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