1. Niloy J. Mitra Leonidas J. Guibas Mark Pauly TU Vienna Stanford University ETH Zurich SIGGRAPH 2007

2.  Invariance under a class of transformations 2

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

4.  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 4

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

6.  Initial pairs are sampled randomly  Pruning based on curvature and normal 6

7.  Use mean-shift algorithm ◦ Non-Parametric Density Estimation 7 The blue data points were traversed by the windows towards the mode

8.  Goal : Extracting the connected components of the model from cluster  Starting with a random point of cluster ◦ Corresponds to a pair (p i, p j ) of points on the model surface  Look at the one-ring neighbors p i and apply T  Check distances of the transformed points to the surface around p j 8

9. Transformation space d 9

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11.  Pairs of sample points define reflective symmetry transform 11

12.  Density plot → accumulation of symmetry evidence 12

13.  Density cluster → reflective symmetry 13

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15. A set of potential corresponding point pairs extracted 15

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17. Cluster contraction Local symmetrization Cluster contraction in transform space Constrained deformation in object space 17

18.  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 18

19. Cluster merging → global symmetrization 19

20. Cluster merging/contraction → Global symmetrization 20

21.  Local Symmetrization ◦ Cluster contraction How to deform in the spatial domain ? Where to move in transform space ?  Global Symmetrization ◦ Cluster merging 21

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

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

24.  Reflection ◦ Minimize energy ◦ Reduced to eigenvalue problem  Rigid Transform ◦ Minimize energy ◦ SVD problem 24

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

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

27.  Local Symmetrization ◦ Cluster contraction Where to move in transform space ? How to deform in the spatial domain ?  Optimal transformation  Global Symmetrization ◦ Cluster merging 27

28.  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] 28 As-Rigid-As-Possible Shape Manipulation [Igarashi 2005] PriMo: Coupled Prisms for Intuitive Surface Modeling [Botsch 2006]

29.  Find sample pairs  Optimize sample positions on surface  Compute the optimal transformation  Update p i :  p are used as deformation constraints  Re-compute the optimal transformation  Find new sample pairs every 5 time step 29

30.  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 30

31.  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 31

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34. 34 Symmetry Based Remeshing [Podolak al SGP 2007]

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38.  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. 38

39.  Symmetrization algorithm ◦ 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 39