Procrustes Analysis and Its Application in Computer Graphics

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Procrustes Analysis and Its Application in
Computer Graphics

• Speaker Lei Zhang
• 2008/10/08

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What is Procrustes Analysis
Procrustes pr?ukr?stiz

• Wikipedia
• ????

Procrustes analysis is the name for the process
of performing a shape-preserving Euclidean
transformation.
Procrustean
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Procrustes Problem
Given
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Procrustes Problem
Given
, find
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Procrustes Problem
Given
, find
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Procrustes Problem

• Orthogonal Procrustes Problem (OPP)

Given
P. H. Schoenemann. A generalized solution of the
orthogonal Procrustes problem. 1966.
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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.
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Procrustes Problem

• Rotation Orthogonal Procrustes Problem

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

• Permutation Procrustes Problem (PPP)

Given
J. C. Gower. Multivariate analysis ordination,
multidimensional scaling and allied topics. 1984.
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Procrustes Problem

• Symmetric Procrustes Problem (SPP)

Given
H. J. Larson. Least squares estimation of the
components of a symmetric matrix. 1966.
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Who is Procrustes

• Greek Mythology
• One who stretches
• A.k.a Polypemon
• A.k.a Damastes

Poseidon
Theseus
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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.
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Applications

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

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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.

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Shape Deformation
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Good Shape Deformation

• Smooth effect on the large scale approximation
• Preserve detail on the local structure

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Direct Local Structure

• Small-sized Cells
• Smooth surface

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Direct Local Structure

• Small-sized Cells
• Discrete surface

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Direct Detail Preserve
Shape-preserving
transformation
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Rotation Transformation
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Rotation Transformation
Rotation Orthogonal Procrustes Problem
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Procrustes Analysis
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Procrustes Analysis
Sigular Value Decomposition
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Procrustes Analysis
Sigular Value Decomposition
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Local Rigidity Energy
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Local Rigidity Energy

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

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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)

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Results
Procrustes in shape deformation
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Shape Registration
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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
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Shape Representation

• Landmarks

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Shape Registration

• Euclidean transformation
• Translation
• Similarity
• Rotation

Landmark correspondence
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Algorithm

• Generalized Orthogonal Procrustes Analysis (GPA)

Initial select default mean shape
Align
Translation

1. Move centroid of each shape to origin
2. Normalize each shapes centroid sized
3. Rotate each shape to approximate the mean shape.

Similarity
Rotation
Calculate the new mean shape
Repeat
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GPA

• Translation

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Algorithm

• Generalized Orthogonal Procrustes Analysis (GPA)

Initial select default mean shape
Align
Translation

1. Move centroid of each shape to origin
2. Normalize each shapes centroid sized
3. Rotate each shape to approximate the mean shape.

Similarity
Rotation
Calculate the new mean shape
Repeat
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GPA

• Similarity

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Algorithm

• Generalized Orthogonal Procrustes Analysis (GPA)

Initial select default mean shape
Align
Translation

1. Move centroid of each shape to origin
2. Normalize each shapes centroid sized
3. Rotate each shape to approximate the mean shape.

Similarity
Rotation
Calculate the new mean shape
Repeat
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GPA
Rotation Orthogonal Procrustes Problem

• Rotation

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Algorithm

• Generalized Orthogonal Procrustes Analysis (GPA)

Initial select default mean shape
Align
Translation

1. Move centroid of each shape to origin
2. Normalize each shapes centroid sized
3. Rotate each shape to approximate the mean shape.

Similarity
Rotation
Calculate the new mean shape
Repeat
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GPA

• Calculate new mean shape

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Algorithm

• Generalized Orthogonal Procrustes Analysis (GPA)

Initial select default mean shape
Align
Translation

1. Move centroid of each shape to origin
2. Normalize each shapes centroid sized
3. Rotate each shape to approximate the mean shape.

Similarity
Rotation
Calculate the new mean shape
Repeat
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Results
Procrustes in shape analysis
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Mesh Parameterization
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Problem Setting
3D mesh
2D parameterization
Keep distortion as minimal as possible
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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
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Distortion Measure
Conformal mapping
Authalic mapping
isometric mapping conformal authalic
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3D mesh
2D parameterization
isometric
Reference triangles
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Procrustes Analysis
Reference triangle
2D parameterization
Procrustes Problem

• Isometric
• Conformal
• Authalic

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Procrustes Analysis
isometric
conformal
authalic
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Shape-preserving
isometric transformation
Rotation Orthogonal Procrustes Problem
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Angle-preserving
conformal transformation
Similarity Procrustes Problem
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Area-preserving
Authalic transformation
Procrustes Problem
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Parameterization
Alternating least-squares (ALS)

• Shape as-rigid-as-possible parameterization
  (ARAP)
• Angle as-similar-as-possible parameterization
  (ASAP)
• Area as-authalic-as-possible parameterization
  (AAAP)

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ARAP
ASAP
AAAP
Model
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ASAP vs. ARAP
ASAP
ARAP
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Insight

• ASAP
• ARAP

Equivalent to LSCM Levy, B., et al. Least
squares conformal maps for atutomatic texture
atlas generation. Siggraph 2002.
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Comparison

• HG99 MIPS an efficient global parameterization
  method. In Proc. Of Curves and Surfaces.
• DMK03 An adaptable surface parameterization
  method. In Proc. Of 12th International Meshing
  Roundtable.

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ARAP 2.06 2.05
ASAP 2.00 88.14
ABF 2.00 2.64

• ABF Sheffa, et al, TOG, 2005
• IC Gu, et al, TVCG, 2008
• CP Gotsman, et al, EG 2008

IC 2.05 2.67
CP 2.00 2.64
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ARAP 2.19 2.11
ASAP 2.05 15.4
ABF 2.12 9.12

• ABF Sheffa, et al, TOG, 2005
• IC Gu, et al, TVCG, 2008
• CP Gotsman, et al, EG 2008

IC 3.09 3.91
CP 2.29 11.9
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ARAP 2.01 2.01
ABF 2.00 2.09
Procrustes in parameterization
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Surface Reconstruction
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Problem Setting
Points Set
Reconstruction
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Meshless Parameterization
Points Set
Parameterization
Reconstruction
Delaunay triangulation
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Local Tangent Flattening
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Rigid Alignment

• For each point

Rotation Orthogonal Procrustes Problem
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Parameterization

• Alternating Least Squares

• B is known, calculate R by Procrustes analysis
• R is known, calculate B by least-squares
  optimization (Laplace equation)

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Initialization

• Affine Alignment

Linear least-squares w.r.t A and a, b, c, d
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Affine Alignment
Points Set
Affine alignment
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Affine alignment
Rigid alignment
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Delaunay Triangulation
Remove redundant triangle
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Results
Floater, et al, CAGD, 2001
Roweis, et al, Science, 2001
Our approach
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Texture Mapping
Floater, et al, CAGD, 2001
Roweis, et al, Science, 2001
Our approach
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Floater, et al, CAGD, 2001
Roweis, et al, Science, 2001
Our approach
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Texture Mapping
Floater, et al, CAGD, 2001
Roweis, et al, Science, 2001
Our approach
Procrustes in surface reconstruction
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Summary

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

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Thanks for your attention!
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QA