Enhancing 3D Geometry Symmetry through Transform Domain Reasoning

1. Symmetrization Niloy J. MitraLeonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007

2. Types of Symmetry • Invariance under a class of transformations

3. Symmetrization • Goal: Symmetrize 3D geometry • Approach: Minimally deform the model in the spatial domain by optimizing the distribution in transformation space

4. Contributions • Given an explicit point‐pairing, a closed form solution for symmetrizing the point set • A symmetrization algorithm that uses transform domain reasoning to guide shape deformation in object domain • Applications: • Extend the types of detected symmetries • Symmetric remeshing • Automatic correspondence for articulated bodies

5. Prior Work: Symmetry Detection • Mitra, Guibas, Pauly: Partial and Approximate Symmetry Detection for 3d Geometry. ACM Trans. Graph. 25, 3, 2006

6. Prior Work: Pair pruning • Initial pairs are sampled randomly • Pruning based on curvature and normal

7. Prior Work: Clustering • Use mean-shift algorithm • Non-Parametric Density Estimation Tessellate the space with windows The blue data points were traversed by the windows towards the mode Run the procedure in parallel

8. Prior Work: Verification • Goal : Extracting the connected components of the model from cluster • Starting with a random point of cluster • Corresponds to a pair (pi, pj) of points on the model surface • Look at the one-ring neighbors pi and apply T • Check distances of the transformed points to the surface around pj

9. 2D Example: Symmetry Detection Transformation space d

10. 2D Example: Another point‐pair votes

11. 2D Example: Voting Continues • Pairs of sample points define reflective symmetry transform

12. 2D Example: Density Plot • Density plot → accumulation of symmetry evidence

13. 2D Example: Density Peaks • Density cluster → reflective symmetry

14. 2D Example: Symmetry Detection

15. 2D Example: Symmetry Detection A set of potential corresponding point pairs extracted

16. 2D Example: Local Symmetrization

17. 2D Example: Local Symmetrization Cluster contraction Local symmetrization Cluster contraction in transform space Constrained deformation in object space

18. Recap • Object space point pairs → points in transform space • Cluster in transform space corresponds to approximate symmetry • Cluster contraction in transform space corresponds to constrained in deformation in object space that enhances object symmetry

19. 2D Example: Global Symmetrization Cluster merging → global symmetrization

20. 2D Example: Global Symmetrization Cluster merging/contraction → Global symmetrization

21. Sub‐problems • Local Symmetrization • Cluster contraction How to deform in the spatial domain ? Where to move in transform space ? • Global Symmetrization • Cluster merging

22. Optimal Displacements • Goal: Minimally displace two points to make them symmetric with respect to a given transformation [Zabrodsky et al. 1997]

23. Optimal Transformation • Goal: Find optimal transformation and minimal displacements for a set of point‐pairs

24. Optimal Transformation • Reflection • Minimize energy • Reduced to eigenvalue problem • Rigid Transform • Minimize energy • SVD problem

25. Optimization • Initial random sampling does not respect symmetries. • The correspondences estimated during the symmetry detection stage are potentially inaccurate and incomplete

26. Optimizing Sample Positions • Every sample p shifted in the direction of displacement dp (white circle) • Project them onto the surface (colored square) • The procedure is iterated until the variance of the cluster is no longer reduced.

27. Sub‐problems • Local Symmetrization • Cluster contraction Where to move in transform space ? How to deform in the spatial domain ?  Optimal transformation • Global Symmetrization • Cluster merging

28. Symmetrizing Deformation • Using existing shape deformation method • Symmetrizing displacements  positional constraints • 2D : As-rigid-as-possible shape manipulation method[Igarashi et al.2005] • 3D : Non-linear PriMo deformation model [Botsch et al. 2006] As-Rigid-As-Possible Shape Manipulation [Igarashi 2005] PriMo: Coupled Prisms for Intuitive Surface Modeling [Botsch 2006]

29. Contracting Clusters • Find sample pairs • Optimize sample positions on surface • Compute the optimaltransformation • Update pi : • pare used as deformation constraints • Re-compute the optimal transformation • Find new sample pairs every 5 time step

30. Merging Clusters • Sort clusters by height • Select the most pronounced cluster for symmetrization • Apply the symmetrizing deformation • Repeat the process with next biggest cluster • Finally, Merge clusters based on distance greedily

31. Control • User controls the deformation by modifying the stiffness of the shape’s material • Soft materials allow for better symmetrization • Stiffer materials more strongly resist the symmetrizing deformation • System allow spatially varying stiffness • User controls the symmetrization by interactively selecting clusters for contraction or merging

32. Results

33. Results

34. Results - Symmetric Meshing Symmetry Based Remeshing [Podolak al SGP 2007]

35. Results

36. Results –Computation Time

37. Results- Articulated Bodies

38. Limitations • Some case, method is fails to process the entire model • The front feet of the bunny and the right foot of the male character • Small-scale features are sometimes ignored  Insufficient local matching • The deformation model does not respect the semantics of the shape.

39. Conclusion • Symmetrizationalgorithm • Robust and efficient, requires minimal user intervention • Handle both local and global symmetries • Future Work • Symmetry respecting geometry processing • Hierarchical shape semantics • Perception, art, design • Other data, e.g. motion data, derived spaces