234} … Exact Cover Problem

15. SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 15 {1234} … 1 2 3 4 Over-segmentation Candidate parts by merging segments Exact Cover Problem

16. SA2014.SIGGRAPH.ORG SPONSORED BY Key idea 16 {1234} … Pyramidal decomposition Pyramidal Final decomposition Over-segmentation Candidate parts by merging segments Exact Cover Problem

17. SA2014.SIGGRAPH.ORG SPONSORED BY Find an appropriate over-segmentation of the shape Build an appropriate set of candidate pyramidal parts Main challenge 17 Infinite possible pyramidal parts

18. SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts ◦Minimum-sized decompositions ◦Easier to generate 18

19. SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough 19 1 2 34 5 6

20. SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough Combination of intersections 20 1 2 34 5 6

21. SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough Combination of intersections 21 1 2 34 5 6 …

22. SA2014.SIGGRAPH.ORG SPONSORED BY Intuition Large pyramidal parts May not be enough Combination of intersections 22 1 2 34 5 6 …

23. SA2014.SIGGRAPH.ORG SPONSORED BY Algorithm overview 23 Set Collection of subsets Exact cover Intersections

24. SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling ◦Sample a set of directions ◦Sample a set of bases along each direction 24

25. SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection ◦Capacity measure 25 Total pyramidal areas Number of decomposed parts

26. SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection ◦Capacity measure 26

27. SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection ◦Capability measure ◦Base voting 27

28. SA2014.SIGGRAPH.ORG SPONSORED BY Cell generation Base sampling Base selection Cell formation ◦Points voted for the same set of bases 28 Estimation of intersections

29. SA2014.SIGGRAPH.ORG SPONSORED BY Block construction To further reduce the number of candidate parts ◦Cluster cells into blocks ◦Find the boundaries 29

30. SA2014.SIGGRAPH.ORG SPONSORED BY Candidate parts & decomposition Parts are formed by merging blocks Decomposition can be found by solving ECP 30 Blocks Decompositions Candidate parts

31. SA2014.SIGGRAPH.ORG SPONSORED BY Results Datasets ◦149 2D shapes ◦80 3D shapes 31

32. SA2014.SIGGRAPH.ORG SPONSORED BY Results 2D shapes 32

33. SA2014.SIGGRAPH.ORG SPONSORED BY Results 3D shapes 33

34. SA2014.SIGGRAPH.ORG SPONSORED BY Evaluations 2-pyramidality ◦25 2D shapes & 30 3D shapes 34

35. SA2014.SIGGRAPH.ORG SPONSORED BY Evaluations Comparison to greedy approach ◦2-pyramidality (2D shapes) 35 Ours: Greedy:

36. SA2014.SIGGRAPH.ORG SPONSORED BY Evaluations Comparison to human judgement ◦30 users & 20 2D shapes 36 OursUser decomposition

37. SA2014.SIGGRAPH.ORG SPONSORED BY 3D Printing 37

38. SA2014.SIGGRAPH.ORG SPONSORED BY Conclusion A relatively new and unexplored shape property. The first construction algorithm for approximate pyramidal decomposition. The key insight is to turn the problem into a exact cover problem. The bottom-up clustering-based approach is both general and versatile. 38

39. SA2014.SIGGRAPH.ORG SPONSORED BY Limitation & further work Realistic waste estimation for 3D printing Pyramidalization ◦Minimal modification to maximally reduce the size of its pyramidal decomposition 39

40. SA2014.SIGGRAPH.ORG SPONSORED BY Set Collection of subsets Exact cover Intersections Thank you! Approximate Pyramidal Shape Decomposition Ruizhen Hu Honghua Li Hao Zhang Daniel Cohen-Or