The Planar-Reflective Symmetry Transform

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 describesall planar symmetries of a shape ? Input Model Symmetry Transform

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

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

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

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

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

19. Computing Discrete Transform • Brute Force O(n6) • Convolution • Monte-Carlo O(n3) planes X = O(n6) O(n3) dot product n planes

20. Computing Discrete Transform • Brute Force O(n6) • Convolution O(n5Log n) • Monte-Carlo O(n2) normal directions X = O(n5log n) O(n3log n) per direction n planes

21. Computing Discrete Transform • Brute Force O(n6) • Convolution O(n5Log n) • Monte-Carlo O(n4) For 3D meshes • Most of the dot product contains zeros. • Use Monte-Carlo Importance Sampling.

22. Monte Carlo Algorithm Offset Angle Input Model Symmetry Transform

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

24. Monte Carlo Algorithm Offset Angle Input Model Symmetry Transform

25. Offset Angle Monte Carlo Algorithm Input Model Symmetry Transform

26. Offset Angle Monte Carlo Algorithm Input Model Symmetry Transform

27. Offset Angle Monte Carlo Algorithm Input Model Symmetry Transform

28. Offset Angle Monte Carlo Algorithm Input Model Symmetry Transform

29. Weighting Samples Need to weight sample pairs by the inverse of the distance between them P2 d P1

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

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 Approach: • Start from local maxima of discrete transform • Refine iteratively to find local maxima precisely ………. Initial Guess Final Result

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 Results: PCA Alignment Symmetry Alignment

44. Application: Matching Motivation: • Database searching = Query Best Match Database

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

46. Application: Matching Approach: • Use Symmetry transform as shape descriptor = Query Transform Database Best Match

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]