1. Computer Graphics Group Tobias Weyand Mesh-Based Inverse Kinematics Sumner et al 2005 presented by Tobias Weyand

2. Computer Graphics Group Tobias Weyand 2 What is Inverse Kinematics? Articulated body Which angles for a certain configuration? Forward kinematics Specify angles Inverse kinematics Specify limb position 11 22 33 11 22 33

3. Computer Graphics Group Tobias Weyand 3 IK in Computer Graphics © NVidia Modelling Meaningful deformations Animation Realistic movement Problems: Skinning time consuming Not everything has bones

4. Computer Graphics Group Tobias Weyand 4 Mesh-based Inverse Kinematics Idea: Learn deformations from examples

5. Computer Graphics Group Tobias Weyand 5 Overview Introduction Related work MeshIK Feature Vectors Linear Feature Space Nonlinear Feature Space Accelerations Results and Conclusion

6. Computer Graphics Group Tobias Weyand 6 Related work: Deformation Transfer Deformation Transfer for Triangle Meshes Transfer source deformations to target Sumner et al 2004

7. Computer Graphics Group Tobias Weyand 7 Related work: Deformation Transfer Triangle deformations:  Least squares problem: But: Limited to example poses

8. Computer Graphics Group Tobias Weyand 8 Related work: Shape Interpolation As-rigid-as-possible Shape Interpolation Morphing of 2D and 3D meshes Considers mesh interior as rigid Alexa et al. 2000

9. Computer Graphics Group Tobias Weyand 9 Related work: Shape Interpolation Triangulate source and target shapes Find locally optimal triangle interpolations But: Limited to 2 meshes Expensive: Compatible dissection Computations on interior and exterior Better: Use only surface

10. Computer Graphics Group Tobias Weyand 10 Mesh-based Inverse Kinematics Goal: Provide a set of example meshes. MeshIK learns meaningful deformations. Directly move a subset of the mesh vertices. MeshIK finds a suitable deformation according to the example meshes.

11. Computer Graphics Group Tobias Weyand 11 Feature Vectors Given: Base mesh P 0, deformed mesh P Deformation: set of affine mappings Deformation gradient: Jacobian of  Discard translation

12. Computer Graphics Group Tobias Weyand 12 Feature Vectors Feature vector: concatentaion of deformation gradients

13. Computer Graphics Group Tobias Weyand 13 Feature Vectors Calculation of f for mesh P P → mesh-vector x: Construct G such that:

14. Computer Graphics Group Tobias Weyand 14 Feature Vectors Properties of G: - Block-diagonal structure - Sparse - Only depends on P 0

15. Computer Graphics Group Tobias Weyand 15 Feature Vectors Extracting a mesh from a feature vector: Fix one vertex in x: Set corresponding rows in G to 0. Add product of these rows with x.

16. Computer Graphics Group Tobias Weyand 16 Multiple Vertex Constraints Transform to least squares problem:  Properties of x: Close relation to the feature vector Fulfills the vertex constraints

17. Computer Graphics Group Tobias Weyand 17 Linear Feature Space Linear combination of features: Vector notation:  Least squares problem Minimize for optimal weights and mesh

18. Computer Graphics Group Tobias Weyand 18 Linear Feature Space: Problems Unnatural interpolation of rotations Goal: Correctly capture rotations

19. Computer Graphics Group Tobias Weyand 19 Transformation Interpolation Linear Combinations of Transformations Scalar product: Addition: Combination: Alexa 2002

20. Computer Graphics Group Tobias Weyand 20 Possible Nonlinear Feature Space But: Practically produces singularities Linear scales and shears suffice MeshIK interpolation: Scales and skews: linear Rotations: above formula

21. Computer Graphics Group Tobias Weyand 21 Extracting Rotations Matrix Animation and Polar Decomposition Method for factoring a transformation T T=RS  New transformation combination Shoemake and Duff 1992

22. Computer Graphics Group Tobias Weyand 22 Nonlinear Feature Space New nonlinear least squares problem: Properties: Fulfills vertex constraints Close to nonlinear feature space Natural rotation interpolation Efficient solver required!

23. Computer Graphics Group Tobias Weyand 23 Gauss-Newton in MeshIK Goal: minimize Approach with Gauss-Newton: Transform into locally linear equation Solve linear least squares problem

24. Computer Graphics Group Tobias Weyand 24 Gauss-Newton in MeshIK Linearize M(w) with Taylor expansion  Linear least squares problem Reduce to one variable: 

25. Computer Graphics Group Tobias Weyand 25 Gauss-Newton in MeshIK Set = 0 !  Iteration steps: Solve the normal equation for Update But: Far too slow! Optimization needed!

26. Computer Graphics Group Tobias Weyand 26 Cholesky Factorization Linear system: Decompose: Method:

27. Computer Graphics Group Tobias Weyand 27 Cholesky Factorization Further acceleration: Exploit structure of A T A and C T C: Note: R T R=g T g Precompute R T R Only calculate R 1, R 2, R 3, R S  Interactive speed

28. Computer Graphics Group Tobias Weyand 28 Performance MeshTrisExamplesPreprocessSolve Flag932140.0160.020 Lion9,996100.04750.210 Horse16,84640.6100.160 Elephant 84,638413.2490.906

29. Computer Graphics Group Tobias Weyand 29 Results

30. Computer Graphics Group Tobias Weyand 30 Video

31. Computer Graphics Group Tobias Weyand 31 Comparison: Shape Interpolation Arap.MeshIK 2 meshesn meshes Needs compatible dissection of meshes → expensive Needs meshes with same topology Operates on exterior and interior Operates only on surface

32. Computer Graphics Group Tobias Weyand 32 Comparison: Shape Interpolation

33. Computer Graphics Group Tobias Weyand 33 Future work Different mesh representation like subdivision surfaces multiresolution hierarchies Different feature vectors Capture mesh properties better Accelerate system solving Better feature space for many examples Simulate physical effects like inertia

34. Computer Graphics Group Tobias Weyand 34 Conclusion MeshIK provides IK without skeleton more direct and dynamic runs at interactive speeds compares well to other approaches eg As-Rigid-As-Possible shape interpolation

35. Computer Graphics Group Tobias Weyand 35 Thank you for your attention!