Laplacian Surface Editing

1. 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. Differential coordinates • Intrinsic surface representation • Allows various surface editing operations: • Detail-preserving mesh editing

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

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

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

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

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

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

10. 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 • Laplaciansmoothing: Taubin [SIGGRAPH 95] • LaplacianMorphing: Alexa [TVC 03] • Image editing: Perez et al. [SIGGRAPH 03] • Mesh Editing: Yu et al. [SIGGRAPH 04]

11. 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. 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. 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. 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 field not a height field

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

16. Defining the transformations Ti • How to formulate Ti? • Based on the local (1-ring) neighborhood • Linear dependence on the unknown vi’ Members of the 1-ring of i-th vertex

17. Defining the transformations Ti • First attempt: define Ti simply by solving

18. Defining the transformations Ti • Plug the expressions for Ti into the least-squares reconstruction formula: Linear combination of the unknown vi’

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

20. Constraining Ti – 2D case • Easy in 2D:

21. Constraining Ti – 3D case • Not linear in 3D: • Linearize by dropping the quadratic term

22. Adjusting Ti • Due to linearization, Ti scale the space along the h axis by cos • When  is large, this causes anisotropy • Possible correction: • Compute Ti , 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. Some results

24. Some results

25. Some results

26. Some results

27. Some results • Video...

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

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

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

31. Detail transfer and mixing • Changing local frames:

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

33. Examples

34. Examples

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

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

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

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

39. 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. 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. Thank you!

42. Gradual transition