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Symmetry Transforms
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Symmetry Transforms 1 1 Motivation Symmetry is everywhere 2 - - PowerPoint PPT Presentation
Symmetry Transforms 1 1 Motivation Symmetry is everywhere 2 - - PowerPoint PPT Presentation
Symmetry Transforms 1 1 Motivation Symmetry is everywhere 2 Motivation Symmetry is everywhere Perfect Symmetry [Blum 64, 67] [Wolter 85] [Minovic 97] [Martinet 05] 3 Motivation Symmetry is everywhere Local Symmetry
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Motivation Symmetry is everywhere Perfect Symmetry
[Blum ’64, ’67] [Wolter ’85] [Minovic ’97] [Martinet ’05]
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Motivation Symmetry is everywhere Local Symmetry
[Blum ’78] [Thrun ‘05] [Simari ’06]
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Motivation Symmetry is everywhere Partial Symmetry
[Zabrodsky ’95] [Kazhdan ’03]
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Goal A computational representation that describes all planar symmetries of a shape
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Symmetry Transform A computational representation that describes all planar symmetries of a shape
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Symmetry Transform A computational representation that describes all planar symmetries of a shape
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Symmetry = 1.0 Perfect Symmetry
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Symmetry Transform A computational representation that describes all planar symmetries of a shape
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Symmetry = 0.3 Local Symmetry
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Symmetry Transform A computational representation that describes all planar symmetries of a shape
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Symmetry = 0.2 Partial Symmetry
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Symmetry Measure Symmetry of a shape is measured by correlation with its reflection
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Symmetry Measure Symmetry of a shape is measured by correlation with its reflection
) ( ) , ( f f f D γ γ ⋅ =
Symmetry = 0.7
) ( ) , ( f f f D γ γ ⋅ =
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Symmetry Measure Symmetry of a shape is measured by correlation with its reflection
) ( ) , ( f f f D γ γ ⋅ =
Symmetry = 0.3
) ( ) , ( f f f D γ γ ⋅ =
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Symmetry Measure Symmetry of a shape is measured by correlation with its reflection
) ( ) , ( f f f D γ γ ⋅ =
) ( ) , ( f f f D γ γ ⋅ =
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Symmetry Measure Symmetry of a shape is measured by correlation with its reflection
) ( ) , ( f f f D γ γ ⋅ =
Symmetry = 0.1
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Previous Work Zabrodsky ‘95 Kazhdan ‘03 Thrun ‘05 Martinet ‘05
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Symmetry Distance Define the Symmetry Distance of a function f with respect to any transformation γ as the L2 distance between f and the nearest function invariant to γ Can show that Symmetry Measure is related to symmetry distance by
g g g
g f f SD
=
− =
) ( |
min ) , (
γ
γ
) ( ) , ( f f f D γ γ ⋅ =
2 2
2 ) , ( f SD f D + − = γ
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Previous Work Zabrodsky ‘95 Kazhdan ‘03 Thrun ‘05 Martinet ‘05
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Previous Work Zabrodsky ‘95 Kazhdan ‘03 Thrun ‘05 Martinet ‘05
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Previous Work Zabrodsky ‘95 Kazhdan ‘03 Thrun ‘05 Martinet ‘05
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Computing Discrete Transform Brute Force O(n6) Convolution Monte-Carlo
O(n3) planes X O(n3) dot product
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Computing Discrete Transform Brute Force O(n6) Convolution O(n5Log n) Monte-Carlo
O(n2) normal directions X O(n3 log n) per direction
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Computing 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
Offset Angle
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Monte Carlo
Monte Carlo Sample for single plane
Offset Angle
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Monte Carlo
Offset Angle
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Monte Carlo
Offset Angle
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Monte Carlo
Offset Angle
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Monte Carlo
Offset Angle
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Monte Carlo
Offset Angle
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Weighting Samples
Need to weight sample pairs by the inverse of the distance between them
P1 P2 d
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Weighting 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 Samples
Need to weight sample pairs by the inverse of the distance between them
Vertical votes concentrated…
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Weighting Samples
Need to weight sample pairs by the inverse of the distance between them
Horizontal votes diffused…
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Application: Alignment Motivation:
Composition of range scans Morphing
PCA Alignment
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Application: Alignment Approach:
Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
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Application: Alignment Approach:
Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
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Application: Alignment Approach:
Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
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Application: Alignment Approach:
Perpendicular planes with the greatest symmetries create a symmetry-based coordinate system.
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Application: Alignment Symmetry Alignment PCA Alignment
Results:
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Application: Matching Motivation:
Database searching
Database Result Query
=
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Application: Matching Observation:
All chairs display similar principal symmetries
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Application: Matching Approach:
Use Symmetry transform as shape descriptor
Database Result Query
=
Transform
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Application: Matching Results:
Symmetry provides orthogonal information about models and can therefore be combined with other descriptors
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