2.3 deformation graphs (mp) - max planck...
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Deformation Graphs
Mark Pauly
Eurographics 2012, Cagliari, Italy
Deformation Models
Desirable Properties– Generality
• handle different geometry representations• predictable, physically plausible deformation behavior
– Efficiency, scalability• processing of large data sets, realtime feedback
– Robustness• stable even for bad input and drastic deformations
– Simplicity• ease of implementation• adaptability, extensibility, re-use
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Eurographics 2012, Cagliari, Italy
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Surface Deformation Space DeformationLi
near
Non
linea
r
[Botsch & Kobbelt, SIG 04][Kobbelt et al, SIG 98]
[Botsch & Kobbelt, EG 05]
[Sederberg & Parry, SIG 86]
[Hsu et al, SIG 92]
[Huang et al, SIG 06][Botsch et al, SGP 06]
[Shi et al, SIG 07]
[Sorkine et al, SGP 04]
[Lipman et al, SIG 05][Yu et al, SIG 04]
Botsch et al, EG 07
Sumner et al, SIG 07
Eurographics 2012, Cagliari, Italy
Deformation Pipeline
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original embedded object
deformed embedded object
space deformation
deformed reduced model
non-linear optimization
reduced model
space discretization
Eurographics 2012, Cagliari, Italy
Deformation Pipeline
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original embedded object
deformed embedded object
space deformation
deformed reduced model
non-linear optimization
reduced model
space discretization
Eurographics 2012, Cagliari, Italy
Space Discretization
• Begin with an embedded object
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Eurographics 2012, Cagliari, Italy
Space Discretization
• Begin with an embedded object
• Sample the object
• Each node deforms nearby space
• Edges connect nodes of overlapping in!uence
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• Begin with an embedded object
• Sample the object
• Each node deforms nearby space
• Edges connect nodes of overlapping in!uence
Eurographics 2012, Cagliari, Italy
Space Discretization
Deformation Graph
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Eurographics 2012, Cagliari, Italy
Deformation Pipeline
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original embedded object
deformed embedded object
space deformation
deformed reduced model
non-linear optimization
reduced model
space discretization
Eurographics 2012, Cagliari, Italy
Optimization
Every graph node de!nes an affinemapping that transforms nearby space
Eurographics 2012, Cagliari, Italy
Graphparameters
Rotationterm
Regularizationterm
Constraintterm
Optimization
Energy formulation
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Eurographics 2012, Cagliari, Italy
Optimization
Constraint term
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Constrained vertices should move accordingto correspondences
Eurographics 2012, Cagliari, Italy
Optimization
Rotation term
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For detail preservation,features should rotate andnot scale or skew.
Eurographics 2012, Cagliari, Italy
Optimization
Regularization term
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where node j thinksnode k should go
where node kactually goes
Neighboring nodes shouldagree on where they transformeach other.
Eurographics 2012, Cagliari, Italy
Deformation Pipeline
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original embedded object
deformed embedded object
space deformation
deformed reduced model
non-linear optimization
reduced model
space discretization
Eurographics 2012, Cagliari, Italy
Space Deformation: Approach 1
Simple averaging of transformations
For each point p• Pick k closest graph nodes {T1, T2, ..., Tk} (k=4)• Build weighted average of transformations
p �→k�
i=1
wiTi(p)
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Eurographics 2012, Cagliari, Italy
Space Deformation: Approach 2
Each graph node yields displacement
Interpolate by triharmonic RBFs
– Guarantees smooth & fair deformation
– Solve dense linear system for RBF coefficients
di = Ti(ci)− ci
d(x) =n�
i=1
wi · �ci − x�3 + p(x)
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Eurographics 2012, Cagliari, Italy
Adaptive Deformation Graphs
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Eurographics 2012, Cagliari, Italy
Deformation Models
Desirable Properties– Generality
• handle different geometry representations• predictable, physically plausible deformation behavior
– Efficiency, scalability• processing of large data sets, realtime feedback
– Robustness• stable even for bad input and drastic deformations
– Simplicity• ease of implementation• adaptability, extensibility, re-use
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Li, Adams, Guibas, Pauly: Single-View Geometry and Motion Reconstruction, SIGGRAPH ASIA 2009
Sumner, Schmid, Pauly: Embedded Deformation for Shape Manipulation, SIGGRAPH 2007
Botsch, Pauly, Wicke, Gross: Adaptive Space Deformation based on Rigid Cells, Eurographics 2007