niloy j. mitra, leonidas j. guibas, and mark...

Niloy J. Mitra, Leonidas J. Guibas, and Mark PaulyCopyright of figures and other materials in this slide is belongs to original authors.

Presenter: 이성호

Korea UniversityComputer Graphics Lab. 이성호| 10 March 2011 | # 2KUCG |

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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Partial Symmetry DetectionGiven

Shape model (represented as point cloud, mesh, ... )

Identify and extract similar (symmetric) patches of different size across different resolutions

Goal

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Types of Symmetry

Transform Types:• Reflection• Rotation + Translation• Uniform Scaling

Shearing and non-linear transformations?

Shearing and non-linear transformations?

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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 § Complexity depends on the number and extent of symmetries§ low memory requirements

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Related Work

[Loy and Eklundh `06]

Hough transform on feature points

[Gal and Cohen-Or `06]

Based on RANSAC

tradeoff memory for speed

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Problem Characteristics

Difficulties§ Which parts are symmetric

• objects not pre-segmented

§ Space of transforms: rotation + translation + scaling§ 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

• Number of points in a cluster : □• Spread of a cluster : □

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Quiz

• Determine dimension of Γ for§ Reflection§ Translation§ Reflection+Translation§ Rotation§ Scaling+Translation§ Scaling+Rotation+Translation

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Pipeline

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Random Sampling

• Sampling yields a set P of surface points.• Select random subset P’⊂P and find all pairs(p’,p)

§ with p’∈P’ and p∈P.§ Appendix gives theoretical bounds

• on the size of P and P’ required to successfully find symmetries

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Pruning: Local Signatures

• Local signature ! invariant under transforms• Signatures disagree ! points don’t correspond

Use (k1, k2) for curvature based pruning

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Reflection: Normal-based Pruning

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Point Pair Pruning

all pairs curvature based curvature + normal based

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Quiz

• Determine signature of§ Reflection§ Translation§ Rotation+translation§ Scale+rotation+translation

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Signatures

• Principal curvature estimation§ For point , calculate principal curvatures

and principal directions§ [Cohen-Steiner and Morval 2003]

• Scaling estimation

• Rotation

• Translation

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• 7 Dimesion transformation

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Transformations

• Reflection requires point-pairs• Rigid transform requires 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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Define norm of Γ

• Norm of as the weighted sum

• 180 degrees == 0.5*bounding box diagonal== scailing factor of 10

• Metric for Γ can be derived as

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Verification

• Clustering gives a good guess• Verify builds symmetric patches• Locally refine solution using ICP (Iterative Closest Point)

algorithm § [Besl and McKay `92]

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ICP Problem

• Align two partially-overlapping meshesgiven initial guessfor relative transform

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Compound Transforms

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[Magnus et al. 2004][Magnus et al. 2004]

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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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Articulated Motion: Horses

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Performance

model # vertices sign. pairing cluster. verif.Dragon 160,947 3.44 49.24 13.63 7.45

Opera 9,376 0.96 0.02 0.03 0.86

Castle 172,606 5.61 117.81 159.73 5.63

Horse 8,431 0.92 0.01 0.01 1.63

Arch 16,921 0.08 5.86 26.89 2.42

(time in seconds)

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Limitations

• Cannot differentiate between § small sized symmetries and comparable noise§ Pre-smoothing

• Less distinct curvature estimates

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Future Work

• Symmetrization

• Unsupervised registration of§ Partial scan alignment§ Protein docking