2emovaloftopologicalandgeometricalerrors · pdf file · 2015-09-28mario botsch...
TRANSCRIPT
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Mesh Editing
Mario BotschBielefeld University
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Mario Botsch
Mesh Deformation
Global deformationwith intuitive
detail preservation
2
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Mario Botsch
Mesh Deformation
3
Local & globaldeformations
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Mario Botsch
Mesh Deformation
4
Editing of complex meshes
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Mario Botsch
Mesh Deformation
5
Editing of“bad” meshes
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Mario Botsch
Overview
6
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
Overview
7
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
Spline Surfaces
• Tensor product surfaces– Rectangular grid of control points– Rectangular surface patches
8
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Mario Botsch
Spline Surfaces
• Tensor product surfaces– Rectangular grid of control points– Rectangular surface patches
• Problems:– Many patches for complex models– Smoothness across patch boundaries
Use irregular triangle meshes!
9
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Mario Botsch
• Paint three surface regions– Support region (blue)– Fixed vertices (gray)– Handle regions (green)
Modeling Metaphor
10
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Mario Botsch
• Mesh deformation by displacement function d– Interpolate prescribed constraints– Smooth, intuitive deformationPhysically-based principles
Modeling Notation
11
S S ! = p + d (p) |p ! S
d : S ! IR3
d (pi) = di
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Mario Botsch
Physically-Based Deformation
• Non-linear stretching & bending energies
• Linearize energies
12
Ωks
I− I2 + kb
II− II2 dudv
stretching bending
Ωks
du2 + dv2
+ kb
duu2 + 2 duv2 + dvv2
dudv
stretching bending
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Mario Botsch
Physically-Based Deformation
• Minimize linearized bending energy
• Variational calculus, Euler-Lagrange PDE
“Best” deformation that satisfies constraints
13
∆2d := duuuu + 2duuvv + dvvvv = 0
E(d) =
Sduu2 + 2 duv2 + dvv2 dS → minf(x)→ min
f (x) = 0
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Mario Botsch
Deformation Energies
14
(Membrane)
(Thin plate)Initial state
(Membrane)∆2d = 0
∆2p = 0∆p = 0
∆d = 0
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Mario Botsch
Discretization
15
• Laplace discretization
• Sparse linear system
xj
xi
Ai!ij
!ij
∆di =1
2Ai
j∈Ni
(cot αij + cot βij)(dj − di)
∆2
0 I 00 0 I
=:M
...di...
=
00
δhi
∆2di = ∆(∆di)
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Mario Botsch
Discretization
16
• Sparse linear system (19 nz/row)
• Can be turned into symm. pos. def. system– Right hand sides changes each frame!– Use efficient linear solvers...
∆2
0 I 00 0 I
=:M
...di...
=
00
δhi
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Mario Botsch
Sparse SPD Solvers• Dense Cholesky factorization
– Cubic complexity– High memory consumption (doesnʼt exploit sparsity)
• Iterative conjugate gradients– Quadratic complexity– Need sophisticated preconditioning
• Multigrid solvers– Linear complexity– But rather complicated to develop (and to use)
• Sparse Cholesky factorization?17
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Mario Botsch
Dense Cholesky Solver
1. Cholesky factorization
2. Solve system
18
A = LLT
y = L−1b, x = L−T y
Ax = bSolve
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Mario Botsch
Sparse Cholesky Factorization
19
PTAP
Reordering
14k non-zeros
L
36k non-zeros
Cholesky Factorization
A=LLT
500×500 matrix3500 non-zeros
7k non-zeros
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Mario Botsch
Pre-computation
Per-frame computation
Sparse Cholesky Solver
1. Matrix re-ordering
2. Cholesky factorization
3. Solve system
20
A = PTAP
A = LLT
y = L!1PT b, x = PL!T y
Ax = bSolve
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Mario Botsch
Bi-Laplace Systems
21
0s
13s
25s
38s
50s
10k 20k 30k 40k 50k
Conjugate Gradients Multigrid Sparse Cholesky
Setup + Precomp. + 3 Solutions
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Mario Botsch
Bi-Laplace Systems
22
0s
4s
8s
11s
15s
10k 20k 30k 40k 50k
3 Solutions (per frame costs)
Conjugate Gradients Multigrid Sparse Cholesky
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Mario Botsch
0 50 100 150 200 250 300 350 400
8
16
24
32
Laplace System
23
Setup + Precomp. + 3 Solutions
Multigrid
SparseCholesky
[Shi et al, SIGGRAPH 06]
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Mario Botsch
Laplace System
24
0 50 100 150 200 250 300 350 400
1
2
3
4
5
3 Solutions (per frame costs)
Multigrid
SparseCholesky
[Shi et al, SIGGRAPH 06]
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Mario Botsch
Discretization
25
• Sparse linear system
• Can be turned into symm. pos. def. system– Right hand sides changes each frame– Sparse Cholesky factorization– Very efficient implementations publicly available
∆2
0 I 00 0 I
=:M
...di...
=
00
δhi
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Mario Botsch
More Efficient Solution
• Handle is transformed affinely
• Precompute linear basis functions
26
...di...
= M−1
00
δhi
= M−1
00Q
B∈ IRn×4
(δa, δb, δc, δd, )T
(. . . , !hi, . . .) = Q (!a, !b, !c, !d)T
(. . . ,hi, . . .) = Q (a,b, c,d)T
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Mario Botsch
Front Deformation
27
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Mario Botsch
Overview
28
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
• Even pure translations induce local rotations!– Inherently nonlinear coupling
• Or: Linear model + multi-scale decomposition...
Multiresolution Modeling
29
Original Non-linear deform.Linear deform.
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Mario Botsch
Multiresolution Editing
Frequency decomposition
Change lowfrequencies
Add high frequency details, stored in local frames
30
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Mario Botsch
Multiresolution Editing
Multiresolution
Modeling
31
S
Dec
ompo
sitio
n
DetailInformation
B
Freeform
Modeling
B
Reconstruction
S
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Mario Botsch
Normal Displacements
32
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Mario Botsch
Limitations
33
• Neighboring displacements are not coupled– Surface bending changes their angle– Leads to volume changes or self-intersections
Original Normal Displ. Nonlinear
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Mario Botsch
Limitations
34
Normal Displ. NonlinearOriginal
• Neighboring displacements are not coupled– Surface bending changes their angle– Leads to volume changes or self-intersections
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Mario Botsch
Limitations
35
• Neighboring displacements are not coupled– Surface bending changes their angle– Leads to volume changes or self-intersections– See course notes for some other techniques...
• Multiresolution hierarchy difficult to compute for meshes of complex topology / geometry– Might require more hierarchy levels
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Mario Botsch
Overview
36
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
Differential Coordinates
1. Manipulate differential coordinates instead of spatial coordinates– Gradients, Laplacians, ...
2. Then find mesh with desired differential coords– Basically an integration step
37
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Mario Botsch
Gradient-Based Editing
38
Original Rotated DiffCoords Reconstructed Mesh
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Mario Botsch
Gradient-Based Editing
• Manipulate gradient field of a function (surface)
• Find function f’ whose gradient is (close to) g’
• Variational calculus yields Euler-Lagrange PDE
39
g → gg = ∇p
S∇p − g2 dS → min
∆p = div g
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Mario Botsch
Gradient-Based Editing
• Use piecewise linear coordinate function
• Its gradient is
40
p(u, v) =
vi
pi · φi(u, v)
φ2
x1
x2
x3
1 x1
x2
x3
1φ3
x1
x2
x3
1 φ1
∇p(u, v) =
vi
pi ·∇φi(u, v)
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Mario Botsch
Gradient-Based Editing
• Use piecewise linear coordinate function
• Its gradient is
• It is constant per triangle
41
p(u, v) =
vi
pi · φi(u, v)
∇p(u, v) =
vi
pi ·∇φi(u, v)
!p|fj=: Gj " IR
3!3
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Mario Botsch
Gradient-Based Editing
• Constant per triangle
• Manipulate per-face gradients
42
!p|fj=: Gj " IR
3!3
!
"#
G1
.
.
.
GF
$
%& = G
'()*
!IR3F!V
·
!
"#
pT1
.
.
.
pTV
$
%&
Gj !" G!
j
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Mario Botsch
• Reconstruct mesh from changed gradients– Overdetermined problem– Weighted least squares system– Linear Poisson (Laplace) system
divdiv! = !
Gradient-Based Editing
43
GT DG ·
!
"
#
p!
1
T
.
.
.
p!
V
T
$
%
&= GT D ·
!
"
#
G!
1
.
.
.
G!
F
$
%
&
G ! IR3F!V
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Mario Botsch
Gradient-Based Editing
44
Original Rotated DiffCoords Reconstructed Mesh
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Mario Botsch
Construct Scalar Field
• Construct smooth scalar field [0,1]• s(x)=1: ! Full deformation (handle)• s(x)=0: ! No deformation (fixed part)• s(x)∈(0,1):! Damp handle transformation (in between)
45
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Mario Botsch
Construct Scalar Field
• Construct a smooth harmonic field– Solve
– with
46
!(s) = 0
s (p) =
!
1 p ! handle0 p ! fixed
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Mario Botsch
Damp Handle Transformation
• Full handle transformation– Rotation: ! R(c, a, α)– Scaling: ! S(s)
• Damped by scalar λ– Rotation: ! R(c, a, λ·α)– Scaling: ! S(λ·s + (1-λ)·1)
47
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Mario Botsch
Gradient-Based Editing
48
Original Rotated Gradients Reconstructed Mesh
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Mario Botsch
Laplacian-Based Editing
• Manipulate Laplacians of a surface
• Find surface whose Laplacian is (close to) δ’
• Variational calculus yields Euler-Lagrange PDE
49
δi = ∆(pi) , δi → δi
∆2p = ∆δ
S
∆p − δ2 dS → min
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Mario Botsch
Discretization
• Discretize Euler-Lagrange PDE
• Frequently used (wrong) version
50
∆2p = ∆δ L2p = Lδ
LT Lp = LT δδ = Lp δ → δ
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Mario Botsch
Discretization
51
L2p = LδLT Lp = LT δ
Irregular mesh
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Mario Botsch
Connection to Plate Energy?
• Neglect change of Laplacians for a moment...
52
∆2(p + d) = ∆2p
∆2p = ∆δ
p = p + dδ = ∆p
∆2d = 0
• Basic formulations equivalent!• Differ in detail preservation
• Rotation of Laplacians• Multi-scale decomposition
∆p − δ2 → min
duu2 + 2 duv2 + dvv2 → min
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Mario Botsch
Limitations
• Differential coordinates work well for rotations– Apply damped handle rotations to diff. coords.
• Pure translations donʼt have rotation component– Translations donʼt change differential coordinates– “Translation insensitivity”
53
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Mario Botsch
Overview
54
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
Surface-Based Deformation• Problems with
– Highly complex models– Topological inconsistencies– Geometric degeneracies
55
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Mario Botsch
Freeform Deformation
• Deform objectʼs bounding box– Implicitly deforms embedded objects
56
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Mario Botsch
Freeform Deformation
• Deform objectʼs bounding box– Implicitly deforms embedded objects
• Tri-variate tensor-product spline
57
d(u, v, w) =l
i=0
m
j=0
n
k=0
dijkNi(u) Nj(v) Nk(w)
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Mario Botsch
Freeform Deformation
• Deform objectʼs bounding box– Implicitly deforms embedded objects
• Tri-variate tensor-product spline
58
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Mario Botsch
Freeform Deformation
• Deform objectʼs bounding box– Implicitly deforms embedded objects
• Tri-variate tensor-product spline– Aliasing artifacts
• Interpolate deformation constraints?– Only in least squares sense
59
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Mario Botsch
Overview
60
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
• Mesh deformation by displacement function d– Interpolate prescribed constraints– Smooth, intuitive deformationPhysically-based principles
Space Deformation
d (pi) = di
61
p → p + d(p)
d : IR3 → IR3
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Mario Botsch
Volumetric Energy Minimization
• Minimize similar energies to surface case
• But displacements function lives in 3D...– Need a volumetric space tessellation?– No, same functionality provided by RBFs
62
IR3duu2 + duv2 + . . . + dww2 dV → min
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Mario Botsch
Radial Basis Functions
• Represent deformation by RBFs
• Triharmonic basis function– C2 boundary constraints– Highly smooth / fair interpolation
63
d (x) =!
j
wj · ! (!cj " x!) + p (x)
! (r) = r3
IR3duuu2 + dvuu2 + . . . + dwww2 du dv dw → min
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Mario Botsch
RBF Fitting
• Represent deformation by RBFs
• RBF fitting– Interpolate displacement constraints– Solve linear system for wj and p
64
d (x) =!
j
wj · ! (!cj " x!) + p (x)
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Mario Botsch
RBF Fitting
• Represent deformation by RBFs
• RBF evaluation– Function d transforms points– Jacobian ∇d transforms normals– Precompute basis functions– Evaluate on the GPU!
65
d (x) =!
j
wj · ! (!cj " x!) + p (x)
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Mario Botsch
RBF Deformation
66
1M vertices
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Mario Botsch
“Bad Meshes”
• 3M triangles• 10k components• Not oriented• Not manifold
67
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Mario Botsch
Overview
68
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
Comparison
69
NonlinearOriginal Bending Min. Gradient
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Mario Botsch
Linear vs. Nonlinear
70
[Botsch & Sorkine, TVCG 08]
• Analyze existing methods– Some work for translations– Some work for rotations– No method works for both
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Mario Botsch
GradientBending Min.
Linear vs. Nonlinear
71
Nonlinear
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Mario Botsch
Nonlinear Deformation?
72
• Sounds easy: “Just donʼt linearize.”
• Not so easy though...– Solve nonlinear problems (Newton, Gauss-Newton)– No convergence guarantees– Robustness issues– Considerably slower
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Mario Botsch
Subspace Gradient Domain Mesh Deformation
• Nonlinear Laplacian coordinates
• Least squares solution on coarse cage subspace
73[Huang et al, SIGGRAPH 06]
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Mario Botsch
Mesh Puppetry
• Skeletons and Laplacian coordinates
• Cascading optimization
74[Shi et al, SIGGRAPH 07]
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Mario Botsch
PriMo: Coupled Prisms for Intuitive Surface Modeling
75[Botsch et al, SGP 06]
• Nonlinear shell-like energy
• Rigid cells ensure robustness
• Hierarchical solver
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Mario Botsch
PriMo: Coupled Prisms for Intuitive Surface Modeling
76[Botsch et al, SGP 06]
• Nonlinear shell-like energy
• Rigid cells ensure robustness
• Hierarchical solver
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Mario Botsch
Adaptive Space DeformationBased on Rigid Cells
• Nonlinear elastic energy for solids and shells
• Rigid cells ensure robustness
• Dynamic, adaptive discretization
77[Botsch et al, Eurographics 07]
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Mario Botsch
Embedded Deformations
• As-rigid-as-possible space deformation
• Coarse deformation graph
78[Sumner et al, SIGGRAPH 07]
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Mario Botsch
Overview
79
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods
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Mario Botsch
Summary
80
vs.
Differential Coords• Designed for large
rotations
• Problems with translations
• How to determine local rotations?
Bending Energy• Precise control of
continuity
• Requires multi-resolution hierarchy
• Problems with large rotations
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Mario Botsch
Summary
81
vs.
Space Deformation– Doesnʼt know about
embedded surface
+ Works for complex and “bad” input
Surface-Based+ More precise control
of surface properties
– Depends on surface complexity & quality
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Mario Botsch
Summary
82
vs.
Nonlinear– Numerically much
more complex
+ Easier edits, fewer constraints
Linear+ Highly efficient &
numerically robust
– Many constraints for large-scale edits
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Mario Botsch
Overview
83
• Surface-based deformation– Energy minimization– Multiresolution editing– Differential coordinates
• Space deformation– Freeform deformation– Energy minimization
• Linear vs. nonlinear methods