Discovering Structural Regularity in 3D Geometry

1. Discovering Structural Regularity in 3D Geometry Speaker: JinliangWu Date: 25 / 9 /2008

2. Authors • Mark Pauly ETH Zurich • Niloy J. Mitra IIT Delhi • Johannes Wallner TU Graz • Helmut Pottmann TU Vienna • Leonidas Guibas Stanford University

3. Regular Structures

4. Regular Structures

5. completion geometric edits Motivation compression geometry synthesis Text Motivation

6. Compression

7. Completion

8. Geometry Synthesis

9. Transform Analysis Input Model Transform Clusters Model Estimation Aggregation Regular Structures Transform Generators Structure Discovery spatial domain transform domain Structure Discovery

10. A similarity transformation T Repetitive Structures

11. Repetitive Structures 1-parameter patterns

12. Repetitive Structures 2-parameter commutative patterns

13. Repetitive Structures

14. Repetitive Structures regular structure is a transformation group acting on is a collection of n patches of a given surface S

15. Repetitive Structures In the simplest setting, is a 1-parameter group with generating similarity transformation T . The elements of can be represented as

16. Input Model Transform Clusters Model Estimation Aggregation Regular Structures Transform Generators Structure Discovery Transform Analysis Structure Discovery

17. Transformation Analysis Algorithm for analyzing transformations

18. Transformations spatial domain transformation space pairwise transformations

19. Transformations spatial domain transformation space pairwise transformations

20. Model Estimation origin density plot of pair-wise transformations

21. Model Estimation cluster centers

22. Transformation Analysis Algorithm for analyzing transformations

23. Transformation

24. Model Estimation Is there a Pattern?

25. Model Estimation Yes, there is!

26. Model Estimation Yes, there is!

27. Model Estimation Global, non-linear optimization – simultaneously detects outliers and grid structure

28. grid location generating vectors Model Estimation • Grid fitting – input: cluster centers – unknowns: grid generators

29. data confidence cluster center closest grid point grid confidence grid point closest cluster center Model Estimation • Fitting terms

30. Model Estimation • Fitting terms • Data and grid confidence terms • objective function

31. Model Estimation Global, non-linear optimization – simultaneously detects outliers and grid structure

32. Input Model Transform Clusters Model Estimation Aggregation Regular Structures Transform Generators Structure Discovery Transform Analysis Structure Discovery

33. Aggregation • Region-growing to extract repetitive elements • Simultaneous registration

34. Simultaneous Registration

35. Input Model Transform Clusters Model Estimation Regular Structures Transform Generators Structure Discovery Transform Analysis Structure Discovery Aggregation

36. Results and Applications

37. Robustness

38. Geometry Synthesis

39. Geometry Synthesis

40. Scan Completion

41. Conclusions • Algorithm is fully automatic • Requires no prior information on size, shape, or location of repetitive elements • Robust, efficient, independent of dimension general tool for scientific data analysis