Andrew Nealen TU Berlin Takeo Igarashi The University of Tokyo / PRESTO JST Olga Sorkine

1. Laplacian Mesh Optimization Andrew NealenTU Berlin Takeo IgarashiThe University of Tokyo / PRESTO JST Olga Sorkine Marc Alexa TU Berlin

2. What is it ?

3. Overview • Motivation • Problem formulation • Laplacian mesh processing basics • Laplacian mesh optimization framework • Applications • Triangle shape optimization • Mesh smoothing • Discussion

4. Motivation • Local detail preserving triangle optimization • A Sketch-Based Interface for Detail Preserving Mesh Editing [Nealen et al. 2005]

5. = L x d Motivation • Local detail preserving triangle optimization • A Sketch-Based Interface for Detail Preserving Mesh Editing [Nealen et al. 2005] • Can we perform globaloptimization this way ?

6. L x d Laplacian Mesh Processing • Discrete Laplacians n = duniform: wij = 1 dcotangent: wij = cot aij + cot bij

7. dx dy dz L L L x y x z d Laplacian Mesh Processing • Surface reconstruction n = duniform: wij = 1 dcotangent: wij = cot aij + cot bij

8. dx dy dz 1 1 c2 c1 1 1 1 1 L L L x z y Laplacian Mesh Processing • Surface reconstruction n = fix edit

9. dx dy dz 1 1 c2 c1 1 1 1 1 L L L z y x ATA x AT b = AT b x (ATA)-1 = A x b = Laplacian Mesh Processing • Least-squares solution n wLi wLi = fix w1 w1 edit Normal Equations w2 w2

10. dx dy dz 1 c1 1 1 L L L L L L x z y Laplacian Mesh Processing • Tangential smoothing n = fix

11. dx dy dz 1 c1 1 1 L L L y z x Laplacian Mesh Processing • Tangential smoothing n = fix

12. dx dy dz 1 c1 1 1 L L L y z x Laplacian Mesh Processing • Tangential smoothing n = fix

13. = L x d More motivation… • So: can we use such a system for globaloptimization ?

14. Our Solution • All vertices are (weighted) anchors • Preserves global shape • Uses existing LS framework • Anchor + Laplacian weights determine result

15. WL WL WP WP p L x f Framework • Detail preserving tri shape optimization for L = Luni and f = dcot(similar to local optimization) • Mesh smoothingL = Lcot (outer fairness) or L = Luni (outer and inner fairness) and f = 0 =

16. WP WP p x d Tri Shape Optimization • Detail preserving tri shape optimization Luni =

17. Positional Weights

18. Constant Weights

19. Linear Weights

20. CDF Weights

21. CDF Weights

22. Sharp Features

23. Sharp Features

24. Sharp Features

25. WL WL WP WP p L x 0 Mesh Smoothing • Mesh smoothing L = Lcot (outer fairness) or L = Lumb (outer and inner fairness) and f = 0 • Controlled by WP and WL (Intensity, Features) • Similar to Least-Squares Meshes [Sorkine et al. 04] =

26. Using WP

27. Using WP and WL

28. Results

29. Noisy

30. Smoothed

31. Original

32. Tri Shape Optimization

33. Smoothing Outer and Inner Fairness (Lumb)

34. Original

35. Tri Shape Optimization

36. SmoothingOuter Fairness only (Lcot)

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. 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