Computer Graphics Group
Tobias Weyand
Mesh
-
Based Inverse
Kinematics
Sumner et al 2005
presented by Tobias Weyand
Computer Graphics Group
Tobias Weyand
2
What is Inverse Kinematics?
•
Articulated body
•
Which angles for a certain
configuration?
•
Forward kinematics
•
Specify angles
•
Inverse kinematics
•
Specify limb position
a
1
a
2
a
3
a
1
a
2
a
3
Computer Graphics Group
Tobias Weyand
3
IK in Computer Graphics
© NVidia
•
Modelling
•
Meaningful deformations
•
Animation
•
Realistic movement
Problems:
•
Skinning time consuming
•
Not everything has bones
Computer Graphics Group
Tobias Weyand
4
Mesh
-
based Inverse Kinematics
Idea: Learn deformations from examples
Computer Graphics Group
Tobias Weyand
5
Overview
•
Introduction
•
Related work
•
MeshIK
•
Feature Vectors
•
Linear Feature Space
•
Nonlinear Feature Space
•
Accelerations
•
Results and Conclusion
Computer Graphics Group
Tobias Weyand
6
Related work: Deformation Transfer
Deformation Transfer for Triangle Meshes
Transfer source deformations to target
Sumner et al 2004
Computer Graphics Group
Tobias Weyand
7
Related work: Deformation Transfer
Triangle deformations:
Least squares problem:
i
i
i
s
S
s
~
i
i
i
t
T
t
~
min
||
||
|
|
1
2
M
i
F
i
i
T
S
But: Limited to example poses
Computer Graphics Group
Tobias Weyand
8
Related work: Shape Interpolation
As
-
rigid
-
as
-
possible Shape Interpolation
•
Morphing of 2D and 3D meshes
•
Considers mesh interior as rigid
Alexa et al. 2000
Computer Graphics Group
Tobias Weyand
9
Related work: Shape Interpolation
•
Triangulate source and target shapes
•
Find locally optimal triangle interpolations
But:
•
Limited to 2 meshes
•
Expensive:
•
Compatible dissection
•
Computations on interior and exterior
Better:
•
Use only surface
Computer Graphics Group
Tobias Weyand
10
Mesh
-
based Inverse Kinematics
Goal:
•
Provide a set of example meshes.
•
MeshIK learns meaningful deformations.
•
Directly move a subset of the mesh
vertices.
•
MeshIK finds a suitable deformation
according to the example meshes.
Computer Graphics Group
Tobias Weyand
11
Feature Vectors
Given: Base mesh P
0
, deformed mesh P
Deformation: set of affine mappings
}
{
j
j
j
j
t
p
T
p
)
(
Deformation gradient: Jacobian of
Discard translation
)
(
p
j
j
j
j
p
j
p
T
t
p
T
D
p
D
)
(
)
(
Computer Graphics Group
Tobias Weyand
12
Feature Vectors
Feature vector: concatentaion of
deformation gradients
3
2
1
3
2
1
3
2
1
z
j
z
j
z
j
y
j
y
j
y
j
x
j
x
j
x
j
j
t
t
t
t
t
t
t
t
t
T
3
1
3
1
3
1
3
2
1
1
1
2
1
1
1
f
z
m
z
y
m
y
x
m
x
x
x
x
t
t
t
t
t
t
t
t
t
Computer Graphics Group
Tobias Weyand
13
Feature Vectors
Calculation of f for mesh P
)
,
(
P
p
p
E
V
T
1
1
1
))
,
,
(
,
),
,
,
((
m
m
m
p
z
y
x
z
y
x
V
P
→
mesh
-
vector x:
T
n
1
n
1
n
3
2
1
)
z
z
y
y
x
x
x
x
(
x
Construct G such that:
Gx
f
Computer Graphics Group
Tobias Weyand
14
Feature Vectors
Properties of G:
g
g
g
G
-
Block
-
diagonal structure
-
Sparse
-
Only depends on P
0
Computer Graphics Group
Tobias Weyand
15
Feature Vectors
Extracting a mesh from a feature vector:
Fix one vertex in x:
•
Set corresponding rows in G to 0.
•
Add product of these rows with x.
c
G
x
~
f
Computer Graphics Group
Tobias Weyand
16
Multiple Vertex Constraints
Transform to least squares problem:
c
G
x
~
f
2
x
f
x
~
min
arg
x
c
G
Properties of x:
•
Close relation to the feature vector
•
Fulfills the vertex constraints
Computer Graphics Group
Tobias Weyand
17
Linear Feature Space
Linear combination of features:
l
l
w
w
w
w
f
f
f
f
2
2
1
1
Vector notation:
w
w
M
f
l
1
f
f
M
T
1
l
w
w
w
Mw
c
Gx
w
x
w
x
,
*
*
min
arg
,
Least squares problem
Minimize for optimal weights and mesh
Computer Graphics Group
Tobias Weyand
18
Linear Feature Space: Problems
Unnatural interpolation of rotations
Goal: Correctly capture rotations
Computer Graphics Group
Tobias Weyand
19
Transformation Interpolation
Linear Combinations of Transformations
Scalar product:
Addition:
Combination:
T
a
B
A
a
T
n
n
n
n
B
A
1
1
lim
T
e
log
a
B
A
e
log
log
n
n
A
A
a
a
1
1
n
n
A
A
e
log
log
1
1
a
a
ˆ
ˆ
ˆ
ˆ
ˆ
Alexa 2002
)
log(
a
T
e
ˆ
Computer Graphics Group
Tobias Weyand
20
Possible Nonlinear Feature Space
l
i
ij
i
T
w
j
e
T
1
log
But:
•
Practically produces singularities
•
Linear scales and shears suffice
MeshIK interpolation:
•
Scales and skews: linear
•
Rotations: above formula
Computer Graphics Group
Tobias Weyand
21
Extracting Rotations
Matrix Animation and Polar Decomposition
Method for factoring a transformation T
T=RS
New transformation combination
Shoemake and Duff 1992
l
i
ij
i
R
w
j
S
w
e
w
T
l
i
ij
i
1
)
log(
1
)
(
Computer Graphics Group
Tobias Weyand
22
Nonlinear Feature Space
)
(
x
min
arg
,
x
,
x
*
*
w
M
c
G
w
w
New nonlinear least squares problem:
Properties:
•
Fulfills vertex constraints
•
Close to nonlinear feature space
•
Natural rotation interpolation
Efficient solver required!
Computer Graphics Group
Tobias Weyand
23
Gauss
-
Newton in MeshIK
Goal: minimize
)
(
x
min
arg
,
x
,
x
*
*
w
M
c
G
w
w
Approach with Gauss
-
Newton:
•
Transform into locally linear equation
•
Solve linear least squares problem
Computer Graphics Group
Tobias Weyand
24
Gauss
-
Newton in MeshIK
Linearize M(w) with Taylor expansion
)
(
)
(
)
(
w
M
D
w
M
w
M
w
||
)
(
)
(
||
min
arg
x
,
x
,
c
w
M
w
M
D
Gx
k
k
w
k
k
Linear least squares problem
x
A
w
M
D
G
k
w
)
(
x
Reduce to one variable:
2
x
,
)
(
min
arg
x
,
c
w
M
x
A
k
k
k
)
(
k
w
w
M
D
G
A
Computer Graphics Group
Tobias Weyand
25
Gauss
-
Newton in MeshIK
Set
)
)
(
(
2
x
2
x
'
T
T
c
w
M
A
A
A
F
k
= 0
!
)
)
(
(
x
T
T
c
w
M
A
A
A
k
Iteration steps:
•
Solve the normal equation for
•
Update
x
k
k
w
w
1
But: Far too slow! Optimization needed!
Computer Graphics Group
Tobias Weyand
26
Cholesky Factorization
Linear system:
)
)
(
(
x
T
T
c
w
M
A
A
A
k
)
)
(
(
x
T
T
c
w
M
A
C
C
k
)
)
(
(
T
T
c
w
M
A
y
C
k
y
C
x
C
C
A
A
T
T
Decompose:
Method:
y
x
Computer Graphics Group
Tobias Weyand
27
Cholesky Factorization
Further acceleration:
Exploit structure of A
T
A and C
T
C:
3
1
T
T
3
T
3
3
T
T
3
T
T
2
T
T
1
T
T
i
i
i
J
J
g
J
g
J
J
g
g
J
g
g
g
J
g
g
g
J
g
g
A
A
3
1
T
T
T
3
T
2
T
1
3
T
T
2
T
T
1
T
T
T
i
i
i
S
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
R
C
C
Note: R
T
R=g
T
g
•
Precompute R
T
R
•
Only calculate R
1
, R
2
, R
3
, R
S
Interactive speed
Computer Graphics Group
Tobias Weyand
28
Performance
Mesh
Tris
Examples
Preprocess
Solve
Flag
932
14
0.016
0.020
Lion
9,996
10
0.0475
0.210
Horse
16,846
4
0.610
0.160
Elephant
84,638
4
13.249
0.906
Computer Graphics Group
Tobias Weyand
29
Results
Computer Graphics Group
Tobias Weyand
30
Video
Computer Graphics Group
Tobias Weyand
31
Comparison: Shape Interpolation
Arap.
MeshIK
2 meshes
n meshes
Needs compatible
dissection of meshes
→
expensive
Needs meshes with
same topology
Operates on exterior
and interior
Operates only on
surface
Computer Graphics Group
Tobias Weyand
32
Comparison: Shape Interpolation
Computer Graphics Group
Tobias Weyand
33
Future work
•
Different mesh representation
•
like subdivision surfaces
•
multiresolution hierarchies
•
Different feature vectors
•
Capture mesh properties better
•
Accelerate system solving
•
Better feature space for many examples
•
Simulate physical effects like inertia
Computer Graphics Group
Tobias Weyand
34
Conclusion
MeshIK
•
provides IK without skeleton
•
more direct and dynamic
•
runs at interactive speeds
•
compares well to other approaches
•
eg As
-
Rigid
-
As
-
Possible shape interpolation
Computer Graphics Group
Tobias Weyand
35
Thank you for your attention!
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