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The Planar-Reflective Symmetry Transform

This study presents a computational representation for all planar symmetries of shapes, with a focus on local and partial symmetries alongside symmetry measurement methods. The algorithm covers computing discrete transforms and pinpointing local maxima precisely, showcasing applications in alignment, segmentation, and matching for shape analysis. Different algorithms like brute force, convolution, and Monte Carlo simulations are discussed, emphasizing the importance of weighting sample pairs and refining local maxima iteratively for accurate results. Techniques for aligning shapes based on symmetries and matching shapes in databases for optimal selection are also addressed.

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The Planar-Reflective Symmetry Transform

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  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]

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