Partial and Approximate Symmetry Detection for 3D Geometry

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Partial and Approximate Symmetry Detection for 3D
Geometry
Niloy J. Mitra Leonidas J. Guibas
Mark Pauly
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Symmetry in Nature
Symmetry is a complexity-reducing concept ...
seek it everywhere. - Alan J. Perlis
"Females of several species, including
humans, prefer symmetrical males." -
Chris Evan
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Symmetry for Geometry Processing
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Partial Symmetry Detection

• Given

Shape model (represented as point cloud, mesh,
... )
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Related Work
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Types of Symmetry

• Transform Types
• Reflection
• Rotation Translation
• Uniform Scaling

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Contributions

• Automatic detection of discrete symmetries !
  reflection, rigid transform, uniform scaling
• Symmetry graphs ! high level structural
  information about object
• Output sensitive algorithms ! low memory
  requirements

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Problem Characteristics

• Difficulties
• Which parts are symmetric ! objects not
  pre-segmented
• Space of transforms rotation translation
• Brute force search is not feasible
• Easy
• Proposed symmetries ! easy to validate

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Reflective Symmetry
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Reflective Symmetry A Pair Votes
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Reflective Symmetry Voting Continues
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Reflective Symmetry Voting Continues
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Reflective Symmetry Largest Cluster

• Height of cluster ! size of patch
• Spread of cluster ! level of approximation

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Pipeline
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Pipeline
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Pruning Local Signatures

• Local signature ! invariant under transforms
• Signatures disagree ! points dont correspond

Use (?1, ?2) for curvature based pruning
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Reflection Normal-based Pruning
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Point Pair Pruning
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Transformations

• Reflection ! point-pairs
• Rigid transform ! more information

Robust estimation of principal curvature frames
Cohen-Steiner et al. 03
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Mean-Shift Clustering

• Kernel
• Radially symmetric
• Radius/spread

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Verification

• Clustering gives a good guess
• Verify ! build symmetric patches
• Locally refine solution using ICP algorithm
  Besl and McKay 92

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Random Sampling

• Height of clusters related to symmetric region
  size
• Random samples ! larger regions likely to be
  detected earlier
• Output sensitive

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Model Reduction Chambord
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Model Reduction Chambord
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Model Reduction Chambord
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Sydney Opera House
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Sydney Opera House
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Approximate Symmetry Dragon
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Limitations
Castro et al. 06

• Cannot differentiate between small sized
  symmetries and comparable noise

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Articulated Motion Horses
symmetry detection between two objects !
registration
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More details in the paper

• Symmetry graph reduction
• Analysis of sampling requirements

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Future Work

• Detect biased deformation
• Pose independent shape matching
• Application to higher dimensional data

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Acknowledgements

• DARPA, NSF, CARGO, ITR, and NIH grants
• Stanford Graduate Fellowship

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Thank you!

• Niloy J. Mitra niloy_at_stanford.edu
• Leonidas J. Guibas guibas_at_cs.stanford.edu
• Mark Pauly pauly_at_inf.ethz.ch

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Performance
(time in seconds)
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Comparison