Symmetrization

1
Symmetrization

• Niloy J. Mitra Leonidas J. Guibas Mark
  Pauly
• TU Vienna Stanford University ETH Zurich
• SIGGRAPH 2007

2
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results, Discussion and Conclusion

3
Abstract

• Enhances symmetries of a model while minimally
  altering its shape.

4
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results, Discussion and Conclusion

5
Introduction

• Symmetrization
• Goal Symmetrize 3D geometry
• ApproachMinimally deform the model in the
  spatial domain by optimizing the destribution in
  transformation space

6
Introduction

• Contributions
• Given an explicit pairing of points , a closed
  form solution for symmetrizing the point set.
• A symmetrization algorithm that uses transform
  domain reasoning to guide shape deformation in
  object domain .
• Applications
• Extend the types of detected symmetries
• Symmetric remeshing

7
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results, Discussion and Conclusion

8
Previous Work

• Symmetry detecting
• MITRA, N. J., GUIBAS, L. J., AND PAULY, M. 2006.
• Partial and approximate symmetry detection for 3d
  geometry. ACM Trans.Graph

9
Previous Work

• Symmetry detecting
• MITRA, N. J., GUIBAS, L. J., AND PAULY, M. 2006.
  Partial and approximate symmetry detection for 3d
  geometry. ACM Trans.Graph

10
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results, Discussion and Conclusion

11
Overview

• 2D Example Symmetry Detection

Mitra 2006
transformation space
d
Y
?
d
?
0
X
12
Overview

• 2D Example Symmetry Detection

transformation space
13
Overview

• 2D Example Symmetry Detection

transformation space
pair of sample points defines reflective symmetry
transform
14
Overview

• 2D Example Symmetry Detection

transformation space
transform domain
object domain
pair of points point
15
Overview

• 2D Example Another point-pair

transformation space
pair of sample points defines reflective symmetry
transform
16
Overview

• 2D Example Another point-pair votes

transformation space
17
Overview

• 2D Example Voting continues

transformation space
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Overview

• 2D Example Voting continues

transformation space
19
Overview

• 2D Example Density plot

transformation space
density plot accumulation of
symmetry evidence
20
Overview

• 2D Example Density peaks

transformation space
density cluster reflective symmetry
21
Overview

• 2D Example Local symmetrization

transformation space
cluster constraction
local symmetrization
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Overview

• 2D Example Local symmetrization

transformation space
cluster constraction
local symmetrization
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Overview

• 2D Example Local symmetrization

transformation space
cluster contraction in transform space
deformation in object space
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Overview

• 2D Example Global symmetrization

After local symmetrized
25
Overview

• 2D Example Global symmetrization

cluster merging global
symmetrization
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Overview

• 2D Example Global symmetrization

cluster merging global
symmetrization
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Overview

• 2D Example Global symmetrization

cluster merging / contraction
global symmetrization
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Overview

• 2D Example Global symmetrization

cluster merging / contraction
global symmetrization
29
Overview

• 2D Example Global symmetrization

cluster merging / contraction
global symmetrization
30
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results, Discussion and Conclusion

31
Optimal Displacements

• given a point pair (p,q)
• define a unique reflective transformation T
• if we want to move T to T
• p,q need to be displaced by vector ,

d
T
p
p
T
T
T
q
q
?
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Optimal Displacements

• We derive

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Optimal Displacements

• Goal Find minimal displacements for a set of
  point pairs

T makes a set of point pairs symmetric with
respect to T such that qi T(pi) for all i.
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Optimal Displacements

• Optimal means transformation T minimizes the cost

35
Optimal Displacements

• Optimal means transformation T minimizes the cost

36
Optimal Displacements

• Optimal means transformation T minimizes the cost

37
Optimal Displacements

• Optimal means transformation T minimizes the cost

38
Optimal Displacements

• Optimal transformation (E 69.4) differs from
  the centroid of the pair transformation (E
  142.6)

39
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results, Discussion and Conclusion

40
Sub-problems

• Optimizing Sample Positions
• Every sample point p will be shifted in the
  direction of its optimal displacement
• Project them back onto the surface
• Re-compute the optimal transformation and
  displacements vectors
• Until the variance is no longer reduced

Iterate
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Sub-problems

• This reduces the variance of a cluster without
  modifying the geometry and thus leads to smaller
  subsequent deformations.

42
Sub-problems

• Contracting Clusters
• Optimize sample positions
• Compute the optimal transformation
• Apply the resulting replacements
• New optimal displacements vectors are computed

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Sub-problems

• Cluster Merging

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Sub-problems

• Cluster Merging

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Sub-problems

• Cluster Merging

46
Sub-problems

• Cluster Merging

47
Sub-problems

• Cluster Merging

moving two points in transformation space to
their centroid
Deformation method
as-rigid-as-possibleIgarashi 2005
non-linear PriMo deformation model
Botsch et al. 2006
48
Outline

• Abstract
• Introduction
• Previous Work
• Overview
• Optimal Displacements
• Sub-problems
• Optimizing Sample Positions
• Contracting Clusters
• Merging Clusters
• Results and Conclusion

49
Results and Conclusion

• Bunny

50
Results and Conclusion

• Symmetrized Bunny

51
Results and Conclusion

• Dragon

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Results and Conclusion

• Symmetrized Dragon

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Results and Conclusion
54
Results and Conclusion

• Symmetric Remeshing

55
Results and Conclusion
56
Results and Conclusion
57
Results and Conclusion

• Articulated Bodies

cluster merging
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Results and Conclusion

• Articulated Bodies

59
Results and Conclusion

• Bunny Feet

60
Results and Conclusion