CS 523: Computer Graphics, Spring 2009 Shape Modeling Skeletal deformation 4/22/2009 1
Believable character animation Computers games and movies Skeleton: intuitive, low dimensional subspace Clip courtesy of Ilya Baran 4/22/2009 2
Discrete representation Skeleton: Skin: collection of line segments discrete samples of the connected by joints surface polygonal mesh 4/22/2009 3
Skin + skeleton Skeleton Seeo defines es the overall motion Skin moves oeswith the skeleton The process of building the skeleton kl and binding it to the skin mesh is called rigging. i 4/22/2009 Clips courtesy of Ilya Baran 4
Skeletal subspace deformation (SSD) The artist needs to specify, for each point on the skin, how much itisinfluencedis influenced by the skeleton bones. 4/22/2009 5
Skeletal subspace deformation (SSD) Affine combination of transformations K v = w T v j kj k j k = 1 v j T 2 T 1 De facto standard for interactive applications simple + fast + works on the GPU T 3 4/22/2009 6
Skeletal subspace deformation (SSD) Hard to set up Visual artifacts No context 4/22/2009 7
Pose space deformation (PSD) [Lewis et al. 2000, Sloan et al. 2001] Each degree of freedom of the skeleton is a dimension: P = ( α, β, γ, α, β, γ, K, α, β, γ ) 1 1 1 2 2 2 K K K P 0 P 2 P P D P 1 4/22/2009 8
Pose space deformation (PSD) [Lewis et al. 2000, Sloan et al. 2001] P0 P1 P 2 PD P Radial Basis functions: a ( P ) = h j D ( P ) + m Φ ( P - P ) j i, j i i i= 1 a 0 a 1 a 2 a D M Blend Linear displacements Deformed Shape 4/22/2009 9
Pose space deformation (PSD) [Lewis et al. 2000, Sloan et al. 2001] 4/22/2009 10
SSD artifacts, requires many examples + setup PSD limitations [Lewis et al. 2000, Sloan et al. 2001] Linear displacements no rotation High memory consumption, performance L 4/22/2009 11
Rotation interpolation and extrapolation 4/22/2009 12
Linear displacements (PSD) 4/22/2009 13
Context Aware Skeletal Shape Deformation Eurographics 2007 Ofir Weber Olga Sorkine Yaron Lipman Craig Gotsman 4/22/2009 14
The contributions Replace SSD by detail preserving pese mesh deformation [Sorkine et al. 2004, Sumner et al. 2004, Yu et al. 2004, Lipman et al. 2005, Zayer et al. 2005] Easy setup Differential morphing Sparse representation of example shapes 4/22/2009 15
Other previous work Pose Space Deformation eo o [Lewis et al. 2000, Sloan et al. 2001, Kry et al. 2002, Kurihara et al. 2004, Rhee et al. 2006] Detail preserving mesh deformation [Sorkine et al. 2004, Sumner et al. 2004, Yu et al. 2004, Lipman et al. 2005, Zayer et al. 2005 ] Survey: [Botsch and Sorkine 2008] MeshIK [Sumner et al. 2005, Der et al. 2006] SCAPE [Anguelov et al. 2005] 4/22/2009 16
Detail preserving Detail preserving deformation 1 Hip 0.5 Knee Ankle Δw k = 0 0 0 Dirichlet boundary y conditions: wk(tn) = 1 for tn Hk wk(tn) = 0 for tn Hl where l k. 4/22/2009 17
Blending rotations For each face t: R(t)= w 1 (t)r 1 w 2 (t)r 2 w K (t)r K : [Buss 93] R 1 R 3 log quaternion R 2 Poisson equation [Yu et al. 2004] Δ [ x y z ] = div[ R ] Sparse linear system 4/22/2009 18
Poisson stitching The Poisson equation averages the different vertex positions Tries to preserve the shape and orientation of thetriangles triangles as much as possible 4/22/2009 19
Poisson stitching The Poisson equation averages the different vertex positions Tries to preserve the shape and orientation of thetriangles triangles as much as possible 4/22/2009 20
Setup 4/22/2009 21
Comparison to SSD Context-aware SSD SSD 4/22/2009 22
Comparison to SSD SSD CASSD Video: 0:0:18 4/22/2009 23
Using context examples 4/22/2009 24
Relative encoding R (t) A (t) T(t) A (t) = T(t)x R (t) T(t)= A (t) x R T (t) 4/22/2009 25
Relative encoding u 2 v 1 n v v u 1 2 R (t) g n n u A (t) = T(t)x R (t) T(t) = A (t) x R T (t) R T (t) applied to the example shape (+ stitched) = T(t) 4/22/2009 26
Blending transformations Polar Decomposition T Q 0 S 0 0 a 0 T 1 Q 1 S 1 a 1 R(t) a 2 T 2 Q 2 S 2 M M M a D t T D Q D S D 4/22/2009 27
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Smooth Difference Deformation without Examples Smooth Example 4/22/2009 29
Compact Representation T Transformations varies smoothly 1 Laplace equation T 2 T 3 T 4 Less than 5% memory Evaluation only at anchors performance Greedy selection ΔTT = 0 Boundary conditions: known T s at anchors See Least squares squares Meshes [Sorkine and Cohen Or 2004] 4/22/2009 30
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One more result Video: 3:55 4/22/2009 32
Conclusions Detail preserving skeletal shape deformation Easy setup No or small number of examples Interpolation ti and meaningful extrapolation ti Sparse representation of examples 4/22/2009 33
Limitations and extensions No dynamics The greedy algorithm is not optimal Map to GPU Wang et al. SIGGRAPH 2007 4/22/2009 34