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 DM

2 T DM CRF

Short Introduction to CRF

CRF-Model for State Estimation

3 A

CRF-Filter Algorithm

Learning the Parameters

4 E R

O

1 I: S E U DM

2 T DM CRF



3 A



4 E R

Intro Transformation of Directed Model to CRF Application Experimental Results

C: S EC DM A S E

ut−2 ut−1

xt−1 xt

z1t−1 z2

t−1 znt−1 z1

t z2t zn

t

. . . . . .

xt−2

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 DM P

p(zt |xt) =∏n

i=1 p(z it |xt ) p(xt+1|xt , u)

u = (δrot1, δrot2, δtrans)executed with gaussiannoise


D DM P

p(zt |xt) =∏n

i=1 p(z it |xt )

P (zit|xt) zi

t zmax

zrand

p(xt+1|xt , u)



D DM P

p(zt |xt) =∏n

i=1 p(z it |xt )

P (zit|xt) zi

t zmax

zrand

p(xt+1|xt , u)



D DM P

p(zt |xt) =∏n

i=1 p(z it |xt )

P (zit|xt) zi

t zmax

zrand

p(xt+1|xt , u)

δrot1

δrot2

δtrans

xt−1

xt



A P D A

xt

zt

p(z it |xt) are not cond. independent

Sensor models can only begenerated seperatly for each beamAssumption that measurementsare independent: “Workssurprisingly well”. . . if. . .

increasing uncertainty (tweaking)using every 10th measurement. . .


A P D A

ut−2 ut−1

xt−1 xt

z1t−1 z2

t−1 znt−1 z1

t z2t zn

t

. . . . . .

xt−2

P (zit|xt) zi

t zmax

zrand


Sensor models can only begenerated seperatly for each beam

Assumption that measurementsare independent: “Workssurprisingly well”. . . if. . .



A P D A

ut−2 ut−1

xt−1 xt

z1t−1 z2

t−1 znt−1 z1

t z2t zn

t

. . . . . .

xt−2

P (zit|xt) zi

t zmax

zrand





A P D A

ut−2 ut−1

xt−1 xt

z1t−1 z2

t−1 znt−1 z1

t z2t zn

t

. . . . . .

xt−2

P (zit|xt) zi

t zmax

zrand




O

1 I: S E U DM

2 T DM CRF



3 A



4 E R

O

1 I: S E U DM

2 T DM CRF



3 A



4 E R


I: CRF

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2

Undirected graphical models

Every (possible) dependencyrepresented by edge

Distribution defined over productsof functions over cliques

Functions are called cliquepotentials

Clique potentials representcompatibility of their variables


I: CRF

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2







I: CRF

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2







I: CRF

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2







I: CRF

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2






O

1 I: S E U DM

2 T DM CRF



3 A



4 E R


CRF-M S E

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2

p(x0:T |z1:T ,u0:T−1) =1

Z(z1:T ,u1:T−1)

T∏t=1

ϕp(xt , xt−1,ut−1)ϕm(xt , zt)

Z(·):∑

all trajectories∏ϕp(·)ϕm(·)

How to define ϕp(·) and ϕm(·)?


CRF-M S E

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2

p(x0:T |z1:T ,u0:T−1) =1

Z(z1:T ,u1:T−1)

T∏t=1


Z(·):∑




CRF-M S E

ut−2 ut−1

xt−1 xt

zt−1 zt

xt−2

p(x0:T |z1:T ,u0:T−1) =1

Z(z1:T ,u1:T−1)

T∏t=1


Z(·):∑




T P P φp

δrot1

δrot2

δtrans

xt−1

xt

ut−1 = (δrot1, δtrans , δrot2) odometry

ut−1 = (δrot1, δtrans , δrot2) derived odometry

. Before: Gaussian noise N(ui

t−1, σi2)

fp(xt , xt−1, ut−1) =

(δrot1 − δrot1)

2

(δtrans − δtrans)2

(δrot2 − δrot2)2

3 features

φp(xt , xt−1, ut−1) = exp{〈wp , fp(xt , xt−1, ut−1)〉

}N

(a,

1σ2

)= exp

(−

(a − a)2

2σ2

)

. Gaussian noise N(ui

t−1,1−2w i

p

)if w i

p < 0


T P P φp

δrot1

δrot2

δtrans

xt−1

xt




t−1, σi2)

fp(xt , xt−1, ut−1) =

(δrot1 − δrot1)

2



3 features


}N

(a,

1σ2

)= exp

(−

(a − a)2

2σ2

)


t−1,1−2w i

p

)if w i

p < 0


T P P φp

δrot1

δrot2

δtrans

xt−1

xt




t−1, σi2)

fp(xt , xt−1, ut−1) =

(δrot1 − δrot1)

2



3 features


}

N

(a,

1σ2

)= exp

(−

(a − a)2

2σ2

)


t−1,1−2w i

p

)if w i

p < 0


T P P φp

δrot1

δrot2

δtrans

xt−1

xt




t−1, σi2)

fp(xt , xt−1, ut−1) =

(δrot1 − δrot1)

2



3 features


}N

(a,

1σ2

)= exp

(−

(a − a)2

2σ2

)


t−1,1−2w i

p

)if w i

p < 0


R: SM N B A

P (zit|xt) zi

t zmax

zrand

p(zt |xt) =n∏

i=1

p(z it |xt)


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mit ∈ {1, 0} measured zmax

mit ∈ {1, 0} expected zmax

c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mi

t ∈ {1, 0} measured zmax


c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mi



c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mi



c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mi



c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mi



c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand


M P φm

φm(xt , zt) = exp

⟨wm,

n∑i=0

f im(zt , xt)

⟩

f im(zt , xt) =

(¬mit ∧ ¬mi

t)cit(z

it − z i

t)2

(¬mit ∧ ¬mi

t)¬c it

(¬mit ∧ mi

t)

( mit ∧ ¬mi

t)

( mit ∧ mi

t)

mi



c it ∈ {1, 0} z i

t − z it < 20cm

P (zit|xt) zi

t zmax

zrand

O

1 I: S E U DM

2 T DM CRF



3 A



4 E R

O

1 I: S E U DM

2 T DM CRF



3 A



4 E R


U CRF P F

At each time step t :Prediction

Move particles according to gaussian noisedetermined by wp

Same as sampling from N(ui

t−1,1−2w i

p

)Correction

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


U CRF P F




t−1,1−2w i

p

)Correction


u


U CRF P F




t−1,1−2w i

p

)Correction


movedparticles


U CRF P F




t−1,1−2w i

p

)Correction


addednoise


U CRF P F




t−1,1−2w i

p

)Correction


...sense...


U CRF P F




t−1,1−2w i

p

)Correction


weights


U CRF P F




t−1,1−2w i

p

)Correction


resample

O

1 I: S E U DM

2 T DM CRF



3 A



4 E R


D wp wm

Drive around in test area

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

Using arbitrary weights, generate trajectory xwith CRF-filter

Use difference of summed features as weightupdate(−):wk = wk−1 + α (

∑f(x∗,u, z) −

∑f(x,u, z))

Decrease α if new Filter cannot track

loop

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


D wp wm





∑f(x∗,u, z) −

∑f(x,u, z))


loop



D wp wm





∑f(x∗,u, z) −

∑f(x,u, z))


loop



D wp wm





∑f(x∗,u, z) −

∑f(x,u, z))


loop



D wp wm





∑f(x∗,u, z) −

∑f(x,u, z))


loop



D wp wm





∑f(x∗,u, z) −

∑f(x,u, z))


loop



L A

Averaged Perceptron Algorithm (Collins 2002) for tagging

wk = wk−1 + α(∑

f(x∗,u, z) −∑

f(x,u, z))

Proven to converge even in presence of errors in training dataIntuition of learning algorithm:

If PF works correctly, then∑f(x∗n , un−1, zn) =

∑f(xn, un−1, zn)

f i occurs less often in x∗ than in x→ decrease influence of f i

on particle filter by decreasing w i

O

1 I: S E U DM

2 T DM CRF



3 A



4 E R


E R

Properties of the learned weightsNorm of weight vector decreases withnumber of laser beams in z

. believes the features/measurements less

. equivalent to initially introduced“tweaking”?!

Two specialized CRF-filters compared togenerative particle filter trained usingexpectation maximization

TrackingError

GlobalLocalizationAccuracy

Generative 7.52 cm 30%CRF-Filter 7.07 cm 96%


E R

Properties of the learned weightsNorm of weight vector decreases withnumber of laser beams in z

. believes the features/measurements less

. equivalent to initially introduced“tweaking”?!

Two specialized CRF-filters compared togenerative particle filter trained usingexpectation maximization

TrackingError

GlobalLocalizationAccuracy

Generative 7.52 cm 30%CRF-Filter 7.07 cm 96%


C

1 A CRF is an alternative, undirected graphical model

2 CRF-Filters use a continuous CRF for recursive stateestimation

3 . . . can be trained to maximize filter performance dependingon the task

4 . . . can deal with correlated measurements5 . . . do not explicitly account for dependencies between sensor

data


C

1 A CRF is an alternative, undirected graphical model2 CRF-Filters use a continuous CRF for recursive state

estimation

3 . . . can be trained to maximize filter performance dependingon the task


data


C


estimation3 . . . can be trained to maximize filter performance depending

on the task


data


C



on the task4 . . . can deal with correlated measurements

5 . . . do not explicitly account for dependencies between sensordata


C



on the task4 . . . can deal with correlated measurements5 . . . do not explicitly account for dependencies between sensor

data

Technology

CRF-Filters: Discriminative Particle Filters for Sequential State Estimation