1. Procrustes Analysis and Its Application in Computer Graphics Speaker: Lei Zhang 2008/10/08

2. What is Procrustes Analysis Wikipedia 削足适履 Procrustes analysis is the name for the process of performing a shape-preserving Euclidean transformation. Procrustes [ pr ə u’kr Λ sti:z ] Procrustean

3. Procrustes Problem Given

4. Procrustes Problem Given, find

5. Procrustes Problem Given, find

6. Procrustes Problem Orthogonal Procrustes Problem (OPP) Given P. H. Schoenemann. A generalized solution of the orthogonal Procrustes problem. 1966.

7. Procrustes Problem Extended Orthogonal Procrustes Problem Given P. H. Schoenemann, R. Carroll. Fitting one matrix to another under choice of a central dilation and a rigid motion. 1970.

8. Procrustes Problem Rotation Orthogonal Procrustes Problem Given G. Wahba. A least squares estimate of satellite attitude. 1966.

9. Procrustes Problem Permutation Procrustes Problem (PPP) Given J. C. Gower. Multivariate analysis: ordination, multidimensional scaling and allied topics. 1984.

10. Procrustes Problem Symmetric Procrustes Problem (SPP) Given H. J. Larson. Least squares estimation of the components of a symmetric matrix. 1966.

11. Who is Procrustes Greek Mythology –One who stretches –A.k.a Polypemon –A.k.a Damastes Theseus Poseidon

12. Peter H. Schonemann Professor At Department of Psychological Science, Purdue University P. H. Schoenemann. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 1966. J. C. Gower, G. B. Dijksterhuis. Procrustes problems. Oxford University Press, 2004.

13. Applications Factor analysis, statistic Satellite tracking Rigid body movement in robotics Structural and system identification Computer graphics Sensor Networks

14. Reference  Olga Sorkine, Marc Alexa. As-rigid-as-possible surface modeling. SGP 2007.  M. B. Stegmann, D. D. Gomez. A brief introduction to statistical shape analysis. Lecture notes. Denmark Technical University.  Ligang Liu, Lei Zhang, Yin Xu, Craig Gotsman, Steven J. Gorlter. A local/global approach to mesh parameterization. SGP 2008.  Lei Zhang, Ligang Liu, Guojin Wang. Meshless parameterization by rigid alignment and surface reconstruction. 2008  Lei Zhang, Ligang Liu, Craig Gotsman, Steven J. Gorlter. An as- rigid-as-possible approach to sensor networks localization. Submitted to IEEE INFOCOM 2009.

15. Shape Deformation

16. Good Shape Deformation Smooth effect on the large scale approximation Preserve detail on the local structure

17. Direct Local Structure Small-sized Cells –Smooth surface

18. Direct Local Structure Small-sized Cells –Discrete surface

19. Direct Detail Preserve Shape-preserving transformation

20. Rotation Transformation

21. Rotation Orthogonal Procrustes Problem

22. Procrustes Analysis

23. S igular V alue D ecomposition

24. Procrustes Analysis S igular V alue D ecomposition

25. Local Rigidity Energy

26. b is known, calculate R by Procrustes analysis R is known, calculate b by least-squares optimization (Laplace equation)

27. Alternating Least-squares Initial guess 1 iterationFinal result b is known, calculate R by Procrustes analysis R is known, calculate b by least-squares optimization (Laplace equation)

28. Results Procrustes in shape deformation

29. Shape Registration

30. What is Shape Shape is all the geometrical information that remains when location, scale and rotational effects are filtered out from an object. --I. L. Dryden and K. V. Mardia. Statistical Shape Analysis. 1998

31. Shape Representation Landmarks

32. Shape Registration Euclidean transformation Translation Similarity Rotation Landmark correspondence

33. Algorithm G eneralized Orthogonal P rocrustes A nalysis (GPA) a)Move centroid of each shape to origin; b)Normalize each shapes centroid sized; c)Rotate each shape to approximate the mean shape. Translation Similarity Rotation Initial: select default mean shape Align: Calculate the new mean shape Repeat

34. GPA Translation

35. Algorithm G eneralized Orthogonal P rocrustes A nalysis (GPA) a)Move centroid of each shape to origin; b)Normalize each shapes centroid sized; c)Rotate each shape to approximate the mean shape. Translation Similarity Rotation Initial: select default mean shape Align: Calculate the new mean shape Repeat

36. GPA Similarity

37. Algorithm G eneralized Orthogonal P rocrustes A nalysis (GPA) a)Move centroid of each shape to origin; b)Normalize each shapes centroid sized; c)Rotate each shape to approximate the mean shape. Translation Similarity Rotation Initial: select default mean shape Align: Calculate the new mean shape Repeat

38. GPA Rotation Rotation Orthogonal Procrustes Problem

39. Algorithm G eneralized Orthogonal P rocrustes A nalysis (GPA) a)Move centroid of each shape to origin; b)Normalize each shapes centroid sized; c)Rotate each shape to approximate the mean shape. Translation Similarity Rotation Initial: select default mean shape Align: Calculate the new mean shape Repeat

40. GPA Calculate new mean shape

41. Algorithm G eneralized Orthogonal P rocrustes A nalysis (GPA) a)Move centroid of each shape to origin; b)Normalize each shapes centroid sized; c)Rotate each shape to approximate the mean shape. Translation Similarity Rotation Initial: select default mean shape Align: Calculate the new mean shape Repeat

42. Results Procrustes in shape analysis

43. Mesh Parameterization

44. Problem Setting 3D mesh2D parameterization Keep distortion as minimal as possible

45. Distortion Measure is Jacobian of, is singular value of 1. Angle-preserving (i.e. conformal mapping) 2. Area-preserving (i.e. authalic mapping) 3. Shape-preserving (i.e. isometric mapping) Floater, M. S. and Hormann, K. Surface parameterization: a tutorial and survey. 2004

46. Distortion Measure Conformal mappingAuthalic mapping isometric mapping = conformal + authalic

47. 3D mesh2D parameterization Reference triangles isometric

48. Procrustes Analysis Reference triangle2D parameterization Procrustes Problem  Isometric  Conformal  Authalic

49. Procrustes Analysis isometricconformalauthalic

50. Shape-preserving isometric transformation Rotation Orthogonal Procrustes Problem

51. Angle-preserving Similarity Procrustes Problem conformal transformation

52. Area-preserving Procrustes Problem Authalic transformation

53. Parameterization  Shape : as-rigid-as-possible parameterization (ARAP)  Angle: as-similar-as-possible parameterization (ASAP)  Area: as-authalic-as-possible parameterization (AAAP) Alternating least - squares ( ALS )

54. Model A R APA S APA A AP

55. ASAP vs. ARAP A S AP A R AP

56. Insight ASAP ARAP *Equivalent to LSCM: Levy, B., et al. Least squares conformal maps for atutomatic texture atlas generation. Siggraph 2002.

57. Comparison [HG99] MIPS: an efficient global parameterization method. In Proc. Of Curves and Surfaces. [DMK03] An adaptable surface parameterization method. In Proc. Of 12 th International Meshing Roundtable.

58. ASAP: 2.00 88.14 ARAP: 2.06 2.05 ABF: 2.00 2.64 IC: 2.05 2.67CP: 2.00 2.64  ABF: Sheffa, et al, TOG, 2005  IC: Gu, et al, TVCG, 2008  CP: Gotsman, et al, EG 2008

59. ASAP: 2.05 15.4 ARAP: 2.19 2.11 ABF: 2.12 9.12 IC: 3.09 3.91CP: 2.29 11.9  ABF: Sheffa, et al, TOG, 2005  IC: Gu, et al, TVCG, 2008  CP: Gotsman, et al, EG 2008

60. ABF: 2.00 2.09 ARAP: 2.01 2.01 Procrustes in parameterization

61. Surface Reconstruction

62. Problem Setting Points SetReconstruction

63. Meshless Parameterization Points Set Reconstruction Parameterization Delaunay triangulation

64. Local Tangent Flattening

65. Rigid Alignment F o r e a c h p o i n t Rotation Orthogonal Procrustes Problem

66. Parameterization Alternating Least Squares B is known, calculate R by Procrustes analysis R is known, calculate B by least-squares optimization (Laplace equation)

67. Initialization Affine Alignment Linear least-squares w.r.t A and a, b, c, d

68. Affine Alignment Points Set Affine alignment

69. Rigid alignment Affine alignment

70. Delaunay Triangulation Remove redundant triangle

71. Results Floater, et al, CAGD, 2001Roweis, et al, Science, 2001Our approach

72. Texture Mapping Floater, et al, CAGD, 2001Roweis, et al, Science, 2001Our approach

73. Floater, et al, CAGD, 2001Roweis, et al, Science, 2001Our approach

74. Texture Mapping Floater, et al, CAGD, 2001Roweis, et al, Science, 2001Our approach Procrustes in surface reconstruction

75. Summary Procrustes Analysis –Euclidean transformation –Direct estimate of shape transformation –Versatile Shape deformation Shape analysis Mesh parameterization Surface reconstruction ……

76. Thanks for your attention!

77. Q&A