Variational Space DeformationVariational Space Deformationwith Barycentric Coordinates

Mirela Ben-ChenStanford University

Joint work withOfir Weber (NYU), Craig Gotsman (Technion)

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The Problem: DeformationThe Problem: Deformation

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Space Deformation

Idea:

Space Deformation

Deformation = map from input domain to some space

x g(x)x

S

g(x)

S

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Space Deformation

Idea:

Space Deformation

Deformation = map from input domain to some space

x g(x)x

S

g(x)

Advantage:

SAdvantage:

Can be applied to any geometry representation4

Which Maps are “Good”?Which Maps are Good ?

Properties of the map Quality of the deformationProperties of the map Quality of the deformation

SourceSource

Good Bad5

Which Maps are “Good”?Which Maps are Good ?

Global Change vs. Local Detail PreservationGlobal Change vs. Local Detail Preservation

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Which Maps are “Good”?Which Maps are Good ?

Easy to use “this point goes there”Easy to use this point goes there

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Which Maps are “Good”?Which Maps are Good ?

Efficient!Efficient!

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How to Find “Good” Maps?How to Find Good Maps?

• All possible maps on a domain = HUGE space

• Instead– Assume map is given on boundary– Infer map in the interior– Called: Cage Based Deformation

• Important properties– Translation invarianceTranslation invariance– Reproduction of identity

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Deformation with a Cage

f

or: Rules of the Game

fi

xi xg(x)

F

i

S

x

S

S = {x x x } x fS = {x1,x2,...,xn} Source polygonSource polygon

xi fi

Target polygonTarget polygon

g(x) = ? Interior?10

Deformation with Barycentric CoordinatesDeformation with Barycentric Coordinates

ffi

xi xg(x)

F

i

S

x

( )iw x n

∑

S

( )i

x 1( ) ( )i i

ig x w x f

=

=∑x

xi1i

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Which Bary Coords to Choose?Which Bary Coords to Choose?

• There’s lots of those around…Bad for deformation

• Real coords (wi (x) ∈ ) are affine invariant

Bad for deformation

• Complex coords (wi (z) ∈ ) better!If holomorphic get– If holomorphic get conformal map*

– Global change / local detail

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An Example: Cauchy-Green CoordinatesAn Example: Cauchy Green Coordinates

w

Take: dw

z1 1

2( , )k

w zz

iw

π −=

Define g as the Cauchy Transform:

1( ) ( )1 dg z f w w= ∫

f(w)

g(z)

Define g as the Cauchy Transform:

( ) ( )2 Si

dw

gz

z f w w−∫π

( ) ( , ) ( )S

g z k w z f w dw= ∫Integrate along edges for a polygon13

Discrete Cauchy-Green CoordinatesDiscrete Cauchy Green Coordinates

• Good:– Holomorphic

– Span all holomorphic maps

Bad• Bad: – Doesn’t interpolate boundary

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Variational Cauchy CoordinatesVariational Cauchy Coordinates

Find best u1,..,un such that

1( ) ( )

n

u j jj

g z C z u=

=∑1j=

Is optimal

Minimize energy functional of gu

Quadratic energy = linear equations in uj15

Szegö Coordinates

Goal: Best fit to target polygon

Szegö Coordinates

Goal: Best fit to target polygon zi

w

fi

g(w) f(w)

2( ) ( ) ( )E f d∫

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Source Cauchy-Green Szegö( ) ( ) ( )

S

E g g w f w ds= −∫

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Point-to-Point CoordinatesPoint to Point Coordinates

r1Goal: Free user from the cage 1

r2r3

Goal: Free user from the cage

f1

f2

g(r1)

ff3

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3D Deformation3D Deformation

• Input:Input: – Model

– Cage (domain Ω)– User Constraints

• Output:– Deformation function

– Fulfills user constraints

– Detail preserving

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3D Deformation

• For variational framework need

3D Deformation

For variational framework need

• Barycentric coordinates

• Energy functional• Energy functional

• Coordinates• Coordinates

• No complex numbers / holomorphic maps in 3Dholomorphic maps in 3D

• Use harmonic maps + additional constraints+ additional constraints

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Finding the Harmonic MapFinding the Harmonic Map

Harmonic maps– Linear subspace of

– Subspace spanned by two maps

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The Deformation FunctionThe Deformation Function

ϕ and ψ known, depend on the domain Ω“Generalized” barycentric coordinates“Generalized” barycentric coordinates

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The EnergyThe Energy

• Can place constraints on deformation function andCan place constraints on deformation function and derivatives – get a linear system in a and b

• We need:– User constraints

Positional + Jacobian

– Detail preservation ( i f lit )(since no conformality)

Jacobian is rotation

– Smoothness

Not linear

Hessian on boundary

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ResultsAs-Rigid-As-Possible (Jacobian is rotation)

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Results As-Similar-As-Possible (Jacobian is scale + rotation)

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ResultsffGiraffe Soup

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Deformation Transfer

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StrategyStrategy

• Cage source and target shapesCage source and target shapes

• Project source deformation on harmonic maps• Project source deformation on harmonic maps

• Compute deformation gradients Jacobians of• Compute deformation gradients Jacobians of deformation function

• Deform target shape to achieve Jacobians– Linear!Linear!

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ResultsP T fPoses Transfer

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ResultsR l i D f i T fReal-time Deformation Transfer

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ResultsA i ti T fAnimation Transfer

Target Source

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ResultsA i ti T fAnimation Transfer

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ConclusionsConclusions

• Variational space deformation provides anVariational space deformation provides an easy-to-use general deformation method

• Efficient linear formulation using barycentric coordinates functions y

• Always looking for “good” maps = good “coordinates”y g g p g– Beyond harmonic and conformal?

– Quasi-conformal?

– Isometric?33

Thank YouThank You34