1. Andrew Nealen, TU Berlin, 2006 1 CG 11 Andrew Nealen TU Berlin Takeo Igarashi The University of Tokyo / PRESTO JST Olga Sorkine Marc Alexa TU Berlin Laplacian Mesh Optimization

2. Andrew Nealen, TU Berlin, 2006 2 CG 22 What is it ?

3. Andrew Nealen, TU Berlin, 2006 3 CG 33 Overview  Motivation Problem formulation Laplacian mesh processing basics  Laplacian mesh optimization framework  Applications Triangle shape optimization Mesh smoothing  Discussion

4. Andrew Nealen, TU Berlin, 2006 4 CG 44 Motivation  Local detail preserving triangle optimization A Sketch-Based Interface for Detail Preserving Mesh Editing [Nealen et al. 2005]

5. Andrew Nealen, TU Berlin, 2006 5 CG 55 Motivation  Local detail preserving triangle optimization A Sketch-Based Interface for Detail Preserving Mesh Editing [Nealen et al. 2005]  Can we perform global optimization this way ? = L x 

6. Andrew Nealen, TU Berlin, 2006 6 CG 66 Laplacian Mesh Processing  Discrete Laplacians = Lx  n  cotangent : w ij = cot  ij + cot  ij  uniform : w ij = 1

7. Andrew Nealen, TU Berlin, 2006 7 CG 77 Laplacian Mesh Processing  Surface reconstruction n  cotangent : w ij = cot  ij + cot  ij  uniform : w ij = 1 = Lx  LL y z x zz yy xx

8. Andrew Nealen, TU Berlin, 2006 8 CG 88 Laplacian Mesh Processing  Surface reconstruction n zz yy xx y z x = L L Lc1c1    fix edit c2c2   

9. Andrew Nealen, TU Berlin, 2006 9 CG 99 Laplacian Mesh Processing  Least-squares solution n zz yy xx y z x = L L Lc1c1    fix edit c2c2    w1w1 w1w1 w2w2 w2w2 w Li Ax = b ATAATAx = bATAT (A T A) -1 x = bATAT Normal Equations

10. Andrew Nealen, TU Berlin, 2006 10 CG 10 Laplacian Mesh Processing  Tangential smoothing n zz yy xx y z x = L L L fix c1c1    L L L

11. Andrew Nealen, TU Berlin, 2006 11 CG 11 L L L Laplacian Mesh Processing  Tangential smoothing n zz yy xx y z x = fix c1c1   

12. Andrew Nealen, TU Berlin, 2006 12 CG 12 L L L Laplacian Mesh Processing  Tangential smoothing n zz yy xx y z x = fix c1c1   

13. Andrew Nealen, TU Berlin, 2006 13 CG 13 More motivation…  So: can we use such a system for global optimization ? =Lx 

14. Andrew Nealen, TU Berlin, 2006 14 CG 14 Our Solution  All vertices are (weighted) anchors  Preserves global shape  Uses existing LS framework  Anchor + Laplacian weights determine result

15. Andrew Nealen, TU Berlin, 2006 15 CG 15 Framework  Detail preserving tri shape optimization for L = L uni and f =  cot  (similar to local optimization)  Mesh smoothing L = L cot (outer fairness) or L = L uni (outer and inner fairness) and f = 0 = Lxf WLWL WLWL p WPWP WPWP

16. Andrew Nealen, TU Berlin, 2006 16 CG 16 Tri Shape Optimization  Detail preserving tri shape optimization = L uni x  p WPWP WPWP

17. Andrew Nealen, TU Berlin, 2006 17 CG 17 Positional Weights

18. Andrew Nealen, TU Berlin, 2006 18 CG 18 Constant Weights

19. Andrew Nealen, TU Berlin, 2006 19 CG 19 Linear Weights

20. Andrew Nealen, TU Berlin, 2006 20 CG 20 CDF Weights

21. Andrew Nealen, TU Berlin, 2006 21 CG 21 CDF Weights

22. Andrew Nealen, TU Berlin, 2006 22 CG 22 Sharp Features

23. Andrew Nealen, TU Berlin, 2006 23 CG 23 Sharp Features

24. Andrew Nealen, TU Berlin, 2006 24 CG 24 Sharp Features

25. Andrew Nealen, TU Berlin, 2006 25 CG 25 Mesh Smoothing  Mesh smoothing L = L cot (outer fairness) or L = L umb (outer and inner fairness) and f = 0  Controlled by W P and W L (Intensity, Features)  Similar to Least-Squares Meshes [Sorkine et al. 04] = Lx0 WLWL WLWL p WPWP WPWP

26. Andrew Nealen, TU Berlin, 2006 26 CG 26 Using W P

27. Andrew Nealen, TU Berlin, 2006 27 CG 27 Using W P and W L

28. Andrew Nealen, TU Berlin, 2006 28 CG 28 Results

29. Andrew Nealen, TU Berlin, 2006 29 CG 29 Noisy

30. Andrew Nealen, TU Berlin, 2006 30 CG 30 Smoothed

31. Andrew Nealen, TU Berlin, 2006 31 CG 31 Original

32. Andrew Nealen, TU Berlin, 2006 32 CG 32 Tri Shape Optimization

33. Andrew Nealen, TU Berlin, 2006 33 CG 33 Smoothing Outer and Inner Fairness (L umb )

34. Andrew Nealen, TU Berlin, 2006 34 CG 34 Original

35. Andrew Nealen, TU Berlin, 2006 35 CG 35 Tri Shape Optimization

36. Andrew Nealen, TU Berlin, 2006 36 CG 36 Smoothing Outer Fairness only (L cot )

37. Andrew Nealen, TU Berlin, 2006 37 CG 37 Discussion  The good... Easily controllable tri shape optimization and smoothing Leverages existing least squares framework Can replace tangential smoothing step for general remeshers ... and the not so good Euclidean distance is not Hausdorff distance, so error control is indirect Does rely on some (user) parameter tweaking

38. Andrew Nealen, TU Berlin, 2006 38 CG 38 Thank you !  Contact info Andrew Nealen nealen@cs.tu-berlin.de Takeo Igarashi takeo@acm.org Olga Sorkine sorkine@cs.tu-berlin.de Marc Alexa marc@cs.tu-berlin.de