Adaptive Space Deformations Based On Rigid Cells Weekly seminar series Visual Media Lab, KAIST, Korea

1. Adaptive Space Deformations Based on Rigid Cells Mario Botsch, Mark Pauly, Martin Wicke and Markus Gross *Mainly based on the authors’ presentation in EG `07

2. Linear Surface Deformation • Minimize quadratic surface energy • Variational calculus • Solve sparse linear system [Sorkine et al, SGP 04] [Lipmanet al, SIG 05] [Kobbelt et al, SIG 98]

3. Problematic Cases Topological inconsistencies Geometric degeneracies Highly complex models

4. Linear Space Deformation • Deform embedding space • Deformation complexity << surface complexity [Sederberg & Parry, SIG 86] [Hsu et al, SIG 92] [Ju et al, SIG 05]

6. Linear vs. Nonlinear Problems with large deformations

7. Nonlinear Surface Deformation • Minimize nonlinear energies – Intuitive large-scale deformation – Robustness issues – Performance issues [Au et al, SIG 07] [Botsch et al, SGP 06]

8. PriMo [Botsch et al, SGP 2006] 1. Extrude triangles to prisms / cells 2. Prescribes position/orientation for cells 3. Find optimal rigid motions per cell 4. Update vertices by averaged cell transformations

10. Adaptive Space Deformations Based on Rigid Cells Design decisions:

11. Adaptive Space Deformations Based on Rigid Cells Design decisions: – Realistic behavior ⇒ Physically plausible

12. Adaptive Space Deformations Based on Rigid Cells Design decisions: – Realistic behavior ⇒ Physically plausible – Large deformations ⇒ Nonlinear energy nonlinear linear

13. Adaptive Space Deformations Based on Rigid Cells Design decisions: – Realistic behavior ⇒ Physically plausible – Large deformations ⇒ Nonlinear energy – Robustness ⇒ Rigid cells

14. Adaptive Space Deformations Based on Rigid Cells Design decisions: – Realistic behavior ⇒ Physically plausible – Large deformations ⇒ Nonlinear energy – Robustness ⇒ Rigid cells – Applicability ⇒ Space deformation

15. AdaptiveSpace Deformations Based on Rigid Cells Design decisions: – Realistic behavior ⇒ Physically plausible – Large deformations ⇒ Nonlinear energy – Robustness ⇒ Rigid cells – Applicability ⇒ Space deformation – Performance ⇒ Adaptive discretization

16. Deformation Pipeline

17. Deformation Energy • Continuum mechanics – Strain energy defined by displacement’s gradient – Local variation of displacement causes stretching • Discrete rigid cells – Each cell stores rigid motion – Local differences of rigid transformations

18. Difference of Transformations • Frobenius norm of matrices – Geometric meaning of matrix elements? • [Pottmann et al, 04] – Difference of images of sample points (which?)

19. Nonlinear Energy • Integrate over neighboring cells’ interiors

20. Nonlinear Energy • Integrate over neighboring cells’ interiors • Accumulated global energy

21. Nonlinear Minimization • Find rigid motion per cell • Generalized shape matching [Botsch et al, SGP 06] – Robust geometric optimization – Nonlinear Newton-type minimization

22. Nonlinear Minimization • Generalized shape matching [Botsch et al, SGP 06] – Affine transform approximation (Linearization) – Nonlinear Newton-type minimization - Linear solve followed by rigid transform projection

23. Robust Optimization

24. Deformation Pipeline

25. Volumetric Discretization • Octree-like adaptive voxelization – Easy to implement, efficient to compute – Analytic integration • T-junctions are no problem !

26. Boundary Refinement

27. Energy-Driven Refinement

28. 3D Adaptive Refinement

29. 3D Adaptive Refinement

30. 3D Adaptive Refinement

31. Deformation Pipeline

32. Deformation Interpolation Final: RBF Interpolation Preview: Averaging / Skinning

33. Complex Models

34. Triangle Soup

35. Aliasing

36. Conclusion • Physically plausible deformation energy – Shells & solids, 2D & 3D – Arbitrary convex cell arrangements – Robust geometric shape matching • Adaptive discretization – Static and dynamic refinement • Smooth space deformation – Meshes, triangle soups, point sets, ...

37. 질문은 손들고