The Planar-Reflective Symmetry Transform Princeton University

Motivation Symmetry is everywhere

Motivation Symmetry is everywhere Perfect Symmetry [Blum ’ 64, ’ 67] [Wolter ’ 85] [Minovic ’ 97] [Martinet ’ 05]

Motivation Symmetry is everywhere Local Symmetry [Blum ’ 78] [Mitra ’ 06] [Simari ’ 06]

Motivation Symmetry is everywhere Partial Symmetry [Zabrodsky ’ 95] [Kazhdan ’ 03]

Goal A computational representation that describes all planar symmetries of a shape ? Input Model

Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Input ModelSymmetry Transform

A computational representation that describes all planar symmetries of a shape ? Symmetry = 1.0Perfect Symmetry

Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Symmetry = 0.0Zero Symmetry

Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Symmetry = 0.3Local Symmetry

Symmetry Transform A computational representation that describes all planar symmetries of a shape ? Symmetry = 0.2Partial Symmetry

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0.7

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0.3

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection

Symmetry Measure Symmetry of a shape is measured by correlation with its reflection Symmetry = 0.1

Outline Introduction Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely Applications – Alignment – Segmentation Summary – Matching – Viewpoint Selection

n planes Computing Discrete Transform Brute Force Convolution Monte-Carlo

n planes Computing Discrete Transform Brute ForceO(n 6 ) Convolution Monte-Carlo O(n 3 ) planes X = O(n 6 ) O(n 3 ) dot product

n planes Computing Discrete Transform Brute ForceO(n 6 ) ConvolutionO(n 5 Log n) Monte-Carlo O(n 2 ) normal directions X = O(n 5 log n) O(n 3 log n) per direction

Computing Discrete Transform Brute ForceO(n 6 ) ConvolutionO(n 5 Log n) Monte-CarloO(n 4 ) For 3D meshes – Most of the dot product contains zeros. – Use Monte-Carlo Importance Sampling.

Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

Monte Carlo Algorithm Offset Angle Monte Carlo sample for single plane Input ModelSymmetry Transform

Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

Monte Carlo Algorithm Offset Angle Input ModelSymmetry Transform

Weighting Samples Need to weight sample pairs by the inverse of the distance between them P1P1 P2P2 d

Weighting Samples Need to weight sample pairs by the inverse of the distance between them Two planes of (equal) perfect symmetry

Weighting Samples Need to weight sample pairs by the inverse of the distance between them Votes for vertical plane…

Weighting Samples Votes for horizontal plane. Need to weight sample pairs by the inverse of the distance between them

Outline Introduction Algorithm – Computing Discrete Transform – Finding Local Maxima Precisely Applications – Alignment – Segmentation Summary – Matching – Viewpoint Selection

Finding Local Maxima Precisely Motivation: Significant symmetries will be local maxima of the transform: the Principal Symmetries of the model Principal Symmetries

Finding Local Maxima Precisely Approach: Start from local maxima of discrete transform

Finding Local Maxima Precisely Initial GuessFinal Result ………. Approach: Start from local maxima of discrete transform Refine iteratively to find local maxima precisely

Outline Introduction Algorithm – Computing discrete transform – Finding Local Maxima Precisely Applications – Alignment – Segmentation Summary – Matching – Viewpoint Selection

Application: Alignment Motivation: Composition of range scans Feature mapping PCA Alignment

Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

Application: Alignment Approach: Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.

Application: Alignment Symmetry Alignment PCA Alignment Results:

Application: Matching Motivation: Database searching Database Best MatchQuery =

Application: Matching Observation: All chairs display similar principal symmetries

Application: Matching Approach: Use Symmetry transform as shape descriptor DatabaseBest MatchQuery = Transform

Application: Matching Results: The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors

Application: Matching Results: The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors

Application: Matching Results: The PRST provides orthogonal information about models and can therefore be combined with other shape descriptors

Application: Segmentation Motivation: Modeling by parts Collision detection [Chazelle ’95][Li ’01] [Mangan ’99][Garland ’01] [Katz ’03]

Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

Application: Segmentation Observation: Components will have strong local symmetries not shared by other components

Application: Segmentation Approach: Cluster points on the surface by how well they support different symmetries Symmetry Vector = { 0.1, 0.5, …., 0.9 } Support = 0.1Support = 0.5Support = 0.9 …..

Application: Segmentation Results:

Application: Viewpoint Selection Motivation: Catalog generation Image Based Rendering [Blanz ’99][Vasquez ’01] [Lee ’05][Abbasi ’00] Picture from Blanz et al. ‘99

Application: Viewpoint Selection Approach: Symmetry represents redundancy in information.

Application: Viewpoint Selection Approach: Symmetry represents redundancy in information Minimize the amount of visible symmetry Every plane of symmetry votes for a viewing direction perpendicular to it Best Viewing Directions

Application: Viewpoint Selection Results: Viewpoint Function

Application: Viewpoint Selection Results: Viewpoint Function Best Viewpoint

Application: Viewpoint Selection Results: Viewpoint Function Best Viewpoint Worst Viewpoint

Application: Viewpoint Selection Results:

Summary Symmetry Transform – Symmetry measure for all planes in space Algorithms – Discrete set of planes – Finding local maxima precisely Applications – Alignment – Matching – Segmentation – Viewpoint Selection

Future Work Extended forms of symmetry Rotational symmetry Point symmetry General transform symmetry Signal processing Further applications Compression Constrained editing Etc.

Acknowledgements: Princeton Graphics Chris DeCoro Michael Kazhdan Funding Air Force Research Lab grant #FA NSF grant #CCF NSF grant #CCR NSF grant #IIS The Sloan Foundation

The End

Comparison Podolak et al.Mitra et al. GoalTransformDiscrete symmetry SamplingUniform gridClustering VotingPoints only Points, normals, curvature Symmetr y Types Planar reflectionReflection/Rotation/etc. Detection Types Perfect, partial, and continuous symmetries Perfect and Partial and approximate symmetries