partial and approximate symmetry detection for 3d geometry mark pauly niloy j. mitra leonidas j....
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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 [email protected] J. Guibas [email protected] Mark Pauly [email protected]
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