EG 4.2 SymmetrySymmetrization - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011... · 2011. 4. 19. · Eurographics$2011$Course$$–$CompungCorrespondencesinGeometricDataSets

Compu&ng Correspondences in Geometric Datasets

4.2 Symmetry & Symmetriza/on

Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets

• Invariance under a class of transformations

– global vs. partial– exact vs. approximative

Symmetry

Reflection Translation Rotation Reflection + Translation + Rotation + Scaling

2


Symmetry• Example: Reflection in 2D

x

y

3


Symmetry

x

y

4

• Example: Reflection in 2D


Symmetry

x

y

5



Symmetry

x

y

6



φ

Symmetry

x

y

d

φ

d

spatial domain transformation space

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Symmetry

x

y

d

φ

d

φ

d

φ


8



Symmetry• Local analysis: Symmetry as a pair relation

x

y

spatial domain

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Symmetry

x

y φ

transformation space dspatial domain

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• Local analysis: Symmetry as a pair relation


Symmetry

x

y

d

φ


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Symmetry

x

y

d

φ


12



Symmetry

x

y

d

φ

local evidence for symmetry plane


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Symmetry• Accumulation of local evidence

x

y

d

φ



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– clustering to extract symmetry transformation – verification to extract symmetric patches

x

y

d

φ

dφ


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– stochastic Algorithm with provable guarantees

x

y

d

φ

dφ


E(n, µ,∆σ) <�1−

�−2 log α/np

�np/2d

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Sydney Opera House

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Sydney Opera House

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• Random samples on two poses– Correspondences between points are not known

Articulated Shapes

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• Correspondence candidates

Articulated Shapes

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segmentation

Articulated Shapes

transform plot

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Symmetrization• Goal: Symmetrize 3D geometry

• Applications– reverse engineering– recognition, retrieval– compression– symmetric meshing, etc.

• Approach– Minimally deform the model by optimizing the

distribution in transformation space

22


Optimal Displacements• Find minimal displacements that make two points

symmetric with respect to a given transformation

pi

pj

p�i

p�j

pi + p�j2

pj + p�i2

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Optimal Displacements

�di�2 + �dj�2

di

dj

pj

pi

minimize

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• Find minimal displacements that make two points symmetric with respect to a given transformation


Optimal Displacements• Find optimal transformation and minimal

displacements for a set of corresponding points

–

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• Find optimal transformation and minimal displacements for a set of corresponding points

–



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• Find optimal transformation and minimal displacements for a set of corresponding points

–


Optimal Displacements• Find optimal transformation and minimal

displacements for a set of corresponding points

– closed form solution exists!

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Optimization• Embedded deformation

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30



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32



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• 2D Example

Symmetrization

original

transformation space

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• 2D Example

Symmetrization

original


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• 2D Example

Symmetrization

original


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• 2D Example

Symmetrization


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• 2D Example

Symmetrization


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• 2D Example

Symmetrization


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• 2D Example

Symmetrization


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• 2D Example

Symmetrization


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Dragon

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Dragon

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Dragon

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Symmetrizing the Dragon

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Shape Matching

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References

• Mitra, Guibas, Pauly: Partial and Approximate Symmetry Detection for 3D Geometry, ACM Transactions on Graphics (Proceedings of SIGGRAPH), 2006

• Mitra, Guibas, Pauly: Symmetrization, ACM Transactions on Graphics (Proceedings of SIGGRAPH), 2007

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Documents

EG 4.2 SymmetrySymmetrization - Max Planck Societyresources.mpi-inf.mpg.de/deformableShapeMatching/EG2011... · 2011. 4. 19. · Eurographics$2011$Course$$–$CompungCorrespondencesinGeometricDataSets