As-Rigid-As-Possible Surface Modeling

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) • 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” M. Botsch and O. Sorkine to appear at IEEE TVCG NEW! Previous work • Surface-based approaches • Multiresolution modelingZorin et al. 97, Kobbelt et al. 98, Lee 98, Guskov et al. 99, Botsch and Kobbelt 04, … • Differential coordinates – linear optimizationLipman 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

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 v׳j1 v׳i vi Ri vj1 Riis 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׳

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 energyminimization 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!