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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
12 - Spatial And Skeletal Deformations
![Page 2: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/2.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Space Deformations
![Page 3: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/3.jpg)
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]
![Page 4: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/4.jpg)
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
![Page 5: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/5.jpg)
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
![Page 6: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/6.jpg)
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
![Page 7: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/7.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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
object • Smooth deformations around the curve with decreasing
influence
http://www.dgp.toronto.edu/~karan/pdf/ksinghpaperwire.pdf
![Page 9: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/9.jpg)
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
![Page 10: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/10.jpg)
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
![Page 11: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/11.jpg)
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
![Page 12: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/12.jpg)
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 evaluation • Function d transforms points • Jacobian-T ∇d-T transforms normals • Precompute basis functions• Evaluate on the GPU!
![Page 13: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/13.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Local & Global Deformations [Real-Time Shape Editing using Radial Basis Functions, Botsch and Kobbelt, EUROGRAPHICS 2005]
![Page 14: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/14.jpg)
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
![Page 15: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/15.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Space DeformationsSummary so far
• Handle arbitrary input • Meshes (also non-manifold) • Point sets • Polygonal soups • …
• Complexity mainly depends on the control object, not the surface
▪ 3M triangles ▪ 10k components ▪ Not oriented ▪ Not manifold
![Page 16: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/16.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Space DeformationsSummary so far
• Handle arbitrary input • Meshes (also non-manifold) • Point sets • Polygonal soups • …
![Page 17: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/17.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Space DeformationsSummary 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
![Page 18: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/18.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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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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. to the cage elements using coordinate functions
x pi
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]
pʹix pi
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]
x pi
pʹi
xʹ
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]
x pi
pʹi
xʹ
http://www.cs.wustl.edu/~taoju/research/meanvalue.pdf
![Page 24: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/24.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Generalized Barycentric Coordinates
• Lagrange property:
• Reproduction:
• Partition of unity:
![Page 25: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/25.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
2D Mean Value Coordinates
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
3D Mean Value Coordinates
Mean Value Coordinates for Closed Triangular Meshes Tao Ju, Scott Schaefer, Joe Warren
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Coordinate Functions
• Mean-value coordinates [Floater 2003, Ju et al. 2005] • Not necessarily positive on non-
convex domains
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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 HChttp://www.cs.jhu.edu/~misha/Fall07/Papers/Joshi07.pdf
![Page 30: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/30.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Coordinate Functions
• Harmonic coordinates (Joshi et al. 2007)
MVC HC
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Coordinate Functions
• Green coordinates (Lipman et al. 2008)
• Correction: Make the coordinates depend on the cage faces as well
![Page 34: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/34.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Coordinate Functions
• Green coordinates (Lipman et al. 2008)
• Closed-form solution
• Conformal in 2D, quasi-conformal in 3D
MVCGC GC
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CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Cage-based methods: Summary
Pros:
• Nice control over volume • Squish/stretch
Cons:
• Hard to control details of embedded surface
![Page 36: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/36.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Linear Blend Skinning (LBS)
Acknowledgement: Alec Jacobson
![Page 37: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/37.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
LBS generalizes to different handle types
skeletons regions points cages
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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
![Page 39: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/39.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Linear Blend Skinning rigging preferred for its real-time performance
place handles in shape paint weights
![Page 40: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/40.jpg)
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
![Page 41: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/41.jpg)
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
![Page 42: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/42.jpg)
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
![Page 43: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/43.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
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
![Page 44: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/44.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Properties of the Weights
Handle vertices
Interpolation of handlesPartition of unity
is linear along cage faces
![Page 45: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/45.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Weights Should Be Positive
Unconstrained biharmonic [Botsch and Kobbelt 2004]
Bounded Biharmonic Weights[Jacobson et al. 2011]
![Page 46: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/46.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Weights Should Be Smooth
Bounded Biharmonic Weights
Extension of Harmonic Coordinates [Joshi et al. 2005]
[Jacobson et al. 2011]
![Page 47: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/47.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Weights Should Be Smooth
Bounded Biharmonic Weights Extension of Harmonic Coordinates [Joshi et al. 2005]
![Page 48: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/48.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Bounded biharmonic weights enforce properties as constraints to minimization
is linear along cage faces
![Page 49: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/49.jpg)
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
Partition of unity
![Page 50: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/50.jpg)
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
![Page 51: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/51.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Some examples of LBS in action
![Page 52: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/52.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Some examples of LBS in action
![Page 53: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/53.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Some examples of LBS in action
![Page 54: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/54.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
3D Characters
![Page 55: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/55.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
Mixing different handle types
![Page 56: 12 - Spatial and Skeletal Deformationspanozzo/cg18/Slides/12 - Spatial and... · 2018. 9. 23. · CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo Freeform Deformations •](https://reader036.vdocuments.net/reader036/viewer/2022071505/61262e0dde97a33bb81e0b83/html5/thumbnails/56.jpg)
CSCI-GA.2270-001 - Computer Graphics - Daniele Panozzo
ReferencesFundamentals of Computer Graphics, Fourth Edition 4th Edition by Steve Marschner, Peter ShirleyChapter 16
Skinning: Real-time Shape DeformationACM SIGGRAPH 2014 Coursehttp://skinning.org