der, sumner, and popović inverse kinematics for reduced deformable models kevin g. der robert w....

Der, Sumner, and Popović

Inverse Kinematics for Reduced Deformable Models

Kevin G. Der Robert W. Sumner1 Jovan Popović

Computer Science and Artificial Intelligence Laboratory

Massachusetts Institute of Technology

1ETH Zürich

Der, Sumner, and Popović

Our Goal: Interaction

• mesh manipulation• direct, intuitive control• speed

Der, Sumner, and Popović

Related Work

• mesh editing[Zorin et al. 1997, Kobbelt et al. 1998,

Sorkine 2005]

• Mesh-based inverse kinematics[Sumner et al. 2005]

• Skinning mesh animations[ James and Twigg 2005]

Der, Sumner, and Popović

Our Method

Reduced Deformabl

e Model

Mesh Examples

InverseKinematic

s

decouple deformation complexity from geometric complexity!

- completely automated animation pipeline

Der, Sumner, and Popović

Reduced Deformable Model

• expresses deformations compactly

• automatically constructed

Control parameters:

Examples Reduced Deformabl

e Model

[James and Twigg 2005]

• non-rigid• no hierarchy

Skinning weights:(Using mean-shift

clustering)

Der, Sumner, and Popović

RDM: Mesh Reconstruction

Control parameters:

Skinning weights:

20 to 50

huge number

different values estimate each example

x: before deformationv: after deformation

Der, Sumner, and Popović

Inverse Kinematics

• obtain natural poses by blending examples

• blend in which domain?– vertex positions ?– control parameters ?

• find a semantically proper shape

Desired

Blending function.

Der, Sumner, and Popović

Deformation Gradients• simple 3x3 transformation matrices• describe deformation of each example

Skinning equation.

Vertex deformation

gradient.

deformation function

Der, Sumner, and Popović

Deformation Gradients

flattened deformation gradients of vertices

constant matrix

flattened control matrices

for each example: have t → get f

Der, Sumner, and Popović

Shape Blending

• combine example deformation gradients

Blending function.

• recover the control parameters given deformation gradientsHas closed form

solution.

solving for reduced basis!but still slow

Der, Sumner, and Popović

Proxy Vertices

• evaluating deformation gradients at every vertex is undesirable

• summarize using a few vertices

same deformation gradient!

controlsvertex groups

weighted centroid of the group’s vertices

))(1(

))(1(

iVi

iiVi

X

XXf

Der, Sumner, and Popović

Inverse Kinematics

“ Find best natural pose that meets user constraints”

)( bCt

When there’s only a few constraints:

2

1,

**1 ))((minarg,

*1

bGCmGNttt

1NtbCt

Der, Sumner, and Popović

Results

0

1

2

3

4

MeshIK Our method

seconds

Solving time for interaction

Der, Sumner, and Popović

ResultsSolving time for interaction

Mesh Vertices Example Ct Pxy Total Solve MIK Efull(Eproxy)

Horse 8431 12 33 195 0.019 0.007 0.424 3.01(3.00)

Human 12500 10 42 471 0.031 0.013 0.675 6.01(5.95)

Mouse 13188 26 48 278 0.037 0.019 1.351 1.63(1.73)

Dragon 15560 7 72 380 0.046 0.023 0.590 0.96(0.97)

Gorilla 25436 27 28 198 0.045 0.013 2.962 4.38(4.33)

Elephant 42321 11 23 210 0.058 0.008 1.928 5.38(5.46)

Der, Sumner, and Popović

Conclusion

• interactive control of reduced deformable models

• future work– error correction for new poses– model transfer between meshes– other reduced deformable models