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

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

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

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

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

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

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

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

Peter H. Schonemann Professor At Department of Psychological Science, Purdue University P. H. Schoenemann. A generalized solution of the orthogonal Procrustes problem. Psychometrika, 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  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  Lei Zhang, Ligang Liu, Guojin Wang. Meshless parameterization by rigid alignment and surface reconstruction  Lei Zhang, Ligang Liu, Craig Gotsman, Steven J. Gorlter. An as- rigid-as-possible approach to sensor networks localization. Submitted to IEEE INFOCOM 2009.

Shape Deformation

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

S igular V alue D ecomposition

Procrustes Analysis S igular V alue D ecomposition

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

Results Procrustes in shape deformation

Shape Registration

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 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

GPA Translation

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

GPA Similarity

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

GPA Rotation Rotation Orthogonal Procrustes Problem

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

GPA Calculate new mean shape

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

Results Procrustes in shape analysis

Mesh Parameterization

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

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

Distortion Measure Conformal mappingAuthalic mapping isometric mapping = conformal + authalic

3D mesh2D parameterization Reference triangles isometric

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

Procrustes Analysis isometricconformalauthalic

Shape-preserving isometric transformation Rotation Orthogonal Procrustes Problem

Angle-preserving Similarity Procrustes Problem conformal transformation

Area-preserving Procrustes Problem Authalic transformation

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 )

Model A R APA S APA A AP

ASAP vs. ARAP A S AP A R AP

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

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.

ASAP: ARAP: ABF: IC: CP:  ABF: Sheffa, et al, TOG, 2005  IC: Gu, et al, TVCG, 2008  CP: Gotsman, et al, EG 2008

ASAP: ARAP: ABF: IC: CP:  ABF: Sheffa, et al, TOG, 2005  IC: Gu, et al, TVCG, 2008  CP: Gotsman, et al, EG 2008

ABF: ARAP: Procrustes in parameterization

Surface Reconstruction

Problem Setting Points SetReconstruction

Meshless Parameterization Points Set Reconstruction Parameterization Delaunay triangulation

Local Tangent Flattening

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

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

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

Affine Alignment Points Set Affine alignment

Rigid alignment Affine alignment

Delaunay Triangulation Remove redundant triangle

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

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

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

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

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

Thanks for your attention!

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