2.3 Deformation Graphs (MP) - Max Planck...
Transcript of 2.3 Deformation Graphs (MP) - Max Planck...
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Deformation Graphs
Mark Pauly
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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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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
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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
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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
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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
9
original embedded object
deformed embedded object
space deformation
deformed reduced model
non-linear optimization
reduced model
space discretization
![Page 10: 2.3 Deformation Graphs (MP) - Max Planck Societyresources.mpi-inf.mpg.de/.../2.3_Deformation_Graphs... · Deformation Models Desirable Properties –Generality • handle different](https://reader036.fdocuments.us/reader036/viewer/2022063010/5fc2ee6f2db7212b9f39d92c/html5/thumbnails/10.jpg)
Eurographics 2012, Cagliari, Italy
Optimization
Every graph node de!nes an affinemapping that transforms nearby space
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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
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Eurographics 2012, Cagliari, Italy
Optimization
Rotation term
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For detail preservation,features should rotate andnot scale or skew.
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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.
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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
![Page 16: 2.3 Deformation Graphs (MP) - Max Planck Societyresources.mpi-inf.mpg.de/.../2.3_Deformation_Graphs... · Deformation Models Desirable Properties –Generality • handle different](https://reader036.fdocuments.us/reader036/viewer/2022063010/5fc2ee6f2db7212b9f39d92c/html5/thumbnails/16.jpg)
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