1. Partial and Approximate Symmetry Detection for 3D Geometry Mark Pauly Niloy J. Mitra Leonidas J. Guibas

2. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Symmetry in Nature “Symmetry is a complexity-reducing concept [...]; seek it everywhere.” - Alan J. Perlis "Females of several species, including […] humans, prefer symmetrical males." - Chris Evan

3. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Symmetry for Geometry Processing [Funkhouser et al. `05] [Sharf et al. `04] [Katz and Tal `04] [Khazdan et al. `04]

4. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Partial Symmetry Detection Given Shape model (represented as point cloud, mesh,... ) Identify and extract similar (symmetric) patches of different size across different resolutions Goal

5. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Related Work [Podolak et al. `06][Loy and Eklundh `06] Hough transform on feature points [Gal and Cohen-Or `05] tradeoff memory for speed

6. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Types of Symmetry Transform Types: Reflection Rotation + Translation Uniform Scaling

7. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Contributions Automatic detection of discrete symmetries ! reflection, rigid transform, uniform scaling Symmetry graphs ! high level structural information about object Output sensitive algorithms ! low memory requirements

8. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Problem Characteristics Difficulties Which parts are symmetric ! objects not pre-segmented Space of transforms: rotation + translation Brute force search is not feasible Easy Proposed symmetries ! easy to validate

9. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Reflective Symmetry

10. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Reflective Symmetry: A Pair Votes

11. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Reflective Symmetry: Voting Continues

12. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Reflective Symmetry: Voting Continues

13. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Reflective Symmetry: Largest Cluster Height of cluster ! size of patch Spread of cluster ! level of approximation

14. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Pipeline

15. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Pipeline

16. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Pruning: Local Signatures Local signature ! invariant under transforms Signatures disagree ! points don’t correspond Use (  1,  2 ) for curvature based pruning

17. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Reflection: Normal-based Pruning

18. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Point Pair Pruning all pairs curvature based curvature + normal based

19. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Transformations Reflection ! point-pairs Rigid transform ! more information Robust estimation of principal curvature frames [Cohen-Steiner et al. `03]

20. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Mean-Shift Clustering Kernel: Radially symmetric Radius/spread

21. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Verification Clustering gives a good guess Verify ! build symmetric patches Locally refine solution using ICP algorithm [Besl and McKay `92]

22. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Random Sampling Height of clusters related to symmetric region size Random samples ! larger regions likely to be detected earlier Output sensitive

23. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Model Reduction: Chambord

24. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Model Reduction: Chambord

25. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Model Reduction: Chambord

26. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Sydney Opera House

27. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Sydney Opera House

28. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Approximate Symmetry: Dragon correction field UNITS: fraction of bounding box diagonal detected symmetries

29. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Limitations Cannot differentiate between small sized symmetries and comparable noise [Castro et al. `06]

30. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Articulated Motion: Horses ‘symmetry’ detection between two objects ! registration

31. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra More details in the paper Symmetry graph reduction Analysis of sampling requirements

32. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Future Work Detect biased deformation Pose independent shape matching Application to higher dimensional data

33. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Acknowledgements DARPA, NSF, CARGO, ITR, and NIH grants Stanford Graduate Fellowship Pierre Alliez Mario Botsch Doo Young Kwon Marc Levoy Ren Ng Bob Sumner Dilys Thomas anonymous reviewers

34. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Thank you! Niloy J. Mitraniloy@stanford.edu Leonidas J. Guibasguibas@cs.stanford.edu Mark Paulypauly@inf.ethz.ch

35. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Performance model# verticessign.pairingcluster.verif. Dragon 160,9473.4449.2413.637.45 Opera 9,3760.960.020.030.86 Castle 172,6065.61117.81159.735.63 Horse 8,4310.920.01 1.63 Arch 16,9210.085.8626.892.42 (time in seconds)

36. Partial and Approximation Symmetry Detection for 3D GeometryNiloy J. Mitra Comparison Podolak et al.Mitra et al. GoalTransformDiscrete symmetry SamplingUniform gridClustering VotingPoints onlyPoints, normals, curvature Symmetry types Planar reflectionreflection, rotation, trans., unif. scaling Detection types Perfect, partial, continuous Perfect, partial, approximate