Olga Sorkine, TU Berlin, 2007 1 CG 11 As-Rigid-As-Possible Surface Modeling Olga Sorkine Marc Alexa TU Berlin.

Olga Sorkine, TU Berlin, 2007 1

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As-Rigid-As-Possible Surface Modeling

Olga Sorkine Marc Alexa

TU Berlin


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Surface deformation – motivation

Interactive shape modeling• Digital content creation• Scanned data

Modeling is an interactive, iterative process• Tools need to be intuitive

(interface and outcome)• Allow quick experimentation


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What do we expect from surface deformation?

Smooth effect on the large scale As-rigid-as-possible effect on the small scale

(preserves details)


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Previous work

FFD (space deformation)• Lattice-based (Sederberg & Parry 86, Coquillart 90,

…)• Curve-/handle-based (Singh & Fiume 98, Botsch et al. 05,

…)• Cage-based (Ju et al. 05, Joshi et al. 07, Kopf et al.

07) Pros:

• efficiency almost independent of the surface resolution • possible reuse

Cons:• space warp, so can’t precisely control surface properties

images taken from [Sederberg and Parry 86] and [Ju et al. 05]


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Previous work

Surface-based approaches• Multiresolution modeling

Zorin et al. 97, Kobbelt et al. 98, Lee 98, Guskov et al. 99, Botsch and Kobbelt 04, …

• Differential coordinates – linear optimization Lipman et al. 04, Sorkine et al. 04, Yu et al. 04, Lipman et al. 05, Zayer et al. 05, Botsch et al. 06, Fu et al. 06, …

• Non-linear global optimization approachesKraevoy & Sheffer 04, Sumner et al. 05, Hunag et al. 06, Au et al. 06, Botsch et al. 06, Shi et al. 07, …

“On Linear Variational Surface Deformation Methods”M. Botsch and O. Sorkineto appear at IEEE TVCG

“On Linear Variational Surface Deformation Methods”M. Botsch and O. Sorkineto appear at IEEE TVCG

NEW!

images taken from PriMo, Botsch et al. 06


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Surface-based approaches

Pros:• direct interaction with the surface• control over surface properties

Cons:• linear optimization suffers from artifacts (e.g. translation

insensitivity)• non-linear optimization is more expensive and non-trivial to

implement


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Direct ARAP modeling

Preserve shape of cells covering the surface• Cells should overlap to prevent shearing at the

cells boundaries• Equally-sized cells, or compensate for varying

size


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Direct ARAP modeling

Let’s look at cells on a mesh


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Cell deformation energy

Ask all star edges to transform rigidly by some rotation R, then the shape of the cell is preserved

vi vj1

vj2

2

( )

min ( ) ( )i j i i jj N i

R

v v v v


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Cell deformation energy

If v, v׳ are known then Ri is uniquely defined

So-called shape matching problem• Build covariance matrix S = VV׳T

• SVD: S = UWT

• Ri = UWT (or use [Horn 87])

vi vj1

vj2v׳i v׳j1

v׳j2

Ri

Ri is a non-linear function of v׳Ri is a non-linear function of v׳


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2

1 ( )

min ( ) ( )n

i j i i ji j N i

R

v

v v v v

. . ,j js t j C v c

Total deformation energy

Can formulate overall energy of the deformation:

We will treat v׳ and R as separate sets of variables, to enable a simple optimization process


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Energy minimization

Alternating iterations• Given initial guess v ׳

0 , find optimal rotations Ri– This is a per-cell task! We already showed how to

define Ri when v, v׳ are known

• Given the Ri (fixed), minimize the energy by finding new v׳

2

1 ( )

min ( ) ( )n

i j i i ji j N i

R

v

v v v v


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Energy minimization

Alternating iterations• Given initial guess v ׳

0 , find optimal rotations Ri– This is a per-cell task! We already showed how to

define Ri when v, v׳ are known

• Given the Ri (fixed), minimize the energy by finding new v׳

L v b

Uniform mesh Laplacian


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The advantage

Each iteration decreases the energy (or at least guarantees not to increase it)

The matrix L stays fixed• Precompute Cholesky factorization• Just back-substitute in each iteration (+ the

SVD computations)

L v b


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First results

Non-symmetric results


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Need appropriate weighting

The problem: lack of compensation for varying shapes of the 1-ring

2

( )

( ) ( )cell i j i i jj N i

E R

v v v v


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Need appropriate weighting

Add cotangent weights [Pinkall and Polthier 93]:

Reformulate Ri optimization to include the weights (weighted covariance matrix)

2

( )

( ) ( )cell ij i j i i jj N i

E w R

v v v v

vi

vj

ij

ij

1cot α cot β

2ij ij ijw


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Weighted energyminimization results

This gives symmetric results

n

i iNjjiijiijtotal RwE

1 )(

2)()( vvvv


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Initial guess

Can start from naïve Laplacian editing as initial guess and iterate

initial guess 1 iteration 2 iterations

1 iterations 4 iterationsinitial guess


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Initial guess

Faster convergence when we start from the previous frame (suitable for interactive manipulation)


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Some more results

Demo


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Discussion

Works fine on small meshes• fast propagation of rotations across the mesh

On larger meshes: slow convergence• slow rotation propagation

A multi-res strategy will help• e.g., as in Mean-Value Pyramid Coordinates

[Kraevoy and Sheffer 05] or in PriMo [Botsch et al. 06]


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More discussion

Our technique is good for preserving edge length (relative error is very small)

No notion of volume, however• “thin shells for the poor”?

Can easily extend to volumetric meshes


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Conclusions and future work

Simple formulation of as-rigid-as-possible surface-based deformation

Iterations are guaranteed to reduce the energy Uses the same machinery as Laplacian editing,

very easy to implement No parameters except number of iterations per

frame (can be set based on target frame rate)

Is it possible to find better weights? Modeling different materials – varying rigidity

across the surface


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Acknowledgement

Alexander von Humboldt Foundation

Leif Kobbelt and Mario Botsch SGP Reviewers


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Thank you!

Olga [email protected]

m

Marc [email protected]

berlin.de

Olga Sorkine, TU Berlin, 2007 1 CG 11 As-Rigid-As-Possible Surface Modeling Olga Sorkine Marc Alexa TU Berlin.

Documents

surface cells

surface control

mesh slide

cell deformation energy

surfacebased approaches

surface properties cons

nonlinear function of

total deformation energy