Group Representation of Global Intrinsic Symmetries

1. Pacific Graphics 2017 Group Representation of Global Intrinsic Symmetries • 1.Shijiazhuang Tiedao University • 2. Shenzhen University Hui Wang1,2Hui Huang2

2. Content • Motivation • Theory • Algorithm • C2 shapes • Shapes with more symmetries • Application • Conclusion

3. Representation of Euclidean symmetry 3D Euclidean symmetry can be represented compactly as a combination of an orthogonal matrixand a translation vector. 3D transformation matrix

4. How to represent global intrinsic symmetries?

5. Problems Most previous works are based on point-to-point correspondencesof global intrinsic symmetries: • can onlyfind reflectional symmetries; • are inadequate for describing the structure of a global intrinsic symmetry group.

6. Contribution 1 Representeach global intrinsic symmetry of a compact manifold in any dimension as a linear transformation on the function space defined on the manifold, which can be represented by a matrix under a chosen basis. symmetry composition=product their representation matrix

7. Contribution 2 Prove that each global intrinsic symmetry can be uniquely recovered from some sparse symmetric pairs of points under mild conditions, where the minimal number of pairs is the maximum multiplicity of the eigenvalue of the Laplace-Beltrami operator.

8. Contribution 3 Our proposed algorithm is the first one that can detect both rotational and reflectional global intrinsic symmetries and describe their relations in the symmetry group.

9. Content • Motivation • Theory • Algorithm • C2 shapes • Shapes with more symmetries • Application • Conclusion

10. Symmetry representation

11. Matrix representation

12. Symmetry determination

13. Content • Motivation • Theory • Algorithm • C2 shapes • Shapes with more symmetries • Application • Conclusion

14. Process of C2 shapes Four symmetry pairs Representation matrix Point to point

15. Results on popular data set TOSCA SCAPE

16. Comparisons • For C2 shapes, our proposed method performs comparably or slightly better than state-of-the-arts with much smaller computation cost. TOSCA data sets

17. Content • Motivation • Theory • Algorithm • C2 shapes • Shapes with more symmetries • Application • Conclusion

18. Process on shapes with more symmetries

19. More results Mean geodesic distance errors of the computed twenty global intrinsic symmetries (under each symmetry), which are much smaller than the tolerance error 3.92.

20. Content • Motivation • Theory • Algorithm • C2 shapes • Shapes with more symmetries • Application • Conclusion

21. Group table Characterization of finite global intrinsic symmetry group by its group table.

22. Content • Motivation • Theory • Algorithm • C2 shapes • Shapes with more symmetries • Application • Conclusion

23. Conclusions • A novel group representation is introduced for global intrinsic symmetries on 3D shapes, where each symmetry can be represented as a matrix. • Furthermore, we prove that each global intrinsic symmetry can be uniquely recovered from sparse symmetric pairs of points under mild conditions. • Based on the above solid theories, we have also proposed an effective algorithm to detect both rotational and reflectional global intrinsic symmetries in practice.

24. Limitations • Although the proposed group representation is generally valid for compact manifolds in any dimension and genus, the introduced algorithm fails for shapes with continuous global intrinsic symmetries, such as a sphere and cylinder. • Moreover, if the number of sampling points and the maximum multiplicity are very large, the pairs of points generation and clustering of the representation matrices could be computationally heavy.

25. Future works • We would like to describe the continuous symmetries combined with the theories of the Lie group and the Lie algebra representation. • We also would investigate automatic or data-driven method for optimal parameter selection. • More applications of our methods, such as symmetry-aware 3D shape reconstruction.

26. Sample codes vcc.szu.edu.cn/research/2017/Symmetry/ Thanks for the anonymous reviewers & funding from NSFC, 973 program, STDU, SZU!

27. Thanks! 謝謝!