1. 1 Dr. Scott Schaefer Surface Parameterization

2. Parameterization and Texturing 2/30

3. 3/30 Mercator Projection

4. 4/30 Mercator Projection Image taken from http://idvux.spaces.live.com

5. Types of Distortion Isometric Mappings  Preserve lengths Conformal Mappings  Preserve angles Equiareal Mappings  Preserve area 5/30

6. Mathematics of Parameterization 6/30

7. Mathematics of Parameterization 7/30

8. Mathematics of Parameterization 8/30 singular values

9. Mathematics of Parameterization 9/30 conformal

10. Mathematics of Parameterization 10/30 equiareal

11. Mathematics of Parameterization 11/30 isometric

12. Conformal and Harmonic Mappings 12/30

13. Conformal and Harmonic Mappings 13/30

14. Conformal and Harmonic Mappings 14/30 Conformal => Harmonic

15. Harmonic Maps 15/30

16. Harmonic Maps 16/30

17. Mean Value Map 17/30

18. Mean Value Map 18/30 i th row contains mean value weights for i th vertex

19. Comparison 19/30 OriginalBarycenterHarmonicMean value

20. Conformal Maps 20/30

21. Conformal Maps 21/30

22. Conformal Maps 22/30

23. Conformal Maps 23/30

24. Conformal Maps 24/30

25. Conformal Maps 25/30

26. Conformal Maps 26/30

27. Conformal Maps 27/30

28. Conformal Maps 28/30 Sum over all triangles Linear system of equations!

29. Iterative Optimization 29/30 Stitch globally Input 3D mesh Output 2D parameterization Image taken from A Local/Global Approach to Mesh Parameterization

30. Conclusions Many, many, many methods exist for parameterization Not all guarantee 1-to-1 mappings (fold- overs) Most measure some degree of conformality Need a balance between distortions 30/30