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

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Presentation on theme: "Olga Sorkine, TU Berlin, 2007 1 CG 11 As-Rigid-As-Possible Surface Modeling Olga Sorkine Marc Alexa TU Berlin."— Presentation transcript:

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

2 Olga Sorkine, TU Berlin, 2007 2 CG 22 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 Olga Sorkine, TU Berlin, 2007 3 CG 33 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 Olga Sorkine, TU Berlin, 2007 4 CG 44 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 Olga Sorkine, TU Berlin, 2007 5 CG 55 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 “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 Olga Sorkine, TU Berlin, 2007 6 CG 66 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 Olga Sorkine, TU Berlin, 2007 7 CG 77 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 Olga Sorkine, TU Berlin, 2007 8 CG 88 Direct ARAP modeling  Let’s look at cells on a mesh

9 Olga Sorkine, TU Berlin, 2007 9 CG 99 Cell deformation energy  Ask all star edges to transform rigidly by some rotation R, then the shape of the cell is preserved vivi vj1vj1 vj2vj2

10 Olga Sorkine, TU Berlin, 2007 10 CG 10 Cell deformation energy  If v, v ׳ are known then R i is uniquely defined  So-called shape matching problem Build covariance matrix S = VV ׳ T SVD: S = U  W T R i = UW T (or use [Horn 87]) vivi vj1vj1 vj2vj2 v׳iv׳i v׳j1v׳j1 v׳j2v׳j2 RiRi R i is a non-linear function of v ׳

11 Olga Sorkine, TU Berlin, 2007 11 CG 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 Olga Sorkine, TU Berlin, 2007 12 CG 12 Energy minimization  Alternating iterations Given initial guess v ׳ 0, find optimal rotations R i –This is a per-cell task! We already showed how to define R i when v, v ׳ are known Given the R i (fixed), minimize the energy by finding new v ׳

13 Olga Sorkine, TU Berlin, 2007 13 CG 13 Energy minimization  Alternating iterations Given initial guess v ׳ 0, find optimal rotations R i –This is a per-cell task! We already showed how to define R i when v, v ׳ are known Given the R i (fixed), minimize the energy by finding new v ׳ Uniform mesh Laplacian

14 Olga Sorkine, TU Berlin, 2007 14 CG 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 Olga Sorkine, TU Berlin, 2007 15 CG 15 First results  Non-symmetric results

16 Olga Sorkine, TU Berlin, 2007 16 CG 16 Need appropriate weighting  The problem: lack of compensation for varying shapes of the 1-ring

17 Olga Sorkine, TU Berlin, 2007 17 CG 17 Need appropriate weighting  Add cotangent weights [Pinkall and Polthier 93]:  Reformulate R i optimization to include the weights (weighted covariance matrix) vivi vjvj  ij  ij

18 Olga Sorkine, TU Berlin, 2007 18 CG 18 Weighted energy minimization results  This gives symmetric results

19 Olga Sorkine, TU Berlin, 2007 19 CG 19 Initial guess  Can start from naïve Laplacian editing as initial guess and iterate initial guess1 iteration2 iterations 1 iterations4 iterationsinitial guess

20 Olga Sorkine, TU Berlin, 2007 20 CG 20 Initial guess  Faster convergence when we start from the previous frame (suitable for interactive manipulation)

21 Olga Sorkine, TU Berlin, 2007 21 CG 21 Some more results  Demo

22 Olga Sorkine, TU Berlin, 2007 22 CG 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 Olga Sorkine, TU Berlin, 2007 23 CG 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 Olga Sorkine, TU Berlin, 2007 24 CG 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 Olga Sorkine, TU Berlin, 2007 25 CG 25 Acknowledgement  Alexander von Humboldt Foundation  Leif Kobbelt and Mario Botsch  SGP Reviewers

26 Olga Sorkine, TU Berlin, 2007 26 CG 26 Thank you! Olga Sorkine sorkine@gmail.com Marc Alexa marc@cs.tu- berlin.de


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