Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes
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1 Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes Alec Jacobson ETH Zurich June 9, 2013
2 Deformation is an important phase in the life of a shape creation analysis and manipulation consumption modeling in Maya skinning deformation Image courtesy Romain Prévost 3d printing June 9, 2013 Alec Jacobson # 2
3 Deformation is an important phase in the life of a shape creation analysis and manipulation consumption modeling in Maya skinning deformation Image courtesy Romain Prévost 3d printing June 9, 2013 Alec Jacobson # 3
4 User constraints drive deformation toward goal procedural constraints explicit constraints June 9, 2013 Alec Jacobson # 4
5 User constraints drive deformation toward goal procedural constraints explicit constraints June 9, 2013 Alec Jacobson # 5
6 User constraints drive deformation toward goal procedural constraints explicit constraints June 9, 2013 Alec Jacobson # 6
7 Deformation applies to images as planar shapes non-convex cut-out cartoons entire image rectangle June 9, 2013 Alec Jacobson # 7
8 Deformation applies to images as planar shapes non-convex cut-out cartoons entire image rectangle June 9, 2013 Alec Jacobson # 8
9 Real-time performance critical for interactive design and animation 2D 3D June 9, 2013 Alec Jacobson # 9
10 Real-time performance critical for interactive design and animation 2D 3D June 9, 2013 Alec Jacobson # 10
11 Real-time performance critical for interactive design and animation June 9, 2013 Alec Jacobson # 11
12 Many previous techniques provide quality, but not speed high-quality solutions to nonlinear elasticity energy minimizations: ~seconds e.g. [Botsch et al. 2006] physically accurate muscle systems require off-line simulation e.g. [Teran et al. 2005] Image courtesy Joseph Teran June 9, 2013 Alec Jacobson # 12
13 Other techniques limit user interface June 9, 2013 Alec Jacobson # 13
14 Other techniques limit user interface e.g. [Ju et al. 2005] June 9, 2013 Alec Jacobson # 14
15 Other techniques limit user interface June 9, 2013 Alec Jacobson # 15
16 Other techniques limit user interface June 9, 2013 Alec Jacobson # 16
17 Other techniques limit user interface June 9, 2013 Alec Jacobson # 17
18 Each handle type has a specific task, more than just different modeling metaphor June 9, 2013 Alec Jacobson # 18
19 Each handle type has a specific task, more than just different modeling metaphor June 9, 2013 Alec Jacobson # 19
20 Each handle type has a specific task, more than just different modeling metaphor June 9, 2013 Alec Jacobson # 20
21 Real-time applications deserve ample interfaces and high-quality deformations SGP 2010 SIGGRAPH 2011 SGP 2012 June 9, 2013 Alec Jacobson # 21
22 Real-time applications deserve ample interfaces and high-quality deformations SGP 2010 SIGGRAPH 2011 SGP 2012 SIGGRAPH Asia 2011 June 9, 2013 Alec Jacobson # 22
23 Real-time applications deserve ample interfaces and high-quality deformations SGP 2010 SIGGRAPH 2011 SGP 2012 SIGGRAPH Asia 2011 SIGGRAPH 2012 June 9, 2013 Alec Jacobson # 23
24 Real-time applications deserve ample interfaces and high-quality deformations SGP 2010 SIGGRAPH 2011 SGP 2012 SIGGRAPH Asia 2011 SIGGRAPH 2012 SIGGRAPH 2013 June 9, 2013 Alec Jacobson # 24
25 Real-time applications deserve ample interfaces and high-quality deformations SGP 2010 SIGGRAPH 2011 SGP 2012 SIGGRAPH Asia 2011 SIGGRAPH 2012 SIGGRAPH 2013 June 9, 2013 Alec Jacobson # 25
26 Linear Blend Skinning has known artifacts but makes up in real-time performance load shape June 9, 2013 Alec Jacobson # 26
27 Linear Blend Skinning has known artifacts but makes up in real-time performance load shape place handles June 9, 2013 Alec Jacobson # 27
28 Linear Blend Skinning has known artifacts but makes up in real-time performance load shape place handles define weights (bind time) June 9, 2013 Alec Jacobson # 28
29 Linear Blend Skinning has known artifacts but makes up in real-time performance load shape place handles define weights (bind time) deform shape (pose time) June 9, 2013 Alec Jacobson # 29
30 Linear Blend Skinning has known artifacts but makes up in real-time performance load shape place handles define weights (bind time) w j T j June 9, 2013 Alec Jacobson # 30
31 Warping and deformation tasks require different user interactions points skeletons cages June 9, 2013 Alec Jacobson # 31
32 Weights should be smooth, shape-aware, positive and intuitive June 9, 2013 Alec Jacobson # 32
33 Weights should be smooth, shape-aware, positive and intuitive June 9, 2013 Alec Jacobson # 33
![Weights must be smooth everywhere, especially at handles our method [SIGGRAPH 2011]](/docs-images/84/90836151/images/34-0.jpg "extension of Harmonic Coordinates [Joshi et al.")
34 Weights must be smooth everywhere, especially at handles our method [SIGGRAPH 2011] extension of Harmonic Coordinates [Joshi et al. 2005] June 9, 2013 Alec Jacobson # 34
![Weights must be smooth everywhere, especially at handles our method [SIGGRAPH 2011]](/docs-images/84/90836151/images/35-0.jpg "extension of Harmonic Coordinates [Joshi et al.")
35 Weights must be smooth everywhere, especially at handles our method [SIGGRAPH 2011] extension of Harmonic Coordinates [Joshi et al. 2005] June 9, 2013 Alec Jacobson # 35
![2011] extension of Harmonic Coordinates](/docs-images/84/90836151/images/36-1.jpg "[Joshi et al.")
36 Weights must be smooth everywhere, especially at handles our method [SIGGRAPH 2011] extension of Harmonic Coordinates [Joshi et al. 2005] June 9, 2013 Alec Jacobson # 36
37 Shape-awareness ensures respect of domain s features our method non-shape-aware methods e.g. [Schaefer et al. 2006] June 9, 2013 Alec Jacobson # 37
38 Shape-awareness ensures respect of domain s features our method non-shape-aware methods e.g. [Schaefer et al. 2006] June 9, 2013 Alec Jacobson # 38
39 Non-negative weights are mandatory our method unconstrained biharmonic [Botsch & Kobbelt 2004] June 9, 2013 Alec Jacobson # 39
40 Non-negative weights are mandatory our method unconstrained biharmonic [Botsch & Kobbelt 2004] June 9, 2013 Alec Jacobson # 40
41 Spurious extrema cause distracting artifacts unconstrained ext. of [Botsch & Kobbelt 2004] local max local min June 9, 2013 Alec Jacobson # 41
![[SIGGRAPH 2011] local max](/docs-images/84/90836151/images/42-2.jpg "local min June 9, 2013 Alec")
42 Spurious extrema cause distracting artifacts bounded [SIGGRAPH 2011] local max local min June 9, 2013 Alec Jacobson # 42
43 We explicitly prohibit spurious extrema our improved [SGP 2012] local max local min June 9, 2013 Alec Jacobson # 43
44 Constrained optimization ensures satisfaction of all properties argmin w j,j=1,...,m mx klw j k 2 j=1 ext. of [Botsch & Kobbelt 2004] + shape-aware June 9, 2013 Alec Jacobson # 44
45 Constrained optimization ensures satisfaction of all properties argmin w j,j=1,...,m mx j=1 Z ( w j ) 2 dv + shape-aware + smoothness + mesh independence [SGP 2010] June 9, 2013 Alec Jacobson # 45
46 Constrained optimization ensures satisfaction of all properties argmin w j,j=1,...,m mx j=1 Z ( w j ) 2 dv + shape-aware + smoothness + mesh independence + non-negativity + locality + arbitrary handles 0 apple w j apple 1, j =1,...,m [SIGGRAPH 2011] June 9, 2013 Alec Jacobson # 46
47 Constrained optimization ensures satisfaction of all properties argmin w j,j=1,...,m mx j=1 Z [SGP 2012] ( w j ) 2 dv rw j ru j > 0, j =1,...,m + shape-aware + smoothness + mesh independence + non-negativity + locality + arbitrary handles + monotonicity June 9, 2013 Alec Jacobson # 47
48 Weights in 3D retain nice properties Demo June 9, 2013 Alec Jacobson # 48
49 Weights in 3D retain nice properties Demo June 9, 2013 Alec Jacobson # 49
50 Variational formulation allows additional, problem-specific constraints June 9, 2013 Alec Jacobson # 50
51 Variational formulation allows additional, problem-specific constraints June 9, 2013 Alec Jacobson # 51
52 Linear blending subspace is still too small June 9, 2013 Alec Jacobson # 52
53 Linear blend skinning etc. are not rich enough to stretch and twist along bones LBS our extension [SIGGRAPH Asia 2011] June 9, 2013 Alec Jacobson # 53
54 Linear blend skinning etc. are not rich enough to stretch and twist along bones LBS our extension [SIGGRAPH Asia 2011] June 9, 2013 Alec Jacobson # 54
55 Second set of our weights expands skinning subspace June 9, 2013 Alec Jacobson # 55
56 Stretching facilitates exaggeration, a basic principle of life-like animation June 9, 2013 Alec Jacobson # 56
57 Stretching facilitates exaggeration, a basic principle of life-like animation June 9, 2013 Alec Jacobson # 57
58 Good weights alone aren t enough to guarantee intuitive results pseudo-edges [SIGGRAPH June 9, ] Alec Jacobson our improved method # 58 [SIGGRAPH 2012]
59 Good weights alone aren t enough to guarantee intuitive results pseudo-edges [SIGGRAPH June 9, ] Alec Jacobson our improved method # 59 [SIGGRAPH 2012]
60 Good weights alone aren t enough to guarantee intuitive results pseudo-edges [SIGGRAPH June 9, ] Alec Jacobson our improved method # 60 [SIGGRAPH 2012]
![2006] physically accurate muscle systems require off-line simulation e.g.](/docs-images/84/90836151/images/61-1.jpg "[Teran et al.")
61 Many previous techniques provide quality, but not speed high-quality solutions to nonlinear elasticity energy minimizations: ~seconds e.g. [Botsch et al. 2006] physically accurate muscle systems require off-line simulation e.g. [Teran et al. 2005] Image courtesy Joseph Teran June 9, 2013 Alec Jacobson # 61
![2006] significant performance tuning, but grid determines complexity](/docs-images/84/90836151/images/62-1.jpg "~milliseconds e.g. [McAdams et al. 2011] June 9, 2013 Alec Jacobson # 62")
62 Other reduced models employ skinning, but still too slow needs examples, choice of energy complicates per-frame computation ~milliseconds e.g. [Der et al. 2006] significant performance tuning, but grid determines complexity ~milliseconds e.g. [McAdams et al. 2011] June 9, 2013 Alec Jacobson # 62
63 We achieve speeds measured in microseconds 80k triangles 20µs per iteration June 9, 2013 Alec Jacobson # 63
64 We achieve speeds measured in microseconds 80k triangles 20µs per iteration June 9, 2013 Alec Jacobson # 64
65 We achieve speeds measured in microseconds 80k triangles 20µs per iteration June 9, 2013 Alec Jacobson # 65
66 We achieve speeds measured in microseconds 80k triangles 20µs per iteration June 9, 2013 Alec Jacobson # 66
67 This means speed comparable to rendering 30fps June 9, 2013 Alec Jacobson # 67
68 This means speed comparable to rendering 30fps June 9, 2013 Alec Jacobson # 68
69 User specifies subset of parameters, optimize to find remaining ones full optimization mesh vertex positions June 9, 2013 Alec Jacobson # 69
70 User specifies subset of parameters, optimize to find remaining ones full optimization reduced model skinning degrees of freedom June 9, 2013 Alec Jacobson # 70
71 User specifies subset of parameters, optimize to find remaining ones full optimization reduced model matrix form June 9, 2013 Alec Jacobson # 71
72 User specifies subset of parameters, optimize to find remaining ones full optimization reduced model matrix form reduced optimization June 9, 2013 Alec Jacobson # 72
73 Enforce user constraints as linear equalities reduced optimization user constraints full position only unconstrained June 9, 2013 Alec Jacobson # 73
74 Enforce user constraints as linear equalities reduced optimization user constraints full position only unconstrained June 9, 2013 Alec Jacobson # 74
75 We reduce any as-rigid-as-possible energy full energies June 9, 2013 Alec Jacobson # 75
76 We reduce any as-rigid-as-possible energy full energies triangles [Liu et al. 2008] tetrahedra [Chao et al. 2010] spokes [Sorkine & Alexa 2007] spokes and rims [Chao et al. 2010] June 9, 2013 Alec Jacobson # 76
77 We reduce any as-rigid-as-possible energy full energies local/global optimization global step: fix R, minimize with respect to V 0 local step: fix V, 0 minimize with respect to R June 9, 2013 Alec Jacobson # 77
78 We reduce any as-rigid-as-possible energy full energies local/global optimization precompute global step: large, sparse linear solve V 0 = A 1 b local step: fix V, 0 minimize with respect to R June 9, 2013 Alec Jacobson # 78
79 We reduce any as-rigid-as-possible energy full energies local/global optimization global step: large, sparse linear solve V 0 = A 1 b local step: 3x3 SVD for each rotation in R June 9, 2013 Alec Jacobson # 79
80 We reduce any as-rigid-as-possible energy full energies local/global optimization precompute global step: small, dense linear solve T = Ã 1 b substitute local step: 3x3 SVD for each rotation in R similar to: [Huang et al. 06] [Der et al. 06] [Au et al. 07] [Hildebrandt et al. 12] June 9, 2013 Alec Jacobson # 80
81 Direct reduction of elastic energies brings speed up and regularization June 9, 2013 Alec Jacobson # 81
82 Direct reduction of elastic energies brings speed up and regularization full ARAP solution June 9, 2013 Alec Jacobson # 82
83 Direct reduction of elastic energies brings speed up and regularization full ARAP solution our smooth subspace solution June 9, 2013 Alec Jacobson # 83
84 We reduce any as-rigid-as-possible energy full energies local/global optimization global step: small, dense linear solve T = Ã 1 b substitute local step: 3x3 SVD for each rotation in R but #rotations ~ full mesh discretization June 9, 2013 Alec Jacobson # 84
85 We reduce any as-rigid-as-possible energy full energies local/global optimization global step: small, dense linear solve local step: 3x3 SVD for each rotation in T = Ã 1 b R substitute Cluster E k June 9, 2013 Alec Jacobson # 85
86 Rotation evaluations may be reduced by clustering in weight space full energies triangles [Liu et al. 2008] tetrahedra [Chao et al. 2010] spokes [Sorkine & Alexa 2007] spokes and rims [Chao et al. 2010] June 9, 2013 Alec Jacobson # 86
87 Rotation evaluations may be reduced by k-means clustering in weight space full energies x j = weight space w 1 (v j ) w 2 (v j ).. w m (v j ) June 9, 2013 Alec Jacobson # 87
88 Rotation evaluations may be reduced by clustering in weight space full energies June 9, 2013 Alec Jacobson # 88
89 Rotation evaluations may be reduced by clustering in weight space full energies r = 2 r = 4 r = 64 June 9, 2013 Alec Jacobson # 89
90 We reduce any as-rigid-as-possible energy full energies local/global optimization global step: small, dense linear solve local step: 3x3 SVD for each rotation in T = Ã 1 b R #rotations ~ #T, independent of full mesh resolution substitute Cluster E k June 9, 2013 Alec Jacobson # 90
91 With more and more user constraints we fall back to standard skinning June 9, 2013 Alec Jacobson # 91
92 With more and more user constraints we fall back to standard skinning June 9, 2013 Alec Jacobson # 92
93 With more and more user constraints we fall back to standard skinning June 9, 2013 Alec Jacobson # 93
94 Extra weights expand deformation subspace no extra weights 15 extra weights June 9, 2013 Alec Jacobson # 94
95 Extra weights expand deformation subspace no extra weights 15 extra weights June 9, 2013 Alec Jacobson # 95
96 Subspace now rich enough for fast variational modeling full non-linear optimization [Botsch et al. 2006] our reduced method June 9, 2013 Alec Jacobson # 96
97 Subspace now rich enough for fast variational modeling full non-linear optimization [Botsch et al. 2006] our reduced method June 9, 2013 Alec Jacobson # 97
98 Subspace now rich enough for fast variational modeling full non-linear optimization [Botsch et al. 2006] our reduced method June 9, 2013 Alec Jacobson # 98
99 Subspace now rich enough for fast variational modeling full non-linear optimization [Botsch et al. 2006] our reduced method June 9, 2013 Alec Jacobson # 99
100 Subspace now rich enough for fast variational modeling full non-linear optimization [Botsch et al. 2006] our reduced method June 9, 2013 Alec Jacobson # 100
101 Subspace now rich enough for fast variational modeling full non-linear optimization [Botsch et al. 2006] our reduced method June 9, 2013 Alec Jacobson # 101
102 Extra weights and disjoint skeletons make flexible control easy From Cartoon Animation by Preston Blair June 9, 2013 Alec Jacobson # 102
103 Extra weights and disjoint skeletons make flexible control easy June 9, 2013 Alec Jacobson # 103
104 Extra weights and disjoint skeletons make flexible control easy June 9, 2013 Alec Jacobson # 104
105 Physical dynamics also benefit from our reduction June 9, 2013 Alec Jacobson # 105 Demo
106 Physical dynamics also benefit from our reduction June 9, 2013 Alec Jacobson # 106 Demo
107 Our reduction preserves nature of different energies, at no extra cost Surface ARAP Volumetric ARAP V 0 surf = M surf T V 0 vol = M vol T June 9, 2013 Alec Jacobson # 107
108 Our reduction preserves nature of different energies, at no extra cost Surface ARAP Volumetric ARAP V 0 surf = M surf T V 0 vol = M vol T June 9, 2013 Alec Jacobson # 108
109 Volumetric deformation differs drastically from surface-based June 9, 2013 Alec Jacobson # 109
110 Volumetric deformation differs drastically from surface-based June 9, 2013 Alec Jacobson # 110
111 Surface artifacts prevent volume meshing, prevent volumetric processing June 9, 2013 Alec Jacobson # 111
112 Continuous winding number for watertight surfaces generalizes to unclean triangle meshes w(p) = 1 4 ZZ S sin( )d d w(p) = 1 4 mx f=1 f June 9, 2013 Alec Jacobson # 112 [SIGGRAPH 2013]
113 Generalized winding number is ideal indicator for graphcut segmentation of convex hull w(p) = 1 4 ZZ S sin( )d d w(p) = 1 4 mx f=1 f June 9, 2013 Alec Jacobson # 113 [SIGGRAPH 2013]
114 Each advance inspires new improvements, new interfaces SGP 2010 SIGGRAPH 2011 SGP 2012 SIGGRAPH Asia 2011 SIGGRAPH 2012 SIGGRAPH 2013 June 9, 2013 Alec Jacobson # 114
115 We derive a powerful subspace via input shape and handle descriptions Intrinsics from shape s geometry Semantics from handles Fully automatic New interfaces Real-time as an invariant June 9, 2013 Alec Jacobson # 115
116 Future outlook More semantics: large data, collisions, etc. June 9, 2013 Alec Jacobson # 116
117 Future outlook More semantics: large data, collisions, etc. Physical interfaces June 9, 2013 Alec Jacobson # 117
118 Acknowledgements Coauthors: Ladislav Kavan, Ilya Baran, Jovan Popović, Kaan Yücer, Alex Sorkine-Hornung, Tino Weinkauf, Denis Zorin, Olga Sorkine-Hornung Colleagues at NYU and ETH Friends and family This thesis was supported in part by NSF award IIS , ERC grant imodel (StG ), SNF award _137879, the Intel Doctoral Fellowship and a gift from Adobe Systems. June 9, 2013 Alec Jacobson # 118
119 Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes Papers, videos, code: people.inf.ethz.ch/~jalec/ Alec Jacobson June 9, 2013
120
121 Weights should be smooth everywhere, especially at handles Our method Extension of Harmonic Coordinates [Joshi et al. 2005] June 9, 2013 Alec Jacobson ##121
122 Open cages allow arbitrary line constraints June 9, 2013 Alec Jacobson # 122
123 Sparsity helps maintain local influence Smoothed extension of Harmonic Coordinates [Joshi et al. 2005] Our method June 9, 2013 Alec Jacobson ##123
124 Sparsity helps maintain local influence Smoothed extension of Harmonic Coordinates [Joshi et al. 2005] Our method June 9, 2013 Alec Jacobson ##124
125 Sparsity helps maintain local influence Smoothed extension of Harmonic Coordinates [Joshi et al. 2005] Our method June 9, 2013 Alec Jacobson ##125
126 Sparsity helps maintain local influence Smoothed extension of Harmonic Coordinates [Joshi et al. 2005] Our method June 9, 2013 Alec Jacobson ##126
127 Sparsity helps maintain local influence Smoothed extension of Harmonic Coordinates [Joshi et al. 2005] Our method June 9, 2013 Alec Jacobson ##127
![2005] Our method June 9,](/docs-images/84/90836151/images/128-5.jpg "2013 Alec Jacobson ##128")
128 Sparsity helps maintain local influence Smoothed extension of Harmonic Coordinates [Joshi et al. 2005] Our method June 9, 2013 Alec Jacobson ##128
129 Boundedness also helps maintain local influence Unconstrained biharmonic [Botsch and Kobbelt 2004] June 9, 2013 Alec Jacobson ##129
130 Boundedness also helps maintain local influence Our method Unconstrained biharmonic [Botsch and Kobbelt 2004] June 9, 2013 Alec Jacobson ##130
131 Spurious local maxima also cause unintuitive response Our method Extension of unconstrained biharmonic [Botsch and Kobbelt 2004] June 9, 2013 Alec Jacobson ##131
132 Weights propagate transformations at handles to shape in real-time Translate Rotate Scale June 9, 2013 Alec Jacobson ##132
133 Dropping partition of unity as explicit constraint does not effect quality original weights faster weights June 9, 2013 Alec Jacobson ##133
134 Weights may also define an intuitive, shape-aware depth ordering in 2D z z June 9, 2013 Alec Jacobson ##134
135 Weights may also define an intuitive, shape-aware depth ordering in 2D June 9, 2013 Alec Jacobson ##135
136 Rotations at point handles may be computed automatically based on translations June 9, 2013 Alec Jacobson ##136
137 Alternative skinning methods may also take advantage of bounded biharmonic weights Linear blend skinning Dual quaternion skinning June 9, 2013 Alec Jacobson ##137
138 Same weights can interpolate colors HX x 0 = w j (x i )T j x i j=1 June 9, 2013 Alec Jacobson # 138
139 Same weights can interpolate colors HX c i = w j (x i )c j j=1 June 9, 2013 Alec Jacobson # 139
![2011] c i = HX w j (x i )c j j=1 Image](/docs-images/84/90836151/images/140-1.jpg "courtesy Mark Finch June 9, 2013 Alec")
140 Same functions used for color interpolation unconstrained [Finch et al. 2011] c i = HX w j (x i )c j j=1 Image courtesy Mark Finch June 9, 2013 Alec Jacobson # 140
141 Same functions used for color interpolation unconstrained [Finch et al. 2011] c i = HX w j (x i )c j j=1 June 9, 2013 Alec Jacobson # 141
142 Same functions used for color interpolation unconstrained [Finch et al. 2011] c i = HX w j (x i )c j Our [SGP 2012] j=1 June 9, 2013 Alec Jacobson # 142
![al. 2011] Our extended](/docs-images/84/90836151/images/143-2.jpg "method [SGP 2012] June 9,")
143 Without constraints, biharmonic unintuitively interpolates colors Unconstrained Δ 2 [Finch et al. 2011] Our extended method [SGP 2012] June 9, 2013 Alec Jacobson # 143
144 Skinning weights tether optimization over multiple-component meshes June 9, 2013 Alec Jacobson # 144
145 Skinning weights tether optimization over multiple-component meshes Mesh-energy methods e.g. Igarashi et al. 2004, Sumner et al. 2005, Botsch et al. 2006, Sorkine and Alexa 2007, Shi et al. 2007, Solomon et al Our skinning-based method June 9, 2013 Alec Jacobson # 145
![Kobbelt 2004] weight of middle](/docs-images/84/90836151/images/146-3.jpg "point June 9, 2013 Alec")
146 Extrema distort small features Unconstrained [Botsch & Kobbelt 2004] weight of middle point June 9, 2013 Alec Jacobson # 146
147 Deformation applies to images as planar shapes non-convex cut-out cartoons entire image rectangle June 9, 2013 Alec Jacobson # 147
148 Extra weights would expand subspace June 9, 2013 Alec Jacobson # 148
149 Extra weights would expand subspace June 9, 2013 Alec Jacobson # 149
150 Extra weights would expand subspace June 9, 2013 Alec Jacobson # 150
151 Real-time automatic degrees of freedom June 9, 2013 Alec Jacobson # 151
152 Overlapping b-spline bumps in weight space x j = weight space w 1 (v j ) w 2 (v j ). w m (v j ) farthest point sampling June 9, 2013 Alec Jacobson # 152
153 Overlapping b-spline bumps in weight space in weight space x j = weight space w 1 (v j ) w 2 (v j ). w m (v j ) b-spline basis parameterized by distance in weight space June 9, 2013 Alec Jacobson # 153
154 Overlapping b-spline bumps in weight space in weight space x j = weight space w 1 (v j ) w 2 (v j ). w m (v j ) b-spline basis parameterized by distance in weight space June 9, 2013 Alec Jacobson # 154
155 Final algorithm is simple and FAST Precomputation per shape+rig - Compute any additional weights - Construct, prefactor system matrices For a 50K triangle mesh: 12 seconds 2.7 seconds June 9, 2013 Alec Jacobson # 155
156 Final algorithm is simple and FAST Precomputation per shape+rig - Compute any additional weights - Construct, prefactor system matrices For a 50K triangle mesh: 12 seconds 2.7 seconds Precomputation when switching constraint type - Re-factor global step system 6 milliseconds June 9, 2013 Alec Jacobson # 156
157 Final algorithm is simple and FAST Precomputation per shape+rig - Compute any additional weights - Construct, prefactor system matrices For a 50K triangle mesh: 12 seconds 2.7 seconds Precomputation when switching constraint type - Re-factor global step system 6 milliseconds ~30 iterations 22 microseconds global: #weights by #weights linear solve local: #rotations SVDs [McAdams et al. 2011] June 9, 2013 Alec Jacobson # 157
158 Simple drag-only interface for point handles June 9, 2013 Alec Jacobson # 158
159 Extrema distort small features Bounded [Jacobson et al. 2011] weight of middle point June 9, 2013 Alec Jacobson # 159
160 Monotonicity helps preserve small features Bounded [Jacobson et al. 2011] Our June 9, 2013 Alec Jacobson # 160
161 Spurious extrema are unstable, may flip slightly larger region June 9, 2013 Alec Jacobson # 161
162 Spurious extrema are unstable, may flip slightly larger region June 9, 2013 Alec Jacobson # 162
163 Spurious extrema are unstable, may flip Unconstrained [Botsch & Kobbelt, 2004] #
164 Spurious extrema are unstable, may flip Bounded #
165 Lack of extrema leads to more stability Our #
166 In 2D, stretching manipulates foreshortening LBS with rigid bones STBS with stretchable bones June 9, 2013 Alec Jacobson # 166
167 Behavior depends heavily on parameterization and boundary conditions x = 0 2 x = 0 3 x = 0 June 9, 2013 Alec Jacobson # 167
168 Behavior depends heavily on parameterization and boundary conditions Parameter Domain Straight Tube Boundary Conditions x = 0 2 x = 0 3 x = 0 June 9, 2013 Alec Jacobson # 168
169 Behavior depends heavily on parameterization and boundary conditions Torus Boundary Conditions (but connected to straight tubes) June 9, 2013 Alec Jacobson # 169
170 Traditional winding number determines amount of insideness 0 1 June 9, 2013 Alec Jacobson # 170
171 Traditional winding number determines amount of insideness June 9, 2013 Alec Jacobson # 171
172 Winding number jumps across boundaries, harmonic otherwise 2 1½ 1 ½ 0 -½ June 9, 2013 Alec Jacobson # 172
173 Winding number jumps across boundaries, harmonic otherwise 2 1½ 1 ½ 0 -½ June 9, 2013 Alec Jacobson # 173
174 Sum of harmonic functions is harmonic 1 0 ½ -½ = + = June 9, 2013 Alec Jacobson # 174
175 Winding number gracefully tends toward indicator function 1 ½ 0 -½ June 9, 2013 Alec Jacobson # 175
176 Hierarchical evaluation performs asymptotically better Winding number computation time (subdivided Dino) Seconds 1e0 0 Naive 1e-2 1e-2 1e-3 Hierarchical 1e3 1e4 1e5 Number of input facets, m 1e6 1e7 June 9, 2013 Alec Jacobson # 176
177 Hierarchical evaluation performs asymptotically better Winding number computation time (SHREC Dataset) Seconds 1e-2 Naive 1e-3 1e-4 Hierarchical 1e2 1e3 1e4 Number of input facets, m 1e5 June 9, 2013 Alec Jacobson # 177
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