1. As-Rigid-As-Possible Surface Modeling
Olga Sorkine Marc Alexa
TU Berlin

2. 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

3. What do we expect from surface deformation?
Smooth effect on the large scale
As-rigid-as-possible effect on the small scale (preserves details)

4. Previous work FFD (space deformation) Pros: Cons:
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]

5. “On Linear Variational Surface Deformation Methods”
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 approaches Kraevoy & 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. Sorkine
to appear at IEEE TVCG
NEW!
images taken from PriMo, Botsch et al. 06

6. 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

7. 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

8. Direct ARAP modeling
Let’s look at cells on a mesh

9. 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

10. 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 vi
v׳i
v׳j1 Ri
vj1
Ri is a non-linear function of v׳

11. 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

12. 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׳

13. Uniform mesh Laplacian
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׳
Uniform mesh Laplacian

14. 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)

15. First results
Non-symmetric results

16. Need appropriate weighting
The problem: lack of compensation for varying shapes of the 1-ring

17. Need appropriate weighting
Add cotangent weights [Pinkall and Polthier 93]:
Reformulate Ri optimization to include the weights (weighted covariance matrix) vi
ij
ij vj

18. Weighted energy minimization results
This gives symmetric results

19. Initial guess
Can start from naïve Laplacian editing as initial guess and iterate
initial guess
1 iteration
2 iterations
initial guess
1 iterations
4 iterations

20. Initial guess
Faster convergence when we start from the previous frame (suitable for interactive manipulation)

21. Some more results
Demo

22. 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]

23. 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

24. 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

25. Acknowledgement Alexander von Humboldt Foundation
Leif Kobbelt and Mario Botsch
SGP Reviewers

26. Thank you! Olga Sorkine sorkine@gmail.com Marc Alexa