The Planar-Reflective Symmetry Transform Princeton University.
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Transcript of The Planar-Reflective Symmetry Transform Princeton University.
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The Planar-Reflective Symmetry TransformThe Planar-Reflective Symmetry Transform
Princeton University
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MotivationMotivation
Symmetry is everywhere
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MotivationMotivation
Symmetry is everywhere
Perfect Symmetry[Blum ’64, ’67][Wolter ’85][Minovic ’97] [Martinet ’05]
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MotivationMotivation
Symmetry is everywhere
Local Symmetry[Blum ’78][Mitra ’06] [Simari ’06]
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MotivationMotivation
Symmetry is everywhere
Partial Symmetry[Zabrodsky ’95][Kazhdan ’03]
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GoalGoal
A computational representation that describes all planar symmetries of a shape
?Input Model
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Symmetry TransformSymmetry Transform
A computational representation that describesall planar symmetries of a shape
?Input Model Symmetry Transform
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Symmetry TransformSymmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 1.0Perfect Symmetry
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Symmetry TransformSymmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 0.0Zero Symmetry
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Symmetry TransformSymmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 0.3Local Symmetry
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Symmetry TransformSymmetry Transform
A computational representation that describesall planar symmetries of a shape
?Symmetry = 0.2Partial Symmetry
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Symmetry MeasureSymmetry Measure
Symmetry of a shape is measured by correlation with its reflection
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Symmetry MeasureSymmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD Symmetry = 0.7
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Symmetry MeasureSymmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD Symmetry = 0.3
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Symmetry MeasureSymmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD
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Symmetry MeasureSymmetry Measure
Symmetry of a shape is measured by correlation with its reflection
)(),( fffD Symmetry = 0.1
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OutlineOutline
• Introduction• Algorithm
– Computing Discrete Transform– Finding Local Maxima Precisely
• Applications– Alignment– Segmentation
• Summary
– Matching– Viewpoint Selection
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n p
lan
esComputing Discrete TransformComputing Discrete Transform
• Brute Force• Convolution• Monte-Carlo
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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
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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
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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.
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Input Model Symmetry Transform
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Monte Carlo sample for single plane
Input Model Symmetry Transform
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Input Model Symmetry Transform
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Input Model Symmetry Transform
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Input Model Symmetry Transform
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Input Model Symmetry Transform
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Monte Carlo AlgorithmMonte Carlo Algorithm
Off
set
Angle
Input Model Symmetry Transform
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Weighting SamplesWeighting Samples
Need to weight sample pairs by the inverse of the distance between them
P1
P2
d
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Weighting SamplesWeighting Samples
Need to weight sample pairs by the inverse of the distance between them
Two planes of (equal) perfect symmetry
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Weighting SamplesWeighting Samples
Need to weight sample pairs by the inverse of the distance between them
Votes for vertical plane…
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Weighting SamplesWeighting Samples
Votes for horizontal plane.
Need to weight sample pairs by the inverse of the distance between them
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OutlineOutline
• Introduction• Algorithm
– Computing Discrete Transform– Finding Local Maxima Precisely
• Applications– Alignment– Segmentation
• Summary
– Matching– Viewpoint Selection
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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
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Finding Local Maxima PreciselyFinding Local Maxima Precisely
Approach:• Start from local maxima of discrete transform
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Finding Local Maxima PreciselyFinding Local Maxima Precisely
Initial Guess Final Result
……….
Approach:• Start from local maxima of discrete transform• Refine iteratively to find local maxima precisely
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OutlineOutline
• Introduction• Algorithm
– Computing discrete transform– Finding Local Maxima Precisely
• Applications– Alignment– Segmentation
• Summary
– Matching– Viewpoint Selection
![Page 38: The Planar-Reflective Symmetry Transform Princeton University.](https://reader035.fdocuments.net/reader035/viewer/2022062409/56649d775503460f94a587ac/html5/thumbnails/38.jpg)
Application: AlignmentApplication: Alignment
Motivation:• Composition of range scans• Feature mapping
PCA Alignment
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Application: AlignmentApplication: Alignment
Approach:• Perpendicular planes with the greatest symmetries
create a symmetry-based coordinate system.
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Application: AlignmentApplication: Alignment
Approach:• Perpendicular planes with the greatest symmetries
create a symmetry-based coordinate system.
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Application: AlignmentApplication: Alignment
Approach:• Perpendicular planes with the greatest symmetries
create a symmetry-based coordinate system.
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Application: AlignmentApplication: Alignment
Approach:• Perpendicular planes with the greatest symmetries
create a symmetry-based coordinate system.
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Application: AlignmentApplication: Alignment
Symmetry Alignment
PCA Alignment
Results:
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Application: MatchingApplication: Matching
Motivation:• Database searching
Database Best MatchQuery
=
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Application: MatchingApplication: Matching
Observation:• All chairs display similar principal symmetries
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Application: MatchingApplication: Matching
Approach:• Use Symmetry transform as shape descriptor
Database Best MatchQuery
=
Transform
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Application: MatchingApplication: Matching
Results: • The PRST provides orthogonal information about
models and can therefore be combined with other shape descriptors
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Application: MatchingApplication: Matching
Results: • The PRST provides orthogonal information about
models and can therefore be combined with other shape descriptors
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Application: MatchingApplication: Matching
Results: • The PRST provides orthogonal information about
models and can therefore be combined with other shape descriptors
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Application: SegmentationApplication: Segmentation
Motivation:• Modeling by parts• Collision detection
[Chazelle ’95][Li ’01][Mangan ’99][Garland ’01]
[Katz ’03]
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Application: SegmentationApplication: Segmentation
Observation:• Components will have strong local symmetries not
shared by other components
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Application: SegmentationApplication: Segmentation
Observation:• Components will have strong local symmetries not
shared by other components
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Application: SegmentationApplication: Segmentation
Observation:• Components will have strong local symmetries not
shared by other components
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Application: SegmentationApplication: Segmentation
Observation:• Components will have strong local symmetries not
shared by other components
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Application: SegmentationApplication: Segmentation
Observation:• Components will have strong local symmetries not
shared by other components
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Application: SegmentationApplication: Segmentation
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
…..
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Application: SegmentationApplication: Segmentation
Results:
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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
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Application: Viewpoint SelectionApplication: Viewpoint Selection
Approach:• Symmetry represents redundancy in information.
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Application: Viewpoint SelectionApplication: 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
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Application: Viewpoint SelectionApplication: Viewpoint Selection
Results:
Viewpoint Function
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Application: Viewpoint SelectionApplication: Viewpoint Selection
Results:
Viewpoint Function
Best Viewpoint
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Application: Viewpoint SelectionApplication: Viewpoint Selection
Results:
Viewpoint Function
Best Viewpoint
Worst Viewpoint
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Application: Viewpoint SelectionApplication: Viewpoint Selection
Results:
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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
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Future WorkFuture Work
Extended forms of symmetry• Rotational symmetry• Point symmetry• General transform symmetrySignal processing
Further applications • Compression• Constrained editing• Etc.
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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
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The End
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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