1. INFORMATIK Laplacian Surface Editing Olga Sorkine Daniel Cohen-Or Yaron Lipman Tel Aviv University Marc Alexa TU Darmstadt Christian Rössl Hans-Peter Seidel Max-Planck Institut für Informatik

2. INFORMATIK Differential coordinates Intrinsic surface representation Allows various surface editing operations: –Detail-preserving mesh editing

3. INFORMATIK Differential coordinates Intrinsic surface representation Allows various surface editing operations: –Detail-preserving mesh editing –Coating transfer

4. INFORMATIK Differential coordinates Intrinsic surface representation Allows various surface editing operations: –Detail-preserving mesh editing –Coating transfer –Mesh transplanting

5. INFORMATIK What is it? Differential coordinates are defined by the discrete Laplacian operator: For highly irregular meshes: cotangent weights [Desbrun et al. 99] average of the neighbors

6. INFORMATIK Why differential coordinates? They represent the local detail / local shape description –The direction approximates the normal –The size approximates the mean curvature 

7. INFORMATIK Why differential coordinates? Local detail representation – enables detail preservation through various modeling tasks Representation with sparse matrices Efficient linear surface reconstruction

8. INFORMATIK Overall framework Compute differential representation Pose modeling constraints Reconstruct the surface – in least-squares sense

9. INFORMATIK Overall framework ROI is bounded by a belt (static anchors) Manipulation through handle(s)

10. INFORMATIK Related work Multi-resolution: [Zorin el al. 97], [Kobbelt et al. 98], [Guskov et al. 99], [Boier-Martin et al. 04], [Botsch and Kobbelt 04]  2 Laplacian smoothing: Taubin [SIGGRAPH 95] Laplacian Morphing: Alexa [TVC 03] Image editing: Perez et al. [SIGGRAPH 03] Mesh Editing: Yu et al. [SIGGRAPH 04]

11. INFORMATIK Problem: invariance to transformations The basic Laplacian operator is translation-invariant, but not rotation- and scale-invariant Reconstruction attempts to preserve the original global orientation of the details

12. INFORMATIK Invariance – solutions Explicit transformation of the differential coordinates prior to surface reconstruction –Lipman, Sorkine, Cohen-Or, Levin, Rössl and Seidel, “Differential Coordinates for Interactive Mesh Editing“, SMI 2004 Estimation of rotations from naive reconstruction –Yu, Zhou, Xu, Shi, Bao, Guo and Shum, “Mesh Editing With Poisson-Based Gradient Field Manipulation“, SIGGRAPH 2004 Propagation of handle transformation to the rest of the ROI

13. INFORMATIK Estimation of rotations [Lipman et al. 2004] estimate rotation of local frames –Reconstruct the surface with the original Laplacians –Estimate the normals of underlying smooth surface –Rotate the Laplacians and reconstruct again

14. INFORMATIK Explicit assignment of rotations Disadvantages: –Heuristic estimation of the rotations –Speed depends on the support of the smooth normal estimation operator; for highly detailed surfaces it must be large almost a height fieldnot a height field

15. INFORMATIK Implicit definition of transformations The idea: solve for local transformations AND the edited surface simultaneously! Transformation of the local frame

16. INFORMATIK Defining the transformations T i How to formulate T i ? –Based on the local (1-ring) neighborhood –Linear dependence on the unknown v i ’ Members of the 1-ring of i-th vertex

17. INFORMATIK Defining the transformations T i First attempt: define T i simply by solving

18. INFORMATIK Defining the transformations T i Plug the expressions for T i into the least-squares reconstruction formula: Linear combination of the unknown v i ’

19. INFORMATIK Constraining T i Trivial solution for T i will result in membrane surface reconstruction To preserve the shape of the details we constrain T i to rotations, uniform scales and translations Linear constraints on t lm so that T i is rotation+scale+translation ??

20. INFORMATIK Constraining T i – 2D case Easy in 2D:

21. INFORMATIK Constraining T i – 3D case Not linear in 3D: Linearize by dropping the quadratic term

22. INFORMATIK Adjusting T i Due to linearization, T i scale the space along the h axis by cos  When  is large, this causes anisotropy Possible correction: –Compute T i, remove the scaling component and reconstruct the surface again from the corrected  i –Apply our technique from [Lipman et al. 04] first, and then the current technique – with small .

23. INFORMATIK Some results

24. INFORMATIK Some results

25. INFORMATIK Some results

26. INFORMATIK Some results

27. INFORMATIK Some results Video...

28. INFORMATIK Detail transfer and mixing “Peel“ the coating of one surface and transfer to another

29. INFORMATIK Detail transfer and mixing Correspondence: –Parameterization onto a common domain and elastic warp to align the features, if needed

30. INFORMATIK Detail transfer and mixing Detail peeling: Smoothing by [Desbrun et al.99]

31. INFORMATIK Detail transfer and mixing Changing local frames:

32. INFORMATIK Detail transfer and mixing Reconstruction of target surface from :

33. INFORMATIK Examples

34. INFORMATIK Examples

35. INFORMATIK Mixing Laplacians Taking weighted average of  i and  ‘ i

36. INFORMATIK Mesh transplanting The user defines –Part to transplant –Where to transplant –Spatial orientation and scale Topological stitching Geometrical stitching via Laplacian mixing

37. INFORMATIK Mesh transplanting Details gradually change in the transition area

38. INFORMATIK Mesh transplanting Details gradually change in the transition area

39. INFORMATIK Conclusions Differential coordinates are useful for applications that need to preserve local details Reconstruction by linear least-squares – smoothly distributes the error across the domain Linearization of 3D rotations was needed in order to solve for optimal local transformations – can we do better?

40. INFORMATIK Acknowledgments German Israel Foundation (GIF) Israel Science Foundation (founded by the Israel Academy of Sciences and Humanities) Israeli Ministry of Science Bunny, Dragon, Feline courtesy of Stanford University Octopus courtesy of Mark Pauly

41. INFORMATIK Thank you!

42. INFORMATIK Gradual transition