Post on 27-Jan-2021
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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
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry• Example: Reflection in 2D
x
y
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
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• Example: Reflection in 2D
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
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• Example: Reflection in 2D
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
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• Example: Reflection in 2D
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
φ
Symmetry
x
y
d
φ
d
spatial domain transformation space
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• Example: Reflection in 2D
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
d
φ
d
φ
d
φ
spatial domain transformation space
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• Example: Reflection in 2D
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry• Local analysis: Symmetry as a pair relation
x
y
spatial domain
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y φ
transformation space dspatial domain
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• Local analysis: Symmetry as a pair relation
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
d
φ
spatial domain transformation space
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• Local analysis: Symmetry as a pair relation
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
d
φ
spatial domain transformation space
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• Local analysis: Symmetry as a pair relation
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry
x
y
d
φ
local evidence for symmetry plane
spatial domain transformation space
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• Local analysis: Symmetry as a pair relation
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry• Accumulation of local evidence
x
y
d
φ
local evidence for symmetry plane
spatial domain transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry• Accumulation of local evidence
– clustering to extract symmetry transformation – verification to extract symmetric patches
x
y
d
φ
dφ
local evidence for symmetry plane
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetry• Accumulation of local evidence
– stochastic Algorithm with provable guarantees
x
y
d
φ
dφ
local evidence for symmetry plane
E(n, µ,∆σ) <�1−
�−2 log α/np
�np/2d
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Sydney Opera House
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Sydney Opera House
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• Random samples on two poses– Correspondences between points are not known
Articulated Shapes
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• Correspondence candidates
Articulated Shapes
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
segmentation
Articulated Shapes
transform plot
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
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
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
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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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
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
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimal Displacements• Find optimal transformation and minimal
displacements for a set of corresponding points
–
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimal Displacements
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• Find optimal transformation and minimal displacements for a set of corresponding points
–
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimal Displacements
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• Find optimal transformation and minimal displacements for a set of corresponding points
–
Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimal Displacements• Find optimal transformation and minimal
displacements for a set of corresponding points
– closed form solution exists!
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimization• Embedded deformation
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimization• Embedded deformation
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimization• Embedded deformation
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimization• Embedded deformation
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Optimization• Embedded deformation
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
original
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
original
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
original
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
• 2D Example
Symmetrization
transformation space
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Dragon
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Dragon
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Dragon
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Symmetrizing the Dragon
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
Shape Matching
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Eurographics 2011 Course – Compu/ng Correspondences in Geometric Data Sets
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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