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Symmetry Analysis
Ligang LiuGraphics&Geometric Computing Lab
USTChttp://staff.ustc.edu.cn/~lgliu
Ack: T. Funkhouser et al.
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Motivation• Most real‐world objects are symmetric(at least approximately and/or partially)
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Symmetry everywhere
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Reflectional Symmetry
• Symmetry with respect to reflection– Mirror symmetry
• The most fundamental form of symmetry observed
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Many Types of Symmetry
• Rotational symmetry
• Translational symmetry
• Helical symmetry
• … …
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Why Symmetry Important?
• Human cognition– fundamental principle of perception
– important cue for shape recognition
• Shape analysis– high‐level shape information
• Geometry processing– process half of the mesh
– segmentation, remeshing, simplification, …
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Symmetry Research in Graphics
• Symmetry descriptor– [Kazhdan et al. 2004, Zabrodsky et al. 2005, Kazhdan et al. 2006, Podolak et al.
2007]
• Symmetry detection– [Atallah 1985, Mitra et al. 2006, Loy et al. 2006, Martinet et al. 2006, Simari et
al. 2006, Raviv et al. 2007, Ovsjanikov et al. 2008, Pauly et al. 2008, Bokeloh et al. 2009, Yeh et al. 2009, Bronstein et al. 2009, Xu et al. 2009]
• Symmetry aware geometry processing– [Golovinski et al. 2007, Mitra et al. 2007]
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Mathematically, symmetry concept is general…
• Symmetry in math notation– Two objects are symmetric to each other if one is obtained from the other by some mathematical operations.
– The set of operations form a group.
:g G X X× →
( )g x y=( )g x x=
x and y are symmetrical to each other
x is a symmetric object
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Specifically, symmetry is
• an invariant under a class of transformations
g x x⋅ =
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Reflectional symmetries
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Problem• Geometric processing algorithms generally ignore (and sometimes destroy) symmetries, causing …– Inefficient processing
– Rendering artifacts
– Simulation errors
– etc.
[Cohen-Steiner04]
Symmetric Input Mesh
Asymmetric Output Mesh
Simplify
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Goal• We need methods to measure, analyze, represent, and exploit symmetries in 3D models
Symmetric Input Mesh
Symmetric Output Mesh
Representationof Symmetries
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This Talk
• We need methods to measure, analyze, represent, and exploit symmetries in 3D models
Symmetric Input Mesh
Symmetric Output Mesh
Representationof Symmetries
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Outline
• Introduction
• Symmetry measurement
• Symmetry representations
• Applications
• Conclusion
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Outline
• Introduction
• Symmetry measurement
• Symmetry representations
• Applications
• Conclusion
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Symmetry Detection
• Definition: a shape S is symmetric with respect to transform T if and only if S=T(S)
T = PlaneReflect(p)
T = LineReflect(l)
l
l
l
p
T = Rotate(l,180°)
T = Rotate(l,90°)
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Symmetry Measurement• How symmetric is shape S to transform T?
e.g., T = Rotate(p,120)?
Sp
InputShape
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Symmetry Measurement• How symmetric is shape S to transform T?
pS
InputShape
Closest ShapeSymmetric to T
SymmetryDistance
p
Zabrodsky: L2 distance between S and closest shape symmetric to T
T = Rotate(p,120)
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Symmetry Measurement• How symmetric is shape S to transform T?
SymmetryDistance
l
S
Closest ShapeSymmetric to T
Zabrodsky: L2 distance between S and closest shape symmetric to T
T = Reflect(l)
InputShape
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Symmetry Measurement• How symmetric is shape S to transform T?
InputShape
SymmetryMeasure
l
Zabrodsky: L2 distance between S and closest shape symmetric to TMarola,Sun,Kazhdan: Correlation between f(S) and f(T(S))
f(S) + f(T(S))2
Closest ShapeSymmetric to T
f(S) f(S) · f(T(S))f(S)2
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Symmetry Measurement
lS
Symmetry Measure = 1
• Sym(S,T) provides a continuous measure in [0,1] of how symmetric shape S is to transform T
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Symmetry Measurement• Sym(S,T) provides a continuous measure in [0,1] of how symmetric shape S is to transform T
l
S
Symmetry Measure = 0
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Symmetry Measurement• Sym(S,T) provides a continuous measure in [0,1] of how symmetric shape S is to transform T
S
Symmetry Measure = 0.7
l
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Symmetry Measurement
l
S
Symmetry Measure = 0.3
• Sym(S,T) provides a continuous measure in [0,1] of how symmetric shape S is to transform T
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Symmetry Measurement
S
Symmetry Measure = 0.4
l
• Sym(S,T) provides a continuous measure in [0,1] of how symmetric shape S is to transform T
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Outline
• Introduction
• Symmetry measurement
• Symmetry representations
• Applications
• Conclusion
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Symmetry Representations
• Measure the symmetry of an object with respect to a set of transformations
SymmetryRepresentation
Input Shape
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Principle Symmetries
• Measure the symmetry of an object with respect to its most symmetric transformations
Planar ReflectivePrinciple Symmetries
Input Shape
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Principle Symmetries
• Measure the symmetry of an object with respect to its most symmetric transformations
Planar ReflectivePrinciple Symmetries
Input Shape
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Symmetry Descriptor
• Measure the symmetry of an object with respect to all reflections/rotations around a point
Planar ReflectiveSymmetry Descriptor
Input Shape
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Symmetry Transform
• Measure the symmetry of an object with respect to all transformations
Planar ReflectiveSymmetry Transform
Input Shape
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Symmetry Transform• Example planar reflective symmetry transforms:
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Symmetry Transform• Highlights large symmetric features of shape:
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Outline
• Introduction
• Background
• Symmetry representations
• Applications
• Conclusion
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Applications• General idea: symmetry‐aware processing
Symmetry Analysis
Principal Symmetry Transformations
+ Regions of Support
InputMesh
SymmetryAware
ProcessingOutputMesh
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Recent Work • General idea: symmetry‐aware processing
– Completion – Zabrodsky93, Thrun05
– Alignment – Zabrodsky95, Kazhdan02, Podolak06
– Symmetrization – Zabrodsky97, Mitra07, Golovinskiy07
– Feature detection – Reisfeld95
– Reverse engineering – Mills01
– Instancing ‐Martinet05, Mitra08
– Matching – Kazhdan02, Gal05
– Compression – Simari06
– Smoothing – Golovinskiy07
– Segmentation – Mitra06, Podolak06, Liu07
– Viewpoint selection – Podolak06
– Simplification – Podolak07, Mitra07, Golovinskiy07
– Beautification – Langbein03, Golovinskiy07
– Editing – Mitra06, Golovinskiy07
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Applications
• Alignment
• Matching
• Segmentation
• Simplification
• Beautification
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Applications
• Alignment
• Matching
• Segmentation
• Simplification
• Beautification
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Application: Alignment
• Motivation: registration, modeling, etc.
PCA Alignment
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Application: Alignment
• Approach: align planes with highest symmetries
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Application: Alignment
• Approach: align planes with highest symmetries
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Application: Alignment
• Approach: align planes with highest symmetries
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Application: Alignment
• Approach: align planes with highest symmetries
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Application: Alignment• Results:
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Application: Alignment• Results:
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Applications
• Alignment
• Matching
• Segmentation
• Simplification
• Beautification
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Application: Matching
• Motivation: similarity search of database
Database Best MatchQuery
=
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Application: Matching• Observation: symmetry is more consistent than shape for some object classes
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Application: Matching
• Approach: use symmetry transform (or descriptor) as shape descriptor
Database Best MatchQuery
=
SymmetryTransform
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Application: Matching
• Results: symmetry is not as discriminating as shape
Shape X SymmetryShape AloneSymmetry Alone
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Application: Matching• Results: symmetry is not as discriminating as shape, but it is better for some classes
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Application: Matching• Results: symmetry is not as discriminating as shape, but it is better for some classes, and sothe two together are better than either alone
Shape X SymmetryShape AloneSymmetry Alone
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Applications
• Alignment
• Matching
• Segmentation
• Simplification
• Beautification
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Application: Segmentation
• Motivation: animation, modeling by parts, etc.
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Application: Segmentation
• Observation: distinct parts have strong local symmetries not shared by other parts
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Application: Segmentation
• Observation: distinct parts have strong local symmetries not shared by other parts
![Page 57: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/57.jpg)
Application: Segmentation
• Observation: distinct parts have strong local symmetries not shared by other parts
![Page 58: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/58.jpg)
Application: Segmentation
• Observation: distinct parts have strong local symmetries not shared by other parts
![Page 59: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/59.jpg)
Application: Segmentation
• Observation: distinct parts have strong local symmetries not shared by other parts
![Page 60: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/60.jpg)
Application: Segmentation
• Approach: cluster points on the surface by how much they support different symmetries
Symmetry Vector = { 0.1 , 0.5 , …. , 0.9 }
Support = 0.1 Support = 0.5 Support = 0.9
…..
![Page 61: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/61.jpg)
Application: Segmentation• Approach: cluster points on the surface by how much they support different symmetries
![Page 62: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/62.jpg)
Application: Segmentation
• Results:
![Page 63: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/63.jpg)
Applications
• Alignment
• Matching
• Segmentation
• Simplification
• Beautification
![Page 64: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/64.jpg)
Application: Simplification
• Motivation: maintain symmetries as simplify mesh
InputMesh
StandardSimplification
SymmetricSimplification
![Page 65: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/65.jpg)
Application: Simplification• Approach: detect (approximate) symmetries as part of the simplification/approximation process
Symmetry-Aware Variatonal Shape ApproximationPodolak et al.
Original 600 Proxies 200 Proxies 50 Proxies
![Page 66: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/66.jpg)
Application: Simplification
• Results:
Original 100 ProxiesOriginal 100 Proxies
![Page 67: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/67.jpg)
Application: Simplification
• Results:
Input Model Simplified Models
![Page 68: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/68.jpg)
Application: Simplification• Results:
![Page 69: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/69.jpg)
Application: Parameterization• Preserves symmetries in base domain
Input Mesh
Base Domain Parameterization
![Page 70: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/70.jpg)
Applications
• Alignment
• Matching
• Segmentation
• Simplification
• Beautification
![Page 71: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/71.jpg)
Application: Beautification• Goal: make meshes of symmetric objectsperfectly symmetric
Input Mesh Symmetric Mesh
![Page 72: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/72.jpg)
Application: Beautification
• Approach: iterative non‐rigid deformation to align symmetric points (symmetrization)
Input shape CorrespondencesBetween
SymmetricPoints
Shape PreservingDeformation to
Enhance Symmetry
Iterate
![Page 73: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/73.jpg)
Application: Beautification• Results:
Input Mesh
![Page 74: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/74.jpg)
Application: Beautification
• Results:
PrincipalSymmetry Planes
![Page 75: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/75.jpg)
Application: Beautification
• Results:
Output Mesh
![Page 76: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/76.jpg)
Application: Beautification
• Results:
Output Mesh
![Page 77: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/77.jpg)
Application: Beautification
• Results:
Input Mesh
![Page 78: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/78.jpg)
Application: Denoising• Results:
InputMesh
SymmetrizedMesh
BilateralFiltering
![Page 79: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/79.jpg)
More symmetries
![Page 80: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/80.jpg)
Symmetric or not?
![Page 81: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/81.jpg)
Intrinsic Symmetry:symmetry under isometry
![Page 82: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/82.jpg)
Intrinsic Symmetry
• Isometric homeomorphism– preserving all the geodesic distances
:T M M→
( ) ( ), ( ), ( ) , ,d p q d T p T q
p q M=
∀ ∈
p
q
( )T p
( )T q
![Page 83: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/83.jpg)
Intrinsic Symmetry
• More difficult to represent and compute
• Only a few papers so far– [Raviv et al. 2007]
– [Ovsjanikov et al. 2008]
p
q
( )T p
( )T q
![Page 84: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/84.jpg)
Partial Symmetry
• Symmetry plays a fundamental role in cognitive grouping and object identification.
It is more natural to separate the two arms and decompose the shape into two symmetric sub-parts.
![Page 85: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/85.jpg)
Partial Symmetry
![Page 86: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/86.jpg)
Partial Symmetry
![Page 87: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/87.jpg)
Partial Symmetries
Partial extrinsic symmetries Partial intrinsic symmetries
![Page 88: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/88.jpg)
Symmetry Detections
Extrinsic Intrinsic
Global [Atallah 1985][Martinet et al. 2006][Kazhdan et al. 2006][Pauly et al. 2008][Bokeloh et al. 2009]
[Raviv et al. 2007][Ovsjanikov et al. 2008]
Partial [Mitra et al. 2006][Simari et al. 2006]
[Xu et al. 2009]
![Page 89: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/89.jpg)
More…
• Symmetry in 3D Geometry: Extraction and Applications– Niloy J. Mitra, Mark Pauly, Michael Wand, Duygu Ceylan
– State‐of‐the‐art Report of EUROGRAPHICS 2012
![Page 90: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/90.jpg)
Summary
• Representations– Principal symmetries
– Symmetry descriptor
– Symmetry transform
• Applications– Alignment
– Matching
– Segmentation
– Simplification (parameterization)
– Beautification (denoising)
![Page 91: Ligang Liu Graphics&GeometricComputing Lab USTC …• We need methods to measure, analyze, represent, and ... – Beautification – Langbein03, Golovinskiy07 – Editing – Mitra06,](https://reader031.vdocuments.net/reader031/viewer/2022030419/5aa66fb87f8b9ae7438ec27f/html5/thumbnails/91.jpg)
Discussion