12 - Spatial And Skeletal Deformations. CSCI-GA Computer Graphics - Fall 16 - Daniele Panozzo

Transcription

1 12 - Spatial And Skeletal Deformations

2 Space Deformations

3 Space Deformation Displacement function defined on the ambient space Evaluate the function on the points of the shape embedded in the space Twist warp Global and local deformation of solids [A. Barr, SIGGRAPH 84]

4 Freeform Deformations Control object User defines displacements d i for each element of the control object Displacements are interpolated to the entire space using basis functions Basis functions should be smooth for aesthetic results

5 Freeform Deformation [Sederberg and Parry 86] Control object = lattice Basis functions B i (x) are trivariate tensor-product splines:

6 Freeform Deformation [Sederberg and Parry 86] Aliasing artifacts Interpolate deformation constraints? Only in least squares sense

7 Limitations of Lattices as Control Objects Difficult to manipulate The control object is not related to the shape of the edited object Parts of the shape in close Euclidean distance always deform similarly, even if geodesically far

8 Wires [Singh and Fiume 98] Control objects are arbitrary space curves Can place curves along meaningful features of the edited object Smooth deformations around the curve with decreasing influence

9 Handle Metaphor [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005] Wish list for the displacement function d(x) : Interpolate prescribed constraints Smooth, intuitive deformation

10 Radial Basis Functions [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005] Represent deformation by RBFs Triharmonic basis function ϕ (r) = r 3 C 2 boundary constraints Highly smooth / fair interpolation

11 Radial Basis Functions [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005] Represent deformation by RBFs RBF fitting Interpolate displacement constraints Solve linear system for w j and p

12 Radial Basis Functions [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005] Represent deformation by RBFs RBF evaluation Function d transforms points Jacobian -T d -T transforms normals Precompute basis functions Evaluate on the GPU!

13 Local & Global Deformations [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005]

14 Local & Global Deformations [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005] 1M vertices

15 Space Deformations Summary so far Handle arbitrary input Meshes (also non-manifold) Point sets Polygonal soups Complexity mainly depends on the control object, not 3M triangles 10k components Not oriented Not manifold the surface

16 Space Deformations Summary so far Handle arbitrary input Meshes (also non-manifold) Point sets Polygonal soups

17 Space Deformations Summary so far The deformation is only loosely aware of the shape that is being edited Small Euclidean distance similar deformation Local surface detail may be distorted

18 Cage-based Deformations [Ju et al. 2005] Cage = crude version of the input shape Polytope (not a lattice)

19 Cage-based Deformations [Ju et al. 2005] Each point x in space is represented w.r.t. the cage elements using coordinate functions v x p i

20 Cage-based Deformations [Ju et al. 2005] Each point x in space is represented w.r.t. to the cage elements using coordinate functions x p i

21 Cage-based Deformations [Ju et al. 2005] x p i pʹi

22 Cage-based Deformations [Ju et al. 2005] xʹ x p i pʹi

23 Cage-based Deformations [Ju et al. 2005] xʹ x p i pʹi

24 Generalized Barycentric Coordinates Lagrange property: Reproduction: Partition of unity:

25 Coordinate Functions Mean-value coordinates [Floater 2003 *, Ju et al. 2005] Generalization of barycentric coordinates Closed-form solution for w (x) i * Michael Floater, Mean value coordinates, CAGD 20(1), 2003

26 2D Mean Value Coordinates

27 3D Mean Value Coordinates Mean Value Coordinates for Closed Triangular Meshes Tao Ju, Scott Schaefer, Joe Warren

28 Coordinate Functions Mean-value coordinates [Floater 2003, Ju et al. 2005] Not necessarily positive on nonconvex domains

29 Coordinate Functions Harmonic coordinates (Joshi et al. 2007) Harmonic functions h i (x) for each cage vertex p i Solve Δ h = 0 subject to: h i linear on the boundary s.t. h i (p j ) = δ ij MVC HC

30 Coordinate Functions Harmonic coordinates (Joshi et al. 2007) Harmonic functions h i (x) for each cage vertex p i Solve Δ h = 0 subject to: h i linear on the boundary s.t. h i (p j ) = δ ij Volumetric Laplace equation Discretization, no closed-form

31 Coordinate Functions Harmonic coordinates (Joshi et al. 2007) MVC HC

32 Coordinate Functions Green coordinates (Lipman et al. 2008) Observation: previous vertex-based basis functions always lead to affine-invariance!

33 Coordinate Functions Green coordinates (Lipman et al. 2008) Correction: Make the coordinates depend on the cage faces as well

34 Coordinate Functions Green coordinates (Lipman et al. 2008) Closed-form solution Conformal in 2D, quasi-conformal in 3D GC MVC GC

35 Cage-based methods: Summary Pros: Nice control over volume Squish/stretch Cons: Hard to control details of embedded surface

36 Linear Blend Skinning (LBS) Acknowledgement: Alec Jacobson

37 LBS generalizes to different handle types skeletons regions points cages

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

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

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

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

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

43 Challenges with LBS Weight functions w j Can be manually painted or automatically generated Degrees of freedom T j Exposed to the user (possibly with a kinematic chain) Richness of achievable deformations Want to avoid common pitfalls candy wrapper, collapses

44 Properties of the Weights Handle vertices is linear along cage faces Partition of unity Interpolation of handles

45 Weights Should Be Positive Bounded Biharmonic Weights [Jacobson et al. 2011] Unconstrained biharmonic [Botsch and Kobbelt 2004]

46 Weights Should Be Smooth Bounded Biharmonic Weights Extension of Harmonic Coordinates [Joshi et al. 2005] [Jacobson et al. 2011]

47 Weights Should Be Smooth Bounded Biharmonic Weights Extension of Harmonic Coordinates [Joshi et al. 2005]

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

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

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

51 Some examples of LBS in action

52 Some examples of LBS in action

53 Some examples of LBS in action

54 3D Characters

55 Mixing different handle types

56 References Fundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter Shirley Chapter 16 Skinning: Real-time Shape Deformation ACM SIGGRAPH 2014 Course