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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 Speaker: Lei Zhang 2008/10/08
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
Procrustes Problem Given
Procrustes Problem Given , find
Procrustes Problem Given , find
Procrustes Problem • Orthogonal Procrustes Problem (OPP) Given P. H. Schoenemann. A generalized solution of the orthogonal Procrustes problem. 1966.
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.
Procrustes Problem • Rotation Orthogonal Procrustes Problem Given G. Wahba. A least squares estimate of satellite attitude. 1966.
Procrustes Problem • Permutation Procrustes Problem (PPP) Given J. C. Gower. Multivariate analysis: ordination, multidimensional scaling and allied topics. 1984.
Procrustes Problem • Symmetric Procrustes Problem (SPP) Given H. J. Larson. Least squares estimation of the components of a symmetric matrix. 1966.
Who isProcrustes • Greek Mythology • One who stretches • A.k.a Polypemon • A.k.a Damastes Poseidon Theseus
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.
Applications • Factor analysis, statistic • Satellite tracking • Rigid body movement in robotics • Structural and system identification • Computer graphics • Sensor Networks
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.
Good Shape Deformation • Smooth effect on the large scale approximation • Preserve detail on the local structure
Direct Local Structure • Small-sized Cells • Smooth surface
Direct Local Structure • Small-sized Cells • Discrete surface
Direct Detail Preserve Shape-preserving transformation
Rotation Transformation Rotation Orthogonal Procrustes Problem
Procrustes Analysis Sigular Value Decomposition
Procrustes Analysis Sigular Value Decomposition
Local Rigidity Energy • b is known, calculate R by Procrustes analysis • R is known, calculate b by least-squares optimization (Laplace equation)
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)
Results Procrustes in shape deformation
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
Shape Representation • Landmarks
Shape Registration • Euclidean transformation • Translation • Similarity • Rotation Landmark correspondence
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
GPA • Translation
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
GPA • Similarity
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
GPA Rotation Orthogonal Procrustes Problem • Rotation
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
GPA • Calculate new mean shape
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
Results Procrustes in shape analysis
Problem Setting 3D mesh 2D parameterization Keep distortion as minimal as possible
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
Distortion Measure Conformal mapping Authalic mapping isometric mapping = conformal + authalic
3D mesh 2D parameterization isometric Reference triangles
Procrustes Analysis Reference triangle 2D parameterization Procrustes Problem • Isometric • Conformal • Authalic
Procrustes Analysis isometric conformal authalic
Shape-preserving isometric transformation Rotation Orthogonal Procrustes Problem