CRF-Filters: Discriminative Particle Filters for Sequential State Estimation

CRF-F: D P F 
S S E
B L, D F  L L

Hannes Schulz

University of Freiburg, ACS

Feb 2008

O

1 I: S E U D M

2 T  D M  CRF
Short Introduction to CRF
CRF-Model for State Estimation

3 A
CRF-Filter Algorithm
Learning the Parameters

4 E R

Intro Transformation of Directed Model to CRF Application Experimental Results

C: S E
C D M A  S E

ut−2 ut−1

xt−2 xt−1 xt

... ... n
1 2 n
zt−1 1
zt 2
zt zt
zt−1 zt−1

P (xt |u1:t −1 , z1:t ) = ηP (zt |xt ) P (xt |ut −1 , xt −1 )P (xt −1 |u1:t −2 , z1:t −1 ) dxt −1


D  D M P

p (zt |xt ) = n
i =1 p (zti |xt ) p (xt +1 |xt , u)


D  D M P

p (zt |xt ) = n
i =1 p (zti |xt ) p (xt +1 |xt , u)
i
P (zt |xt) ˆi
zt zmax

zrand


D  D M P

p (zt |xt ) = n
i =1 p (zti |xt ) p (xt +1 |xt , u)
i
P (zt |xt) ˆi
zt zmax
δrot2

zrand
xt

δtrans

δrot1
xt−1

u = (δrot1 , δrot2 , δtrans )
executed with gaussian
noise


A P   D A

p (zti |xt ) are not cond. independent

zt
xt


A P   D A

ut−2 ut−1

xt−2 xt−1 xt p (zti |xt ) are not cond. independent
Sensor models can only be
... ... n
1 2 n
zt−1 1 2 zt
zt−1 zt−1 zt zt generated seperatly for each beam

i
P (zt |xt) ˆi
zt zmax

zrand


A P   D A

ut−2 ut−1

... ... n
1 2 n
zt−1 1 2 zt
Assumption that measurements
are independent: “Works
i
P (zt |xt) ˆi
zt zmax
surprisingly well”. . . if. . .

zrand


A P   D A

ut−2 ut−1

... ... n
1 2 n
zt−1 1 2 zt
Assumption that measurements
are independent: “Works
i
P (zt |xt) ˆi
zt zmax
surprisingly well”. . . if. . .
increasing uncertainty (tweaking)
using every 10th measurement
zrand ...


I: CRF

Undirected graphical models

ut−2 ut−1

xt−2 xt−1 xt

zt−1 zt


I: CRF

Every (possible) dependency
ut−2 ut−1 represented by edge

xt−2 xt−1 xt

zt−1 zt


I: CRF

Distribution deﬁned over products
xt−2 xt−1 xt
of functions over cliques
zt−1 zt


I: CRF

xt−2 xt−1 xt
zt−1 zt Functions are called clique
potentials


I: CRF

xt−2 xt−1 xt
zt−1 zt Functions are called clique
potentials
Clique potentials represent
compatibility of their variables


CRF-M  S E

ut−2 ut−1

xt−2 xt−1 xt

zt−1 zt

T
1
p (x0:T |z1:T , u0:T −1 ) = ϕp (xt , xt −1 , ut −1 )ϕm (xt , zt )
Z (z1:T , u1:T −1 )
t =1


CRF-M  S E

ut−2 ut−1

xt−2 xt−1 xt

zt−1 zt

T
1
Z (z1:T , u1:T −1 )
t =1

Z (·): all trajectories ϕp (·)ϕm (·)


CRF-M  S E

ut−2 ut−1

xt−2 xt−1 xt

zt−1 zt

T
1
Z (z1:T , u1:T −1 )
t =1

Z (·): all trajectories ϕp (·)ϕm (·)
How to deﬁne ϕp (·) and ϕm (·)?


T P P φp

ut −1 = (δrot1 , δtrans , δrot2 ) odometry
ut −1 = (δrot1 , δtrans , δrot2 ) derived odometry
ˆ ˆ ˆ ˆ
δrot2
2
Before: Gaussian noise N uti −1 , σi
xt

δtrans

δrot1
xt−1


T P P φp

ˆ ˆ ˆ ˆ
δrot2
2
xt
 (δrot1 − δrot1 )2
ˆ
 
 

δtrans
 
fp (xt , xt −1 , ut −1 ) =  (δtrans − δtrans )2
 



 ˆ 



 3 features
 
(δrot2 − δrot2 )2
ˆ

 


δrot1
xt−1


T P P φp

ˆ ˆ ˆ ˆ
δrot2
2
xt
ˆ
 
 

δtrans
 
 



 ˆ 



 3 features
 
ˆ

 


δrot1 φp (xt , xt −1 , ut −1 ) = exp wp , fp (xt , xt −1 , ut −1 )
xt−1


T P P φp

ˆ ˆ ˆ ˆ
δrot2
2
xt
ˆ
 
 

δtrans
 
 



 ˆ 



 3 features
 
ˆ

 


δrot1 φp (xt , xt −1 , ut −1 ) = exp wp , fp (xt , xt −1 , ut −1 )
xt−1 1 (a − a )2
ˆ
N a, = exp −
σ2 2σ2

Gaussian noise N uti −1 , 1
−2wpi if wp < 0
i


R: S M   N¨ B A


i
P (zt |xt) ˆi
zt zmax

zrand

n
p (zt |xt ) = p (zti |xt )
i =1


M P φm
i
P (zt |xt) ˆi
zt zmax
n
 
 
φm (xt , zt ) = exp  wm , fm (zt , xt ) 
i

 


 

i =0
zrand

(¬mti ∧ ¬mti )cti (zti − zti )2
ˆ ˆ
 


 


i i i
 




 ˆ
(¬mt ∧ ¬mt )¬ct





fm (zt , xt ) = 
i
 
(¬mti ∧ mti )
ˆ

 




 


( mti ∧ ¬mti )
 




 ˆ 




 
( mti ∧ mti )
ˆ
 


M P φm
i
P (zt |xt) ˆi
zt zmax
n
 
 
i

 


 

i =0
zrand

ˆ ˆ
 


 


i i i
 




 ˆ





fm (zt , xt ) = 
i
 
(¬mti ∧ mti )
ˆ

 




 


( mti ∧ ¬mti )
 




 ˆ 




 
( mti ∧ mti )
ˆ
 

mti ∈ {1, 0} measured zmax


M P φm
i
P (zt |xt) ˆi
zt zmax
n
 
 
i

 


 

i =0
zrand

ˆ ˆ
 


 


i i i
 




 ˆ





fm (zt , xt ) = 
i
 
(¬mti ∧ mti )
ˆ

 




 


( mti ∧ ¬mti )
 




 ˆ 




 
( mti ∧ mti )
ˆ
 

mti ∈ {1, 0} expected zmax
ˆ


M P φm
i
P (zt |xt) ˆi
zt zmax
n
 
 
i

 


 

i =0
zrand

ˆ ˆ
 


 


i i i
 




 ˆ





fm (zt , xt ) = 
i
 
(¬mti ∧ mti )
ˆ

 




 


( mti ∧ ¬mti )
 




 ˆ 




 
( mti ∧ mti )
ˆ
 

mti ∈ {1, 0} expected zmax
ˆ
cti ∈ {1, 0} zti − zti < 20cm
ˆ


U  CRF    P F

At each time step t:
Prediction
Move particles according to gaussian noise
determined by wp
Same as sampling from N uti −1 ,
ˆ 1
i
−2wp

Correction
Particle at xt gets weight φm (xt , zt )
Resample (includes normalization)


U  CRF    P F

Prediction
determined by wp
u
ˆ 1
i
−2wp

Correction


U  CRF    P F

Prediction
determined by wp moved
ˆ 1
−2wpi particles
Correction


U  CRF    P F

Prediction
determined by wp added
Same as sampling from N uti −1 , noise
1
ˆ −2wpi

Correction


U  CRF    P F

Prediction
determined by wp ...sense...
ˆ 1
−2wpi

Correction


U  CRF    P F

Prediction
determined by wp weights
ˆ 1
−2wpi

Correction


U  CRF    P F

Prediction
determined by wp
resample
ˆ 1
i
−2wp

Correction


D     wp  wm

Drive around in test area


D     wp  wm

Use high-quality scanmatcher to generate
“ground truth” trajectory x∗


D     wp  wm

ˆ
Using arbitrary weights, generate trajectory x
with CRF-ﬁlter


D     wp  wm

ˆ
with CRF-ﬁlter
Use difference of summed features as weight
update(−) :
wk = wk −1 + α ( f (x∗ , u, z) − f (x, u, z))
ˆ


D     wp  wm

ˆ
with CRF-ﬁlter
update(−) :
wk = wk −1 + α ( f (x∗ , u, z) − f (x, u, z))
ˆ
Decrease α if new Filter cannot track


D     wp  wm

ˆ
with CRF-ﬁlter
update(−) :
wk = wk −1 + α ( f (x∗ , u, z) − f (x, u, z))
ˆ
Decrease α if new Filter cannot track
loop

Adapts weights to task, sensor dependencies/environment,
sensor noise, particle ﬁlter parameters


L A

Averaged Perceptron Algorithm (Collins 2002) for tagging

w k = w k −1 + α f (x∗ , u, z) − f (x, u, z)
ˆ

Proven to converge even in presence of errors in training data
Intuition of learning algorithm:
If PF works correctly, then

f (xn , un−1 , zn ) =
∗
f (xn , un−1 , zn )
ˆ

f i occurs less often in x∗ than in x → decrease inﬂuence of f i
ˆ
on particle ﬁlter by decreasing w i


E R

Properties of the learned weights
Norm of weight vector decreases with
number of laser beams in z
believes the features/measurements less
equivalent to initially introduced
“tweaking”?!


E R

Properties of the learned weights
Norm of weight vector decreases with
number of laser beams in z
believes the features/measurements less
equivalent to initially introduced
“tweaking”?!
Two specialized CRF-ﬁlters compared to
generative particle ﬁlter trained using
expectation maximization

Tracking Global
Error Localization
Accuracy
Generative 7.52 cm 30%
CRF-Filter 7.07 cm 96%


C

1 A CRF is an alternative, undirected graphical model


C

2 CRF-Filters use a continuous CRF for recursive state
estimation


C

estimation
3 . . . can be trained to maximize ﬁlter performance depending
on the task


C

estimation
on the task
4 . . . can deal with correlated measurements


C

estimation
on the task
4 . . . can deal with correlated measurements
5 . . . do not explicitly account for dependencies between sensor
data

CRF-Filters: Discriminative Particle Filters for Sequential State Estimation

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CRF-Filters: Discriminative Particle Filters for Sequential State Estimation