sppra2010 estimating a rotation matrix r by using higher-order matrices r^n with application to...

Estimating a rotation matrix Rby using higher-order matrices Rn

with application tosupervised pose estimation

Toru Tamaki

Bisser Raytchev

Kazufumi Kaneda

Toshiyuki Amano

Estimating a rotation matrix Rby using higher-order matrices Rn

with application tosupervised pose estimation

Toru Tamaki

Bisser Raytchev

Kazufumi Kaneda

Toshiyuki Amano

Can Rn estimate Rmore accurately than R ?

To  improve  estimates…  Average!

t1

t2

tn

t… …Measure many times

AverageLease-Squares

How improve?

When  measurement  is  only  once…

To  improve  estimates…  Average!

t1

t2

tn

t… …Measure with many timers

AverageLease-Squares

When  measurement  is  only  once…

Improve!

To  improve  estimates…  Average?

…

t

t2

tn

…Measure with many timers

When  measurement  is  only  once…

How improve?

t

AverageLease-Squares

To  improve  estimates…  Average.

t

t2

tn

t… …Measure with many timers

When  measurement  is  only  once…

Improve!

But, What is it?

Our problem: Pose estimation

Pose parameters

Rt

3x3 rotation matrix

3D translation

R

image

R

Regression:Appearance-based / View-based pose estimation

Parametric Eigenspace (Murase et al., 1995)linear regression (Okatani et al., 2000)

kernel CCA (Melzer et al., 2003)SV regression (Ando et al., 2005)

Manifold learning, and others(Rothganger et al., 2006) (Lowe, 2004) (Ferrari et

al., 2006) (Kushal et al., 2006) (Viksten, 2009)

Poseparameters

Our concept

p1 R

Rotationmatrix

Posevectorimage

µ

!1

Training

2µ

!

p2 R22

axis

angle

! µ

Our concept

p1

Posevector

µ

!

2µ

!

p22

1 R

Rotationmatrix

R2

PolarDecomp.

! µaxis angle

! 2µ

EigenDecomp.

Newimage

axis

angle

Our concept

p1 R ! µ

Rotationmatrix axis angle

Posevector

µ

!

2µ

!

p2 R2 ! 2µ2

1Polar

Decomp.Eigen

Decomp.

! µ

Newimage

29 [deg]

62 [deg]

30 [deg]

Examples

210 [deg]

420 [deg]

? [deg]

31 [deg]

=60 [deg]

30 [deg]Div by 2 Div by 2

axis

angle

Surveying

Surveying – ARCHEOSCANhttp://archeoscan.com/16.html

EDMElectronic Distance Measurement

Principle of EDM

µ2

Dist

µ1

Transmitter Receiver

¸1

¸2

targetdevice

Use•Longer wavelength ¸1 first, for a rough phase estimate µ1•Shorter wavelength ¸2 next, for a fine phase estimate µ2

Our concept

p1 R ! µ

Rotationmatrix axis angle

Posevector

µ

!

2µ

!

p2 R2 ! 2µ2

1Polar

Decomp.Eigen

Decomp.

! µ

Newimage

29 [deg]

62 [deg]

30 [deg]

Examples

210 [deg]

420 [deg]

210 [deg]

31 [deg]

=60 [deg]

210 [deg]Div by 2 Div by 2

Simulation 1

!

µ

R

p1

Posevector

p8

+noise

+noise

… R

Rotationmatrix

R8

PolarDecomp.

… !1 µ1

axis angle

!8 µ8

EigenDecomp.

… …R R

R8

…

Measurements

Simulation 2

p1 R !1 µ1

Rotationmatrix axis angle

Posevector

p8 R8 !8 µ8

PolarDecomp.

EigenDecomp.

!

µ

+noise R

R8

R

… … … … …

Error  doesn’t  change…No free lunch!

R

Measurements

Experimental results

p1 R !1 µ1

Rotationmatrix axis angle

Posevector

p8 R8 !8 µ8

PolarDecomp.

EigenDecomp.

!

µ

R

… … … …1

8

•Linear regression•Training withimages and poses

images

Summary

• Improve estimates of a pose R with many measurements R1, R2,  …,  R8• Simulations and experimental results shows that the concept works!