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Procrustes Analysis and Its Application in Computer Graphics

Procrustes Analysis and Its Application in Computer Graphics. Speaker: Lei Zhang 2008/10/08. What is Procrustes Analysis. Procrustes [ prəu’kr Λ sti:z ]. Wikipedia 削足适履. Procrustes analysis is the name for the process of performing a shape-preserving Euclidean transformation.

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Procrustes Analysis and Its Application in Computer Graphics

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

  2. What is Procrustes Analysis Procrustes [prəu’krΛsti:z] • Wikipedia • 削足适履 Procrustes analysis is the name for the process of performing a shape-preserving Euclidean transformation. 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 isProcrustes • Greek Mythology • One who stretches • A.k.a Polypemon • A.k.a Damastes Poseidon Theseus

  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 IEEEINFOCOM 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 Transformation Rotation Orthogonal Procrustes Problem

  22. Procrustes Analysis

  23. Procrustes Analysis Sigular Value Decomposition

  24. Procrustes Analysis Sigular Value Decomposition

  25. Local Rigidity Energy

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

  27. Alternating Least-squares 1 iteration Final result Initial guess • 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 • Generalized Orthogonal Procrustes Analysis (GPA) Initial: select default mean shape Align: Translation Move centroid of each shape to origin; Normalize each shapes centroid sized; Rotate each shape to approximate the mean shape. Similarity Rotation Calculate the new mean shape Repeat

  34. GPA • Translation

  35. Algorithm • Generalized Orthogonal Procrustes Analysis (GPA) Initial: select default mean shape Align: Translation Move centroid of each shape to origin; Normalize each shapes centroid sized; Rotate each shape to approximate the mean shape. Similarity Rotation Calculate the new mean shape Repeat

  36. GPA • Similarity

  37. Algorithm • Generalized Orthogonal Procrustes Analysis (GPA) Initial: select default mean shape Align: Translation Move centroid of each shape to origin; Normalize each shapes centroid sized; Rotate each shape to approximate the mean shape. Similarity Rotation Calculate the new mean shape Repeat

  38. GPA Rotation Orthogonal Procrustes Problem • Rotation

  39. Algorithm • Generalized Orthogonal Procrustes Analysis (GPA) Initial: select default mean shape Align: Translation Move centroid of each shape to origin; Normalize each shapes centroid sized; Rotate each shape to approximate the mean shape. Similarity Rotation Calculate the new mean shape Repeat

  40. GPA • Calculate new mean shape

  41. Algorithm • Generalized Orthogonal Procrustes Analysis (GPA) Initial: select default mean shape Align: Translation Move centroid of each shape to origin; Normalize each shapes centroid sized; Rotate each shape to approximate the mean shape. Similarity Rotation Calculate the new mean shape Repeat

  42. Results Procrustes in shape analysis

  43. Mesh Parameterization

  44. Problem Setting 3D mesh 2D parameterization Keep distortion as minimal as possible

  45. Distortion Measure is singular value of is Jacobian 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 mapping Authalic mapping isometric mapping = conformal + authalic

  47. 3D mesh 2D parameterization isometric Reference triangles

  48. Procrustes Analysis Reference triangle 2D parameterization Procrustes Problem • Isometric • Conformal • Authalic

  49. Procrustes Analysis isometric conformal authalic

  50. Shape-preserving isometric transformation Rotation Orthogonal Procrustes Problem

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