1. Volume Parameterization Reporter : Lei Zhang 10\24\2007

2. What is volume Surface model 2D manifold Solid\Volume model 3D manifold

3. Discrete Representation triangletetrahedron quadranglehexahedron

4. Surface Parameterization Parameterization can be viewed as a one-to-one mapping from a suitable domain to the surface. 1997~ 2007 Huge amount of papers

5. Surface Parameterization Application Re-meshing Morphing Texture mapping

6. Volume Parameterization Parameterization can be viewed as a one-to-one mapping from a suitable domain to the volume. Not proposed formally!

7. Texture Mapping

8. Tetrahedral\Hexhedral Meshing (a)(b)(c)(d)

9. Rendering Acceleration (a) (d) (c) (b)

10. Volume Parameterization Difficulties  Math ground  High dimensional geometry  3 manifold is special  Data structure  Enormous datas  Complex connectivity  Tiny amount of papers ? ? ?

11. Problems Surface:  conformal mapping  harmonic mapping  … … Volume:  ?  … …

12. Harmonic Volume Mapping Harmonic equation

13. Yalin Wang, Xianfeng Gu, and Shing-Tung Yau. Volumetric harmonic map. Communications in Information and Systems, 2004. Yalin Wang, Xianfeng Gu, Tony F. Chan, Paul M. Thompson, and Shing-Tung Yau. Volumetric harmonic brain mapping. IEEE International Symposium on Biomedical Imaging, 2004.

14. Harmonic mapping from 3 manifold to 3D solid sphere Surface conformal mapping

15. Steepest descent method

16. Xin Li, Xiaohu Guo, Hongyu Wang, Ying He, Xianfeng Gu, and Hong Qin. Harmonic volumetric mapping for solid modeling application. SPM, 2007.

17. Problem Formulation

18. Solving Equation self-adjoint Green function Fundamental Solution Method (MFS) G. Fairweather and A. Karageorghis. The method of fundamental solutions for elliptic boundary value problems. Advances in Computational Mathematics, 1998.

19. MFS

20. Electric field Intuitive Explanation e

21. Algorithm

22. Discussion

23. Continuous Volume Mapping M. S. Floater. Mean Value Coordinates. CAGD, 2003,20, 19-27. T. Ju, S. Schaefer and J. Warren. Mean Value Coordinates for Closed Triangular Meshes. Siggraph2005.

24. Mean Value Interpolation

25. Summary  Research Status  Not hot  Parameterization  2 manifold  3 manifold  Application  Maybe not much desirous ?

26. Thanks for your attention!