1. The Planar-Reflective Symmetry Transform Princeton University

2. Motivation Symmetry is everywhere

3. Motivation Symmetry is everywhere Perfect Symmetry [Blum ’ 64, ’ 67] [Wolter ’ 85] [Minovic ’ 97] [Martinet ’ 05]

4. Motivation Symmetry is everywhere Local Symmetry [Blum ’ 78] [Mitra ’ 06] [Simari ’ 06]

5. Motivation Symmetry is everywhere Partial Symmetry [Zabrodsky ’ 95] [Kazhdan ’ 03]

6. Goal A computational representation that describes all planar symmetries of a shape ? Input Model

7. Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Input ModelSymmetry Transform

8. A computational representation that describes all planar symmetries of a shape ? Symmetry = 1.0Perfect Symmetry

9. Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Symmetry = 0.0Zero Symmetry

10. Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Symmetry = 0.3Local Symmetry

11. Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Symmetry = 0.2Partial Symmetry

12. Symmetry Measure Symmetry of a shape is measured by correlation with its reflection

13. Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0.7

14. Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0.3

15. Symmetry Measure Symmetry of a shape is measured by correlation with its reflection

16. Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0.1

17. Outline Introduction Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely Applications – Alignment – Segmentation Summary – Matching – Viewpoint Selection

18. n planes Computing Discrete Transform Brute Force Convolution Monte-Carlo

19. n planes Computing Discrete Transform Brute ForceO(n 6 ) Convolution Monte-Carlo O(n 3 ) planes X = O(n 6 ) O(n 3 ) dot product

20. n planes Computing Discrete Transform Brute ForceO(n 6 ) ConvolutionO(n 5 Log n) Monte-Carlo O(n 2 ) normal directions X = O(n 5 log n) O(n 3 log n) per direction

21. Computing Discrete Transform Brute ForceO(n 6 ) ConvolutionO(n 5 Log n) Monte-CarloO(n 4 ) For 3D meshes – Most of the dot product contains zeros. – Use Monte-Carlo Importance Sampling.

22. Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

23. Monte Carlo Algorithm Offset Angle Monte Carlo sample for single plane Input ModelSymmetry Transform

24. Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

25. Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

26. Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

27. Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

28. Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

29. Weighting Samples Need to weight sample pairs by the inverse of the distance between them P1P1 P2P2 d

30. Weighting Samples Need to weight sample pairs by the inverse of the distance between them Two planes of (equal) perfect symmetry

31. Weighting Samples Need to weight sample pairs by the inverse of the distance between them Votes for vertical plane…

32. Weighting Samples Votes for horizontal plane. Need to weight sample pairs by the inverse of the distance between them

33. Outline Introduction Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely Applications – Alignment – Segmentation Summary – Matching – Viewpoint Selection

34. Finding Local Maxima Precisely Motivation: Significant symmetries will be local maxima of the transform: the Principal Symmetries of the model Principal Symmetries

35. Finding Local Maxima Precisely Approach: Start from local maxima of discrete transform

36. Finding Local Maxima Precisely Initial GuessFinal Result ………. Approach: Start from local maxima of discrete transform Refine iteratively to find local maxima precisely

37. Outline Introduction Algorithm – Computing discrete transform – Finding Local Maxima Precisely Applications – Alignment – Segmentation Summary – Matching – Viewpoint Selection

38. Application: Alignment Motivation: Composition of range scans Feature mapping PCA Alignment

39. Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

40. Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

41. Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

42. Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

43. Application: Alignment Symmetry Alignment PCA Alignment Results:

44. Application: Matching Motivation: Database searching Database Best MatchQuery =

45. Application: Matching Observation: All chairs display similar principal symmetries

46. Application: Matching Approach: Use Symmetry transform as shape descriptor DatabaseBest MatchQuery = Transform

47. Application: Matching Results: The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors

48. Application: Matching Results: The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors

49. Application: Matching Results: The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors

50. Application: Segmentation Motivation: Modeling by parts Collision detection [Chazelle ’95][Li ’01] [Mangan ’99][Garland ’01] [Katz ’03]

51. Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

52. Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

53. Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

54. Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

55. Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

56. Application: Segmentation Approach: Cluster points on the surface by how well they support different symmetries Symmetry Vector = { 0.1, 0.5, …., 0.9 } Support = 0.1Support = 0.5Support = 0.9 …..

57. Application: Segmentation Results:

58. Application: Viewpoint Selection Motivation: Catalog generation Image Based Rendering [Blanz ’99][Vasquez ’01] [Lee ’05][Abbasi ’00] Picture from Blanz et al. ‘99

59. Application: Viewpoint Selection Approach: Symmetry represents redundancy in information.

60. Application: Viewpoint Selection Approach: Symmetry represents redundancy in information Minimize the amount of visible symmetry Every plane of symmetry votes for a viewing direction perpendicular to it Best Viewing Directions

61. Application: Viewpoint Selection Results: Viewpoint Function

62. Application: Viewpoint Selection Results: Viewpoint Function Best Viewpoint

63. Application: Viewpoint Selection Results: Viewpoint Function Best Viewpoint Worst Viewpoint

64. Application: Viewpoint Selection Results:

65. Summary Symmetry Transform – Symmetry measure for all planes in space Algorithms – Discrete set of planes – Finding local maxima precisely Applications – Alignment – Matching – Segmentation – Viewpoint Selection

66. Future Work Extended forms of symmetry Rotational symmetry Point symmetry General transform symmetry Signal processing Further applications Compression Constrained editing Etc.

67. Acknowledgements: Princeton Graphics Chris DeCoro Michael Kazhdan Funding Air Force Research Lab grant #FA8650-04-1-1718 NSF grant #CCF-0347427 NSF grant #CCR-0093343 NSF grant #IIS-0121446 The Sloan Foundation

68. The End

69. Comparison Podolak et al.Mitra et al. GoalTransformDiscrete symmetry SamplingUniform gridClustering VotingPoints only Points, normals, curvature Symmetr y Types Planar reflectionReflection/Rotation/etc. Detection Types Perfect, partial, and continuous symmetries Perfect and Partial and approximate symmetries