1. INFORMATIK Differential Coordinates for Interactive Mesh Editing Yaron Lipman Olga Sorkine Daniel Cohen-Or David Levin Tel-Aviv University Christian Rössl Hans-Peter Seidel Max-Planck Institut für Informatik

2. INFORMATIK Our goal: Edit a surface while retaining its visual appearance

3. INFORMATIK Edit a surface while retaining its visual appearance Original surface The details are deformed The details shape is preserved

4. INFORMATIK Our goal Editing a surface while retaining its visual appearance –Smooth deformation –Smooth transition –Preserve relative local directions of the details –Minimal user interaction –Interactive time response T

5. INFORMATIK Differential coordinates Differential coordinates are defined for triangular mesh vertices average of the neighbors the relative coordinate vector

6. INFORMATIK Differential coordinates Differential coordinates are defined for triangular mesh vertices

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

8. INFORMATIK Related work Multi-resolution: Zorin el al.[97], Kobbelt et al.[98], Guskov et al.[99] Laplacians smoothing Taubin [SIGGRAPH95], Laplacians Morphing Alexa [TVC03] Image editing: Perez et al. [SIGGRAPH03] Mesh Editing: Zhou et al. [SIGGRAPH 04]

9. INFORMATIK Laplacian reconstruction Denote by a triangular mesh with geometry, embedded in R³. For each vertex we define the Laplacian vector: The Laplacians represents the details locally.

10. INFORMATIK Laplacian reconstruction The operator is linear and thus can be represented by the following matrix:

11. INFORMATIK Laplacian reconstruction Transforming the mesh to the differential representation: Note that where

12. INFORMATIK Laplacian reconstruction Thus for reconstructing the mesh from the Laplacian representation: add constraints to get full rank system and therefore unique solution, i.e. unique minimizer to the functional where is the index set of constrained vertices, are weights and are the spatial constraints.

13. INFORMATIK Laplacian reconstruction The use of Laplacian (differential) representation and least squares solution forces local detail preserving

14. INFORMATIK Laplacian reconstruction Laplacian reconstruction gives smooth transformation, interactive time and ease of user interface -using few spatial constraints but doesn’t preserve details orientation and shape

15. INFORMATIK Rotated Laplacian reconstruction We’d like to perform deformation which preserves the detail orientation and shape: We’d like to estimate the target shape Laplacians

16. INFORMATIK Rotated Laplacian reconstruction For each 1-ring we look for rigid affine transformations :

17. INFORMATIK Rotated Laplacian reconstruction The Laplacians are translation invariant:

18. INFORMATIK Rotated Laplacian reconstruction Laplacians are not rotational invariant (they represent detail with orientation) Note that the Laplacian operator commute with linear rotations : R

19. INFORMATIK Rotated Laplacian reconstruction Therefore we get: So all we need is to estimate the local rotations.

20. INFORMATIK Rotated Laplacian reconstruction From our assumption that detail remain with same orientation to the underlying smooth surface: The rotations are defined by the smoothed surface. We use the Laplacian reconstruction to evaluate the smoothed underlying surface normals.

21. INFORMATIK Rotated Laplacian reconstruction In summary we have the following steps: 1. Reconstruct the surface with original Laplacians: 2. Approximate local rotations 3. Rotate each Laplacian coordinate by 4. Reconstruct the edited surface:

22. INFORMATIK Some results

23. INFORMATIK Some results

24. INFORMATIK Some results

25. INFORMATIK Implementation We solve the normal equations via factorization. The factorization is done once for each ROI. And back substitution for each new handle location.

26. INFORMATIK Future work (October 2003) The main problem of the Laplacian coordinates are the need to estimate the rotation explicitly (also in Zhou et al. SIGGRAPH 2004). Instead those rotations can be computed implicitly so that the final shape is defined in one step! To be presented in SGP 2004 in Nice next month…

27. INFORMATIK Differential Coordinates for Interactive Mesh Editing