1. Co-Hierarchical Analysis of Shape Structures Oliver van Kaick 1,4 Kai Xu 2 Hao Zhang 1 Yanzhen Wang 2 Shuyang Sun 1 Ariel Shamir 3 Daniel Cohen-Or 4 4 Tel Aviv University 1 Simon Fraser University 3 The Interdisciplinary Center 2 HPCL, Nat. Univ. of Defense Tech.

2. Shape segmentation 2 Analysis of sets of shapes Joint segmentation Huang et al. 2011 Co-segmentation Sidi et al. 2011 Active co-analysis Wang et al. 2012

3. Shape segmentation 3 Segmentation: a flat representation

4. Part hierarchy 4 Hierarchy: a higher-level organization of shape parts

5. Applications of hierarchies 5 Use the hierarchy for various tasks Structure-aware shape editing [Wang et al. 2011] Hierarchical segmentation

6. Part hierarchies 6 Extraction of hierarchies from individual or pairs of shapes Symmetry hierarchy Wang et al. 2011 Geometry structuring Martinet 2007 Part recombination Jain et al. 2012

7. Co-hierarchical analysis 7 Our goal: Extraction of a unified (binary) hierarchy Through an unsupervised co-analysis of the set

8. Co-hierarchical analysis 8 A unified explanation of the structures Top-down to account for the structural variability

9. Co-hierarchical analysis 9 The co-hierarchy of a set of velocipedes Capturing the functionality of the parts

10. Co-hierarchical analysis 10 The co-hierarchy of a set of velocipedes Capturing the functionality of the parts

11. Co-hierarchical analysis 11 The co-hierarchy of a set of velocipedes Capturing the functionality of the parts

12. Challenge of co-hierarchical analysis 12 Shapes can have many possible hierarchies We need to select one hierarchy per shape …

13. Challenge of co-hierarchical analysis 13 There can be geometric variability in the set We need to compare the shape structures

14. Challenge of co-hierarchical analysis 14 There can also be much structural variability We need to account for that

15. Challenge of co-hierarchical analysis 15 Cluster-and-select scheme: clustering, representative selection, and resampling

16. Overview 16

17. Overview 17

18. Sampling the space of hierarchies We sample the space by sampling the splits Difficult to define a generic splitting criterion Criterion: balance of volume, compactness of parts, normalized cut? We resort to random sampling We sample splits in a top-down manner 18

19. Tree-to-tree distance 19 Tree-to-tree distance: structural differences

20. Node distance 20 Transformation between bounding boxes Bounding boxes focus on the structural similarity

21. Shape distance 21 Shape distance: distance between hierarchies

22. Cluster-and-select motivation 22 Representative selection

23. Cluster-and-select 23 Minimal illustrative example with four shapes

24. Cluster-and-select 24 Multiple possible hierarchies per shape

25. Cluster-and-select 25 Sampling of hierarchies

26. Cluster-and-select 26 Multi-instance clustering

27. Cluster-and-select 27 Representative selection

28. Cluster-and-select 28 Traditional clustering: maximize similarity within clusters and dissimilarity between clusters

29. Cluster-and-select 29 Our problem: maximize similarity within clusters and similarity between clusters

30. Cluster-and-select 30 Samples maximize the similarity within clusters

31. Cluster-and-select 31 Also maximize the similarity between clusters

32. Cluster-and-select 32 Resampling of hierarchies

33. Cluster-and-select 33 Resampling of hierarchies

34. Cluster-and-select 34 Repeat the process: clustering, selection

35. Cluster-and-select 35 Representative movement

36. Results of co-hierarchical analysis 36 The co-hierarchies are shown as a hierarchical segmentation

37. Results of co-hierarchical analysis 37 Hierarchical segmentation results

38. Results of co-hierarchical analysis 38 Hierarchical segmentation results

39. Results of co-hierarchical analysis 39 Hierarchical segmentation results

40. Results of co-hierarchical analysis 40 Consistency of the co-hierarchy [Wang et al. 2011] Ours

41. Results of co-hierarchical analysis 41 Cluster-and-select on a mixed set of shapes

42. Summary of contributions Co-hierarchical analysis of sets of shapes Structure-driven shape analysis – To deal with geometric variability Hierarchical analysis – To deal with structural variability A novel cluster-and-select scheme – To account for both variability and similarity The structural co-hierarchy representation – Unifies the learned structures 42

43. Limitations and future work Co-hierarchical analysis: only a first step More sophisticated node and tree distances Initial random sampling of trees Integrate segmentation and hierarchical analysis Multi-class co-hierarchies Which hierarchy should be selected? 43

44. 44 Co-Hierarchical Analysis of Shape Structures Project page: http://www.cs.sfu.ca/~ovankaic/personal/conshier/http://www.cs.sfu.ca/~ovankaic/personal/conshier/ Thank you for your attention!

45. Appendix 45

46. Tree-to-tree distance 46 Node distance Recursive children distance NiNi NjNj

47. Tree-to-tree distance 47 NiNi NjNj Node distance Recursive children distance

48. Results of co-hierarchical analysis 48 Hierarchical segmentation results

49. Results of co-hierarchical analysis 49 Hierarchical segmentation results: deeper levels

50. Results of co-hierarchical analysis 50 Improvements shown by the cluster-and-select