12 - Spatial And Skeletal Deformations CSCI-GA.2270-001 - Computer - - PowerPoint PPT Presentation

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12 - Spatial And Skeletal Deformations CSCI-GA.2270-001 - Computer - - PowerPoint PPT Presentation

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12 - Spatial And Skeletal Deformations CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Space Deformations CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Space Deformation Displacement function defined on the ambient space

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

12 - Spatial And Skeletal Deformations

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Space Deformations

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

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]

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Freeform Deformations

  • Control object
  • User defines displacements di 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

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Freeform Deformation 


[Sederberg and Parry 86]

  • Control object = lattice
  • Basis functions Bi (x) are 


trivariate tensor-product splines:

http://tom.cs.byu.edu/~tom/papers/ffd.pdf

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Freeform Deformation 


[Sederberg and Parry 86]

  • Aliasing artifacts
  • Interpolate deformation constraints?
  • Only in least squares sense

http://tom.cs.byu.edu/~tom/papers/ffd.pdf

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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

http://tom.cs.byu.edu/~tom/papers/ffd.pdf

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Wires


[Singh and Fiume 98]

  • Control objects are arbitrary space curves
  • Can place curves along meaningful features of the edited
  • bject
  • Smooth deformations around the curve with decreasing

influence

http://www.dgp.toronto.edu/~karan/pdf/ksinghpaperwire.pdf

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

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
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

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
  • C2 boundary constraints
  • Highly smooth / fair interpolation
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

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 wj and p
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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!
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Local & Global Deformations


[Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005]

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Local & Global Deformations


[Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005]

1M vertices

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Space Deformations


Summary so far

  • Handle arbitrary input
  • Meshes (also non-manifold)
  • Point sets
  • Polygonal soups
  • …
  • Complexity mainly depends

  • n the control object, not 


the surface

▪ 3M triangles ▪ 10k components ▪ Not oriented ▪ Not manifold

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Space Deformations


Summary so far

  • Handle arbitrary input
  • Meshes (also non-manifold)
  • Point sets
  • Polygonal soups
  • …
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

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
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Cage-based Deformations


[Ju et al. 2005]

  • Cage = crude version of the input shape
  • Polytope (not a lattice)

http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Cage-based Deformations


[Ju et al. 2005]

  • Each point x in space is represented w.r.t. the cage elements using

coordinate functions

vx

pi

http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf

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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 pi

http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf

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Cage-based Deformations


[Ju et al. 2005]

pʹi x pi

http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf

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Cage-based Deformations


[Ju et al. 2005]

x pi pʹi xʹ

http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf

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Cage-based Deformations


[Ju et al. 2005]

x pi pʹi xʹ

http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf

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Generalized Barycentric Coordinates

  • Lagrange property:
  • Reproduction:
  • Partition of unity:
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Coordinate Functions

  • Mean-value coordinates 


[Floater 2003*, Ju et al. 2005]

  • Generalization of barycentric coordinates
  • Closed-form solution for wi (x)

* Michael Floater, “Mean value coordinates”, CAGD 20(1), 2003

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2D Mean Value Coordinates

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3D Mean Value Coordinates

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

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Coordinate Functions

  • Mean-value coordinates 


[Floater 2003, Ju et al. 2005]

  • Not necessarily positive on non-

convex domains

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Coordinate Functions

  • Harmonic coordinates (Joshi et al. 2007)
  • Harmonic functions hi(x) for each cage vertex pi
  • Solve

subject to: hi linear on the boundary s.t. hi (pj) = δij

Δ h = 0

MVC HC

http://www.cs.jhu.edu/~misha/Fall07/Papers/Joshi07.pdf

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Coordinate Functions

  • Harmonic coordinates (Joshi et al. 2007)
  • Harmonic functions hi(x) for each cage vertex pi
  • Solve

subject to: hi linear on the boundary s.t. hi (pj) = δij

  • Volumetric Laplace equation
  • Discretization, no closed-form

Δ h = 0

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Coordinate Functions

  • Harmonic coordinates (Joshi et al. 2007)

MVC HC

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Coordinate Functions

  • Green coordinates (Lipman et al. 2008)
  • Observation: previous vertex-based basis functions always lead to

affine-invariance!

http://www.wisdom.weizmann.ac.il/~ylipman/GC/gc.htm

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Coordinate Functions

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

  • Green coordinates (Lipman et al. 2008)
  • Closed-form solution
  • Conformal in 2D, quasi-conformal in 3D

MVC GC GC

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Cage-based methods: Summary

Pros:

  • Nice control over volume
  • Squish/stretch

Cons:

  • Hard to control details of embedded surface
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Linear Blend Skinning (LBS)

Acknowledgement: Alec Jacobson

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LBS generalizes to different handle types

skeletons regions points cages

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Linear Blend Skinning rigging preferred for its real- time performance

place handles in shape

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Linear Blend Skinning rigging preferred for its real- time performance

place handles in shape paint weights

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Linear Blend Skinning rigging preferred for its real- time performance

place handles in shape paint weights deform handles

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Linear Blend Skinning rigging preferred for its real- time performance

place handles in shape paint weights deform handles

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Linear Blend Skinning rigging preferred for its real- time performance

place handles in shape paint weights deform handles

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Challenges with LBS

  • Weight functions wj
  • Can be manually painted or

automatically generated

  • Degrees of freedom Tj
  • Exposed to the user (possibly with a

kinematic chain)

  • Richness of achievable 


deformations

  • Want to avoid common pitfalls – candy

wrapper, collapses

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Properties of the Weights

Handle vertices

Interpolation of handles Partition of unity is linear along cage faces

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Weights Should Be Positive

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

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Weights Should Be Smooth

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

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Weights Should Be Smooth

Bounded Biharmonic Weights Extension of Harmonic Coordinates

[Joshi et al. 2005]

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Bounded biharmonic weights enforce properties as constraints to minimization

is linear along cage faces

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Bounded biharmonic weights enforce properties as constraints to minimization

is linear along cage faces

Constant inequality constraints Partition of unity

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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo

Bounded biharmonic weights enforce properties as constraints to minimization

is linear along cage faces

Constant inequality constraints Solve independently and normalize

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Some examples of LBS in action

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Some examples of LBS in action

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Some examples of LBS in action

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3D Characters

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Mixing different handle types

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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 http://skinning.org