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

3

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

4

original embedded object

deformed embedded object

space deformation

deformed reduced model

non-linear optimization

reduced model

space discretization

Eurographics 2012, Cagliari, Italy

Deformation Pipeline

5

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

6

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

8

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

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

12

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

15

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

18

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

19

✓

✓

(✓)

(✓)

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