10 - ARAP and Linear Blend Skinning

Transcription

1 10 - ARAP and Linear Blend Skinning Acknowledgements: Olga Sorkine-Hornung

2 As Rigid As Possible

3 Demo Libigl demo 405

4 As-Rigid-As-Possible Deformation Preserve shape of cells covering the surface Ask each cell i to transform rigidly by best-fitting rotation R i

5 As-Rigid-As-Possible Deformation Optimal R i is uniquely defined by, R i is a nonlinear so-called shape-matching problem, function of x solved by a 3x3 SVD

6 Optimal Rotation Rotation group

7 Shape Matching Problem

8 Shape Matching Problem

9 Shape Matching Problem

10 Shape Matching Problem

11 Shape Matching Problem Align two point sets Find a translation vector t and rotation matrix R so that

12 Shape Matching Solution Solve for translation first (w.r.t. R, p, and q) Point sets {q i } and {Rp i } have the same center of mass

13 Finding the Rotation R To find the optimal R, we bring the centroids of both point sets to the origin We want to find R that minimizes

14 Finding the Rotation R I These terms do not depend on R, so we can ignore them in the minimization

15 Finding the Rotation R

16 Finding the Rotation R R =

17 Finding the Rotation R =

18 Finding the Rotation R Take a look at the Find R that maximizes Matrix Cookbook! SVD: orthonormal matrix

19 Finding the Rotation R We want to maximize M: orthonormal matrix all entries 1

20 Finding the Rotation R Our best shot is m ii = 1, i.e. to make M = I

21 Summary of Rigid Alignment Translate the input points to the centroids Compute the covariance matrix Compute its SVD: The optimal orthonormal R is

22 Sign Correction It is possible that : sometimes reflection is the best orthonormal transform

23 Sign Correction It is possible that : sometimes reflection is the best orthonormal transform

24 Sign Correction To restrict ourselves to rotations only: take the last column of U (corresponding to the smallest singular value) and invert its sign. Why? See

25 As-Rigid-As-Possible Deformation Optimal R i is uniquely defined by, R i is a nonlinear so-called shape-matching problem, function of x solved by a 3x3 SVD

26 As-Rigid-As-Possible Deformation Total ARAP energy: sum up for all the cells i Treat x and R as separate sets of variables Simple local-global iterative optimization process Decreases the energy at each step

27 As-Rigid-As-Possible Deformation Total ARAP energy: sum up for all the cells i Local step: keep x fixed, find optimal R i per cell i Global step: keep R i fixed, solve for x quadratic minimization problem

28 As-Rigid-As-Possible Deformation Total ARAP energy: sum up for all the cells i Local step: keep x fixed, find optimal R per cell i i Global step: keep R fixed, solve for x i quadratic minimization problem The matrix L stays fixed, can pre-factorize

29 Initial Guess Can use naïve Laplacian editing initial guess 1 iteration 2 iterations initial guess 1 iterations 4 iterations

30 Initial Guess Can also use the previous frame Replace all handle vertex positions by the currently prescribed ones Fast convergence

31 Large Rotations Use previous frame as the initial guess

32 Examples

33 Discussion Nonlinear deformation that models a kind of elastic behavior Very simple to implement, no parameters to tune except number of iterations Each step is guaranteed to not increase the energy Compare with Gauss-Newton Each iteration is relatively cheap, no matrix refactorization necessary

34 Discussion Works fine on small meshes On larger meshes: slow convergence Each iteration is more expensive Need more iterations because the conditioning of the system becomes worse as the matrix grows Material stiffness depends on the cell size lots of wrinkles for fine meshes when using 1-rings as cells

35 Acceleration using Subspace Techniques Subspace created by influence weight functions for each handle Drastically reduces the number of degrees of freedom in the optimization Alec Jacobson, Ilya Baran, Ladislav Kavan, Jovan Popović, and Olga Sorkine. Fast Automatic Skinning Transformations, 2012.

36 Acceleration using Subspace Techniques Subspace created by influence weight functions for each handle Drastically reduces the number of degrees of freedom in the optimization Alec Jacobson, Ilya Baran, Ladislav Kavan, Jovan Popović, and Olga Sorkine. Fast Automatic Skinning Transformations, 2012.

37 Acceleration using Subspace Techniques Subspace created by influence weight functions for each handle Drastically reduces the number of degrees of freedom in the optimization Alec Jacobson, Ilya Baran, Ladislav Kavan, Jovan Popović, and Olga Sorkine. Fast Automatic Skinning Transformations, 2012.

38 Demo Libigl demo 406

39 Linear Blend Skinning Acknowledgements: Alec Jacobson

40 LBS generalizes to different handle types skeletons regions points cages

41 Linear Blend Skinning rigging preferred for its realtime performance place handles in shape

42 Linear Blend Skinning rigging preferred for its realtime performance place handles in shape paint weights

43 Linear Blend Skinning rigging preferred for its realtime performance place handles in shape paint weights deform handles

44 Linear Blend Skinning rigging preferred for its realtime performance place handles in shape paint weights deform handles

45 Linear Blend Skinning rigging preferred for its realtime performance place handles in shape paint weights deform handles

46 Challenges with LBS Weight functions w j Need intuitive, general and automatic weights Degrees of freedom T j Let the energy decide! Richness of achievable deformations Want to avoid common LBS pitfalls candy wrapper, collapses

47 Bounded Biharmonic Weights Alec Jacobson, Ilya Baran, Jovan Popović, Olga Sorkine-Hornung

48 Automatic weights that unify points, skeletons and cages

49 Weights should be smooth, shape-aware, positive and intuitive

50 Weights must be smooth everywhere, especially at handles Bounded Biharmonic Weights Extension of Harmonic Coordinates [Joshi et al. 2005]

51 Weights must be smooth everywhere, especially at handles Bounded Biharmonic Weights Extension of Harmonic Coordinates [Joshi et al. 2005]

52 Shape-awareness ensures respect of domain s features Bounded Biharmonic Weights Non-shape-aware methods e.g. [Schaefer et al. 2006]

53 Non-negative weights are necessary for intuitive response Bounded Biharmonic Weights Unconstrained biharmonic [Botsch and Kobbelt 2004]

54 Weights must maintain other simple, but important properties Handle vertices is linear along cage faces Partition of unity Interpolation of handles

55 How about w j (x 0 ) = d (x 0, H j ) 1?

56 Inverse distance methods inherently suffer from fall-off effect

57 Inverse distance methods inherently suffer from fall-off effect

58 Inverse distance methods inherently suffer from fall-off effect Approaching 0.5

59 Inverse distance methods inherently suffer from fall-off effect Inverse- distance weights BBW

60 Bounded biharmonic weights enforce properties as constraints to minimization is linear along cage faces

61 Bounded biharmonic weights enforce properties as constraints to minimization Constant inequality constraints Partition of unity is linear along cage faces

62 Bounded biharmonic weights enforce properties as constraints to minimization Constant inequality constraints Solve independently and normalize is linear along cage faces

63 Weights optimized as precomputation at bind-time is linear along cage faces FEM discretization 2D! Triangle mesh 3D! Tet mesh

64 Weights optimized as precomputation at bind-time is linear along cage faces Sparse quadratic programming with constant inequality constraints 2D! less than second per handle 3D! tens of seconds per handle

65 Some examples of BBW in action

66 Some examples of BBW in action

67 Some examples of BBW in action

68 3D Characters

69 Demo Libigl demo 403

70 Thank you