As-Rigid-As-Possible Surface Modeling

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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 cant 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,

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

• Lets 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

vj2
vi
vj1
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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 U?WT
• Ri UWT (or use Horn 87)

v?j2
vj2
v?i
v?j1
vi
Ri
vj1
Ri is a non-linear function of v?
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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?

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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?

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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)

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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

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Need appropriate weighting

• Add cotangent weights Pinkall and Polthier 93
• Reformulate Ri optimization to include the
  weights (weighted covariance matrix)

vi
?ij
?ij
vj
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Weighted energyminimization results

• This gives symmetric results

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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 iterations
initial 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 Sorkine sorkine_at_gmail.com Marc Alexa marc_at_cs.tu-berlin.de