The Planar-Reflective Symmetry Transform Princeton University.

The Planar-Reflective Symmetry TransformThe Planar-Reflective Symmetry Transform

Princeton University

MotivationMotivation

Symmetry is everywhere



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



Local Symmetry[Blum ’78][Mitra ’06] [Simari ’06]



Partial Symmetry[Zabrodsky ’95][Kazhdan ’03]

GoalGoal

A computational representation that describes all planar symmetries of a shape

?Input Model

Symmetry TransformSymmetry Transform

A computational representation that describesall planar symmetries of a shape

?Input Model Symmetry Transform



?Symmetry = 1.0Perfect Symmetry



?Symmetry = 0.0Zero Symmetry



?Symmetry = 0.3Local Symmetry



?Symmetry = 0.2Partial Symmetry

Symmetry MeasureSymmetry Measure

Symmetry of a shape is measured by correlation with its reflection



)(),( fffD Symmetry = 0.7






)(),( fffD




OutlineOutline

• Introduction• Algorithm

– Computing Discrete Transform– Finding Local Maxima Precisely

• Applications– Alignment– Segmentation

• Summary

– Matching– Viewpoint Selection

n p

lan

esComputing Discrete TransformComputing Discrete Transform

• Brute Force• Convolution• Monte-Carlo

n p

lan

esComputing Discrete TransformComputing Discrete Transform

• Brute Force O(n6)• Convolution• Monte-Carlo

O(n3) planes

X = O(n6)

O(n3) dot product

n p

lan

es

Computing Discrete TransformComputing Discrete Transform

• Brute Force O(n6)• Convolution O(n5Log n)• Monte-Carlo

O(n2) normal directions

X = O(n5log n)

O(n3log n) per direction

Computing Discrete TransformComputing Discrete Transform

• Brute Force O(n6)• Convolution O(n5Log n)• Monte-Carlo O(n4) For 3D meshes

– Most of the dot product contains zeros.– Use Monte-Carlo Importance Sampling.

Monte Carlo AlgorithmMonte Carlo Algorithm

Off

set

Angle

Input Model Symmetry Transform


Off

set

Angle

Monte Carlo sample for single plane



Off

set

Angle



Off

set

Angle



Off

set

Angle



Off

set

Angle



Off

set

Angle


Weighting SamplesWeighting Samples

Need to weight sample pairs by the inverse of the distance between them

P1

P2

d



Two planes of (equal) perfect symmetry



Votes for vertical plane…


Votes for horizontal plane.


OutlineOutline


– Computing Discrete Transform– Finding Local Maxima Precisely


• Summary


Finding Local Maxima PreciselyFinding Local Maxima Precisely

Motivation:• Significant symmetries will be local maxima of the

transform: the Principal Symmetries of the model

Principal Symmetries


Approach:• Start from local maxima of discrete transform


Initial Guess Final Result

……….

Approach:• Start from local maxima of discrete transform• Refine iteratively to find local maxima precisely

OutlineOutline


– Computing discrete transform– Finding Local Maxima Precisely


• Summary


Application: AlignmentApplication: Alignment

Motivation:• Composition of range scans• Feature mapping

PCA Alignment


Approach:• Perpendicular planes with the greatest symmetries

create a symmetry-based coordinate system.











Symmetry Alignment

PCA Alignment

Results:

Application: MatchingApplication: Matching

Motivation:• Database searching

Database Best MatchQuery

=


Observation:• All chairs display similar principal symmetries


Approach:• Use Symmetry transform as shape descriptor

Database Best MatchQuery

=

Transform


Results: • The PRST provides orthogonal information about

models and can therefore be combined with other shape descriptors







Application: SegmentationApplication: Segmentation

Motivation:• Modeling by parts• Collision detection

[Chazelle ’95][Li ’01][Mangan ’99][Garland ’01]

[Katz ’03]


Observation:• Components will have strong local symmetries not

shared by other components














Approach:• Cluster points on the surface by how well they

support different symmetries

Symmetry Vector = { 0.1 , 0.5 , …. , 0.9 }

Support = 0.1 Support = 0.5 Support = 0.9

…..


Results:

Application: Viewpoint SelectionApplication: Viewpoint Selection

Motivation:• Catalog generation• Image Based Rendering

[Blanz ’99][Vasquez ’01][Lee ’05][Abbasi ’00]

Picture from Blanz et al. ‘99


Approach:• Symmetry represents redundancy in information.


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


Results:

Viewpoint Function


Results:

Viewpoint Function

Best Viewpoint


Results:

Viewpoint Function

Best Viewpoint

Worst Viewpoint


Results:

SummarySummary

• 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 WorkFuture Work

Extended forms of symmetry• Rotational symmetry• Point symmetry• General transform symmetrySignal processing

Further applications • Compression• Constrained editing• Etc.

Acknowledgements: Acknowledgements:

Princeton Graphics• Chris DeCoro• Michael Kazhdan

Funding• Air Force Research Lab grant #FA8650-04-1-1718• NSF grant #CCF-0347427• NSF grant #CCR-0093343• NSF grant #IIS-0121446• The Sloan Foundation

The End

ComparisonComparison

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/etc.

Detection Types

Perfect, partial, and continuous symmetries

Perfect and Partial and approximate symmetries

The Planar-Reflective Symmetry Transform Princeton University.

Documents

planar symmetries

weight sample pairs

themp1p2dweighting samplesneed

on5log n on3log n

everywherelocal symmetryblum

everywhereperfect symmetryblum

horizontal plane

d meshesmost