Adaptive Space Deformations Based On Rigid Cells

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

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Adaptive Space Deformations Based On Rigid Cells

  1. 1. Adaptive Space Deformations Based on Rigid Cells<br />Mario Botsch, Mark Pauly, Martin Wicke and Markus Gross<br />*Mainly based on the authors’ presentation in EG `07<br />
  2. 2. Linear Surface Deformation<br />• Minimize quadratic surface energy<br />• Variational calculus<br />• Solve sparse linear system<br />[Sorkine et al, SGP 04]<br />[Lipmanet al, SIG 05]<br />[Kobbelt et al, SIG 98]<br />
  3. 3. Problematic Cases<br />Topological<br />inconsistencies<br />Geometric<br />degeneracies<br />Highly complex<br />models<br />
  4. 4. Linear Space Deformation<br />• Deform embedding space<br />• Deformation complexity << surface complexity<br />[Sederberg & Parry, SIG 86]<br />[Hsu et al, SIG 92]<br />[Ju et al, SIG 05]<br />
  5. 5.
  6. 6. Linear vs. Nonlinear<br />Problems with large deformations<br />
  7. 7. Nonlinear Surface Deformation<br />• Minimize nonlinear energies<br /> – Intuitive large-scale deformation<br /> – Robustness issues<br /> – Performance issues<br />[Au et al, SIG 07]<br />[Botsch et al, SGP 06]<br />
  8. 8. PriMo [Botsch et al, SGP 2006]<br />1. Extrude triangles to prisms / cells<br />2. Prescribes position/orientation for cells<br />3. Find optimal rigid motions per cell<br />4. Update vertices by averaged cell transformations<br />
  9. 9.
  10. 10. Adaptive Space Deformations Based on Rigid Cells<br />Design decisions:<br />
  11. 11. Adaptive Space Deformations Based on Rigid Cells<br />Design decisions:<br />– Realistic behavior ⇒ Physically plausible<br />
  12. 12. Adaptive Space Deformations Based on Rigid Cells<br />Design decisions:<br />– Realistic behavior ⇒ Physically plausible<br />– Large deformations ⇒ Nonlinear energy<br />nonlinear<br />linear<br />
  13. 13. Adaptive Space Deformations Based on Rigid Cells<br />Design decisions:<br />– Realistic behavior ⇒ Physically plausible<br />– Large deformations ⇒ Nonlinear energy<br />– Robustness ⇒ Rigid cells<br />
  14. 14. Adaptive Space Deformations Based on Rigid Cells<br />Design decisions:<br />– Realistic behavior ⇒ Physically plausible<br />– Large deformations ⇒ Nonlinear energy<br />– Robustness ⇒ Rigid cells<br />– Applicability ⇒ Space deformation<br />
  15. 15. AdaptiveSpace Deformations Based on Rigid Cells<br />Design decisions:<br />– Realistic behavior ⇒ Physically plausible<br />– Large deformations ⇒ Nonlinear energy<br />– Robustness ⇒ Rigid cells<br />– Applicability ⇒ Space deformation<br />– Performance ⇒ Adaptive discretization<br />
  16. 16. Deformation Pipeline<br />
  17. 17. Deformation Energy<br />• Continuum mechanics<br />– Strain energy defined by displacement’s gradient<br />– Local variation of displacement causes stretching<br />• Discrete rigid cells<br />– Each cell stores rigid motion<br />– Local differences of rigid transformations<br />
  18. 18. Difference of Transformations<br />• Frobenius norm of matrices<br />– Geometric meaning of matrix elements?<br />• [Pottmann et al, 04]<br />– Difference of images of sample points (which?)<br />
  19. 19. Nonlinear Energy<br />• Integrate over neighboring cells’ interiors<br />
  20. 20. Nonlinear Energy<br />• Integrate over neighboring cells’ interiors<br />• Accumulated global energy<br />
  21. 21. Nonlinear Minimization<br />• Find rigid motion per cell <br />• Generalized shape matching [Botsch et al, SGP 06]<br />– Robust geometric optimization<br />– Nonlinear Newton-type minimization<br />
  22. 22. Nonlinear Minimization<br />• Generalized shape matching [Botsch et al, SGP 06]<br />– Affine transform approximation (Linearization)<br />– Nonlinear Newton-type minimization<br /> - Linear solve followed by rigid transform projection<br />
  23. 23. Robust Optimization<br />
  24. 24. Deformation Pipeline<br />
  25. 25. Volumetric Discretization<br />• Octree-like adaptive voxelization<br /> – Easy to implement, efficient to compute<br /> – Analytic integration<br />• T-junctions are no problem !<br />
  26. 26. Boundary Refinement<br />
  27. 27. Energy-Driven Refinement<br />
  28. 28. 3D Adaptive Refinement<br />
  29. 29. 3D Adaptive Refinement<br />
  30. 30. 3D Adaptive Refinement<br />
  31. 31. Deformation Pipeline<br />
  32. 32. Deformation Interpolation<br />Final:<br />RBF Interpolation<br />Preview:<br />Averaging / Skinning<br />
  33. 33. Complex Models<br />
  34. 34. Triangle Soup<br />
  35. 35. Aliasing<br />
  36. 36. Conclusion<br />• Physically plausible deformation energy<br /> – Shells & solids, 2D & 3D<br /> – Arbitrary convex cell arrangements<br /> – Robust geometric shape matching<br />• Adaptive discretization<br /> – Static and dynamic refinement<br />• Smooth space deformation<br /> – Meshes, triangle soups, point sets, ...<br />
  37. 37. 질문은 손들고<br />

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