computer graphics group tobias weyand mesh-based inverse kinematics sumner et al 2005 presented by...

Computer Graphics GroupTobias Weyand

Mesh-Based Inverse Kinematics

Sumner et al 2005

presented by Tobias Weyand

Computer Graphics GroupTobias Weyand2

What is Inverse Kinematics?

• Articulated body

• Which angles for a certain configuration?

• Forward kinematics• Specify angles

• Inverse kinematics• Specify limb position

1

2

3

1

2

3


IK in Computer Graphics

© NVidia

• Modelling

• Meaningful deformations

• Animation

• Realistic movement

Problems:

• Skinning time consuming

• Not everything has bones


Mesh-based Inverse Kinematics

Idea: Learn deformations from examples


Overview

• Introduction• Related work

• MeshIK• Feature Vectors• Linear Feature Space• Nonlinear Feature Space• Accelerations

• Results and Conclusion


Related work: Deformation Transfer

Deformation Transfer for Triangle Meshes

Transfer source deformations to target

Sumner et al 2004


Related work: Deformation TransferTriangle deformations:

Least squares problem:

iii sSs ~iii tTt ~

min||||||

1

2

M

iFii TS

But: Limited to example poses


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


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


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.


Feature Vectors

Given: Base mesh P0, deformed mesh P

Deformation: set of affine mappings }{ j

jjj tpTp )(

Deformation gradient: Jacobian of

Discard translation

)( pj

jjjpjp TtpTDpD )()(


Feature Vectors

Feature vector: concatentaion of deformation gradients

321

321

321

zj

zj

zj

yj

yj

yj

xj

xj

xj

j

ttt

ttt

ttt

T

313131321112111f z

mzy

myx

mxxxx ttttttttt


Feature Vectors

Calculation of f for mesh P

),(P pp EV T111 )),,(,),,,(( mmmp zyxzyxV

P → mesh-vector x:T

n1n1n321 )zzyyxxxx(x

Construct G such that:

Gxf


Feature Vectors

Properties of G:

g

g

g

G

- Block-diagonal structure- Sparse- Only depends on P0


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.

cG x~

f


Multiple Vertex Constraints

Transform to least squares problem:

cG x~

f2

xfx

~minargx cG

Properties of x:• Close relation to the feature vector• Fulfills the vertex constraints


Linear Feature Space

Linear combination of features:

llw www ffff 2211

Vector notation: ww Mf

l1 ffM T1 lwww

MwcGxwxwx

,

** minarg,

Least squares problem

Minimize for optimal weights and mesh


Linear Feature Space: Problems

Unnatural interpolation of rotations

Goal: Correctly capture rotations


Transformation Interpolation

Linear Combinations of Transformations

Scalar product:

Addition:

Combination:

T

BA

T

n

nn

nBA

11

lim

Te log

BAe loglog

nn AA 11nn AAe loglog 11

Alexa 2002

)log( Te


Possible Nonlinear Feature Space

l

iiji Tw

j eT 1

log

But:• Practically produces singularities• Linear scales and shears suffice

MeshIK interpolation:• Scales and skews: linear• Rotations: above formula


Extracting Rotations

Matrix Animation and Polar Decomposition

Method for factoring a transformation T

T=RS

New transformation combination

Shoemake and Duff 1992

l

iiji

Rw

j SwewT

l

iiji

1

)log(1)(


Nonlinear Feature Space

)(xminarg,x,x

** wMcGww

New nonlinear least squares problem:

Properties:• Fulfills vertex constraints• Close to nonlinear feature space• Natural rotation interpolation

Efficient solver required!


Gauss-Newton in MeshIK

Goal: minimize

)(xminarg,x,x

** wMcGww

Approach with Gauss-Newton:• Transform into locally linear equation • Solve linear least squares problem



Linearize M(w) with Taylor expansion

)()()( wMDwMwM w

||)()(||minargx,x,

cwMwMDGx kkwkk

Linear least squares problem

xAwMDG kw )(x

Reduce to one variable:

2

x,)(minargx, cwMxA kkk

)( kw wMDGA



Set ))((2x2x' TT cwMAAAF k

= 0!

))((x TT cwMAAA k

Iteration steps:• Solve the normal equation for• Update

x

kk ww 1

But: Far too slow! Optimization needed!


Cholesky Factorization

Linear system:

))((x TT cwMAAA k

))((x TT cwMACC k

))((TT cwMAyC k

yC

x

CCAA TT

Decompose:

Method:y

x


Cholesky FactorizationFurther acceleration:

Exploit structure of ATA and CTC:

3

1

TT3

T33

T

T3

T

T2

T

T1

T

T

i ii JJgJgJJg

gJgggJgggJgg

AA

3

1

TTT3

T2

T1

3TT

2TT

1TT

T

i iiS RRRRRRRRRR

RRRRRRRRRRRR

CC

Note: RTR=gTg• Precompute RTR• Only calculate R1, R2, R3, RS

Interactive speed


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


Results


Video


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


Comparison: Shape Interpolation


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


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


Thank you for your attention!

computer graphics group tobias weyand mesh-based inverse kinematics sumner et al 2005 presented by...

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