Partial and Approximate Symmetry Detection for 3D Geometry Mark Pauly Niloy J. Mitra Leonidas J....

Partial and Approximate Symmetry Detection for 3D Geometry

Mark PaulyNiloy J. Mitra

Leonidas J. Guibas

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

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

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Symmetry for Geometry Processing

[Funkhouser et al. `05]

[Sharf et al. `04]

[Katz and Tal `04]

[Khazdan et al. `04]

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Partial Symmetry DetectionGiven

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

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

Goal

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Related Work

[Podolak et al. `06] [Loy and Eklundh `06]

Hough transform on feature points

[Gal and Cohen-Or `05]

tradeoff memory for speed

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Types of Symmetry

Transform Types:ReflectionRotation + TranslationUniform Scaling

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

ContributionsAutomatic detection of discrete symmetries ! reflection, rigid transform, uniform scalingSymmetry graphs ! high level structural information about objectOutput sensitive algorithms ! low memory requirements

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Problem CharacteristicsDifficulties

Which parts are symmetric ! objects not pre-segmentedSpace of transforms: rotation + translationBrute force search is not feasible

EasyProposed symmetries ! easy to validate

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Reflective Symmetry

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Reflective Symmetry: A Pair Votes

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Reflective Symmetry: Voting Continues

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Reflective Symmetry: Voting Continues

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Reflective Symmetry: Largest Cluster

Height of cluster ! size of patchSpread of cluster ! level of approximation

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Pipeline

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Pipeline

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Pruning: Local SignaturesLocal signature ! invariant under transformsSignatures disagree ! points don’t correspond

Use (1, 2) for curvature based pruning

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Reflection: Normal-based Pruning

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Point Pair Pruning

all pairs curvature based curvature + normal based

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

TransformationsReflection ! point-pairsRigid transform ! more information

Robust estimation of principal curvature frames [Cohen-Steiner et al. `03]

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Mean-Shift ClusteringKernel:

Radially symmetricRadius/spread

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

VerificationClustering gives a good guessVerify ! build symmetric patchesLocally refine solution using ICP algorithm [Besl and McKay `92]

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Random SamplingHeight of clusters related to symmetric region sizeRandom samples ! larger regions likely to be detected earlierOutput sensitive

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Model Reduction: Chambord

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Model Reduction: Chambord

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Model Reduction: Chambord

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Sydney Opera House

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Sydney Opera House

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Approximate Symmetry: Dragon

correction fieldUNITS: fraction of bounding box diagonal

detected symmetries

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Limitations

Cannot differentiate between small sized symmetries and comparable noise

[Castro et al. `06]

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Articulated Motion: Horses

‘symmetry’ detection between two objects ! registration

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

More details in the paperSymmetry graph reduction Analysis of sampling requirements

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Future WorkDetect biased deformationPose independent shape matchingApplication to higher dimensional data

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Acknowledgements

DARPA, NSF, CARGO, ITR, and NIH grantsStanford Graduate Fellowship

Pierre AlliezMario BotschDoo Young KwonMarc Levoy

Ren NgBob SumnerDilys Thomasanonymous reviewers

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Thank you!

Niloy J. Mitra niloy@stanford.eduLeonidas J. Guibas guibas@cs.stanford.edu Mark Pauly pauly@inf.ethz.ch

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

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)

Partial and Approximation Symmetry Detection for 3D Geometry Niloy J. Mitra

Comparison

Podolak et al. Mitra et al.

Goal Transform Discrete symmetry

Sampling Uniform grid Clustering

Voting Points only Points, normals, curvature

Symmetry types

Planar reflection reflection, rotation, trans., unif. scaling

Detection types

Perfect, partial,continuous

Perfect, partial, approximate