tracking with focus on the particle filter (part ii) michael rubinstein idc

Trackingwith focus on the particle filter

(part II)Michael Rubinstein

IDC

Last time…

• Background– State space– Dynamic systems– Recursive Bayesian filters– Restrictive cases

• Kalman filter• Grid-based filter

– Suboptimal approximations• Extended Kalman filter

© Michael Rubinstein

This talk

• Particle filtering– MC integration– Sequential Importance Sampling (SIS)– Resampling– PF variants

• Multiple-target tracking– BraMBLe: A Bayesian Multiple-Blob Tracker/ Isard,

MacCormick

© Michael Rubinstein

Dynamic System

State equation: state vector at time instant k state transition function, i.i.d process noise

Observation equation:

observations at time instant kobservation function,i.i.d measurement noise

),( 1 kkkk vxfx

kxkf

xvx NNNk RRRf :

kv

),( kkkk wxhz kz

khkw

zwx NNNk RRRh :

© Michael Rubinstein

k-1 k k+1xk-1 xk xk+1

zk-1 zk zk+1

kf

kh

Stochastic diffusion

Recursive Bayes filter

• Prediction:

• Update:

© Michael Rubinstein

11:1111:1 )|()|()|( kkkkkkk dxzxpxxpzxp (1)

)|(

)|()|()|(

1:1

1:1:1

kk

kkkkkk zzp

zxpxzpzxp

kkkkkkk dxzxpxzpzzp )|()|()|( 1:11:1

(2)

Sample space

)|( :1 kk zxp

Particle filtering

• Many variations, one general concept:

• Randomly Chosen = Monte Carlo (MC)• As the number of samples become very large – the

characterization becomes an equivalent representation of the true pdf

Represent the posterior pdf by a set of randomly chosen weighted samples (particles)

© Michael Rubinstein

Sample space

Posterior

Particle filtering

• Compared to methods we’ve mentioned last time– Can represent any arbitrary distribution

– multimodal support

– Keep track of many hypotheses as there are particles– Approximate representation of complex model

rather than exact representation of simplified model

• The basic building-block: Importance Sampling

© Michael Rubinstein

Monte Carlo integration

• Evaluate complex integrals using probabilistic techniques

• Assume we are trying to estimate a complicated integral of a function f over some domain D:

• Also assume there exists some PDF p defined over D

xdxfFD

)(

© Michael Rubinstein

Monte Carlo integration

• Then

• But

• This is true for any PDF p over D!

xdxpxp

xfxdxfF

DD

)(

)(

)()(

pxxp

xfExdxp

xp

xfD

~,)(

)()(

)(

)(

© Michael Rubinstein

Monte Carlo integration

• Now, if we have i.i.d random samples sampled from p, then we can approximate

by

• Guaranteed by law of large numbers:

Nxx

,...,1

)(

)(

xp

xfE

N

i i

iN xp

xf

NF

1 )(

)(1

Fxp

xfEFN

sa

N

)(

)(,

.

© Michael Rubinstein

Importance Sampling (IS)

• What about ? • If p is very small, can be arbitrarily large,

‘damaging’ the average• Design p such that is bounded• Rule of thumb: take p similar to f as possible

• The effect: get more samples in ‘important’ areas of f, i.e. where f is large

0)( xp

pf /

pf /

© Michael Rubinstein

Importance weights

Importance or proposal density

Convergence of MC integration• Chebyshev’s inequality: let X be a

random variable with expected value μ and std σ. For any real number k>0,

• For example, for , it shows that at least half the values lie in interval

• Let , then MC estimator is)(

)(

i

ii xp

xfy

N

iiN y

NF

1

1

2

1}|Pr{|

kkX

2k)2,2(

Pafnuty Lvovich Chebyshev

© Michael Rubinstein

Convergence of MC integration• By Chebyshev’s,

• Hence, for a fixed threshold, the error decreases at rate

yVN

yVN

yVN

yN

VFVN

ii

N

ii

N

iiN

1111][

12

12

1

}

][|][Pr{|

21

NNN

FVFEF

}

][1|Pr{|

21

yV

NFFN

)/1( k

N/1

© Michael Rubinstein

Convergence of MC integration

• Meaning1. To cut the error in half, it is necessary to

evaluate 4 times as many samples2. Convergence rate is independent of the

integrand dimension!• On contrast, the convergence rate of grid-based

approximations decreases as increases

© Michael Rubinstein

xN

IS for Bayesian estimation

• We characterize the posterior pdf using a set of samples (particles) and their weights

• Then the joint posterior density at time k is approximated by

Ni

ik

ik wx 1:0 },{

N

i

ikk

ikkk xxwzxp

1:0:0:1:0 )()|(

© Michael Rubinstein

kkk

X kk

kkk

kkk

X

k

dxzxqzxq

zxpxf

dxzxpxfXfE

:0:1:0:1:0

:1:0:0

:0:1:0:0

)|()|(

)|()(

)|()())((

IS for Bayesian estimation

• We draw the samples from the importance density with importance weights

• Sequential update (after some calculation…)

)|( :1:0 kk zxq

)|(

)|(

:1:0

:1:0

kk

kkik zxq

zxpw

© Michael Rubinstein

),|(~ 1 kikk

ik zxxqx

Particle update

),|(

)|()|(

1

11

kik

ik

ik

ik

ikki

kik zxxq

xxpxzpww

Weight update

Sequential Importance Sampling (SIS)

• FOR i=1:N– Draw– Update weights

• END• Normalize weights

kNi

ik

ik

Ni

ik

ik zwxwx ,},{SIS},{ 1111

),|(~ 1 kikk

ik zxxqx

),|(

)|()|(

1

11

kik

ik

ik

ik

ikki

kik zxxq

xxpxzpww

© Michael Rubinstein

State estimates

• Any function can be calculated by discrete pdf approximation

)( kxf

N

i

ik

ikk xfw

NxfE

1

)(1

)(

© Michael Rubinstein

MAP Mean

Robust mean• Example:– Mean (simple average)– MAP estimate: particle with

largest weight– Robust mean: mean within

window around MAP estimate

Choice of importance density

© Michael Rubinstein

Hsiao et al.

Choice of importance density

• Most common (suboptimal): the transitional prior

© Michael Rubinstein

)|(),|(

)|()|(

)|(),|(

11

11

11

ikk

ik

kik

ik

ik

ik

ikki

kik

ikkk

ikk

xzpwzxxq

xxpxzpww

xxpzxxq

sN

j

jkk

jkk

ikk

ikki

kk

xzpw

xzpww

11|

1||

)|(

)|(Grid filter weight update:

The degeneracy phenomenon

• Unavoidable problem with SIS: after a few iterations most particles have negligible weights– Large computational effort for updating particles

with very small contribution to

• Measure of degeneracy - the effective sample size:

– Uniform: , severe degeneracy:

)|( :1 kk zxp

N

i

ik

effw

N1

2)(

1

NNeff 1effN

© Michael Rubinstein

Resampling

• The idea: when degeneracy is above some threshold, eliminate particles with low importance weights and multiply particles with high importance weights

• The new set is generated by sampling with replacement from the discrete representation of such that

NiN

ik

Ni

ik

ik xwx 1

1*1 },{},{

)|( :1 kk zxp jk

jk

ik wxx }Pr{ *

© Michael Rubinstein

Resampling

• Generate N i.i.d variables• Sort them in ascending order • Compare them with the cumulative sum of

normalized weights

Ni

ik

ik

Ni

ik

ik wxwx 11* },{RESAMPLE},{

]1,0[~Uui

Ristic et al.© Michael Rubinstein

Resampling• Complexity: O(NlogN)

– O(N) sampling algorithms exist

© Michael RubinsteinHsiao et al.

Generic PF

• Apply SIS filtering• Calculate• IF

• • END

kNi

ik

ik

Ni

ik

ik zwxwx ,},{PF},{ 1111

kNi

ik

ik

Ni

ik

ik zwxwx ,},{SIS},{ 1111

effN

threff NN N

iik

ik

Ni

ik

ik wxwx 11 },{RESAMPLE},{

© Michael Rubinstein

Generic PF},{ 1Nxik

},{ kik wx

},{ 1* Nxik

},{ 11

Nxik

},{ 11 kik wx

Van der Merwe et al.

Uniformly weighted measureApproximates )|( 1:1 kk zxp

Compute for each particle its importance weight to Approximate

(Resample if needed)

Project ahead to approximate

)|( :11 kk zxp

)|( :1 kk zxp

)|( 1:11 kk zxp© Michael Rubinstein

PF variants

• Sampling Importance Resampling (SIR)• Auxiliary Sampling Importance Resampling

(ASIR)• Regularized Particle Filter (RPF)• Local-linearization particle filters• Multiple models particle filters (maneuvering

targets)• …

© Michael Rubinstein

Sampling Importance Resampling (SIR)

• A.K.A Bootstrap filter, Condensation

© Michael Rubinstein

• Initialize from prior distribution • For k > 0 do

• Resample into • Predict • Reweight• Normalize weights • Estimate (for display)

Ni

ii wx 100 },{

Ni

ik

ik wx 111 },{

Ni

ik Nx 1

*1 }

1,{

0X

)|(~ *11

ikkk

ik xxxpx

)|( ikkk

ik xxzpw

kx̂

Intermission

Questions?

© Michael Rubinstein

Multiple Targets (MTT/MOT)• Previous challenges

– Full/partial occlusions– Entering/leaving the scene– …

• And in addition– Estimating the number of objects– Computationally tractable for multiple simultaneous

targets– Interaction between objects

– Many works on multiple single-target filters

© Michael Rubinstein

BraMBLe: A Bayesian Multiple-Blob Tracker

M. Isard and J. MacCormickCompaq Systems Research Center

ICCV 2001

Some slides taken from Qi ZhaoSome images taken from Isard and MacCormick

BraMBLE

• First rigorous particle filter implementation with variable number of targets

• Posterior distribution is constructed over possible object configurations and number

• Sensor: single static camera• Tracking: SIR particle filter• Performance: real-time for 1-2 simultaneous

objects

© Michael Rubinstein

The BraMBLe posterior

© Michael Rubinstein

)|( :1 kk zxp

State at frame k Image Sequence

Number,Positions,Shapes,Velocities,…

State space

• Hypothesis configuration:

• Object configuration:

© Michael Rubinstein

),...,,,( 21 mkkkkk xxxmX

),,,( ik

ik

ik

ik

ikx SVX

identifier

position

),( zxX

velocity

),( zx vvV

shape),,,,,,,( swhswf hwwww S

max131 MN x

Object model

• A person is modeled as a generalized-cylinder with vertical axis in the world coords

© Michael Rubinstein

CalibratedCamera),,,,,,,( swhswf hwwww S

)},(),,(),,(),0,{(),( hwhwhwwyr hsswwfii

Observation likelihood

• Image overlaid with rectangular Grid (e.g. 5 pixels)

© Michael Rubinstein

Y

Cr

Cb Mexican Hat(second deriv of G)

Gaussian

)|( ttp XZ

Observation likelihood

• The response values are assumed conditionally independent given X

© Michael Rubinstein

g ggg g lzpzp )|()|()|p( XXZ

0gl

2gl1gl

3gl

)|( ttp XZ

Appearance models

• GMMs for background and foreground are trained using kmeans

© Michael Rubinstein

Bk Bkg

kggg N

Klz ),(

1)0|p(

Fk

kF

kFgg N

Klz ),(

1)0|p(

4K

16K

Observation likelihood

© Michael Rubinstein

)0|p(

)0|p(log

gg

gg

lz

lz

System (prediction) model• The number of objects can change:

– Each object has a constant probability to remain in the scene.

– At each time step, there is constant probability that a new object will enter the scene.

•

© Michael Rubinstein

r

i

,...)~,(,...)~,( 1,1,'1

'1

'1

nt

nt

nt

nt

nt

nt xmXxmX

)|( 1tt XXp

Predictionfunction Initialization

function

Prediction function

• Motion evolution: damped constant velocity• Shape evolution: 1st order auto-regressive process

model (ARP)

© Michael Rubinstein

1tX 1tV

11 8.0 tt VX

Particles

© Michael Rubinstein

1tX

2tX

1NtX N

tX...N Points:

N Weights:1t

2t

1Nt N

t

Estimate

• Denote the set of existing unique identifiers

© Michael Rubinstein

},...,{ 1 Mt M

Total probability that object i exists

(particle,target)

tX̂

Results

• N=1000 particles• initialization samples always generated

© Michael Rubinstein

Results

• Single foreground model cannot distinguish between overlapping objects – causes id switches

© Michael Rubinstein

Parameters

© Michael Rubinstein

Summary

• The particle filters were shown to produce good approximations under relatively weak assumptions– can deal with nonlinearities– can deal with non-Gaussian noise– Multiple hypotheses– can be implemented in O(N)– easy to implement– Adaptive focus on more probable regions of the

state-space

© Michael Rubinstein

In practice1. State (object) model2. System (evolution) model3. Measurement (likelihood) model4. Initial (prior) state5. State estimate (given the pdf)

6. PF specifics1. Importance density2. Resampling method

• Configurations for specific problems can be found in literature

© Michael Rubinstein

Thank you!

© Michael Rubinstein

References

• Beyond the Kalman filter/ Ristic,Arulamplam,Gordon– Online tutorial: A Tutorial on Particle Filters for Online

Nonlinear/Non-Gaussian Bayesian Tracking/ Arulampalam et al 2002

• Stochastic models, estimation and control/ Peter S. Maybeck

• An Introduction to the Kalman Filter/ Greg Welch, Gary Bishop

• Particle filters an overview/ Matthias Muhlich

© Michael Rubinstein

Sequential derivation 1

• Suppose at time k-1, characterize

• We receive new measurement and need to approximate using new set of samples

• We choose q such that

And we can generate new particles

Ni

ik

ik wx 111:0 },{

)|( 1:11:0 kk zxp

kz)|( :1:0 kk zxp

)|(),|()|( 1:11:0:11:0:1:0 kkkkkkk zxqzxxqzxq

),|(~ :11:0 kikk

ik zxxqx

© Michael Rubinstein

Sequential derivation 2

• For the weight update equation, it can be shown that

And so )|()|()|(

)|()|(

)|()|()|(

1:11:01

1:11:01:1

1:1:0

kkkkkk

kkkk

kkkkkk

zxpxxpxzp

zxpzzp

xxpxzpzxp

),|(

)|()|()|(),|(

)|()|()|(

)|(

)|(

:11:0

11

1:11:0:11:0

1:11:01

:1:0

:1:0

kik

ik

ik

ik

ikki

k

kkkkk

kkkkkk

kk

kkik

zxxq

xxpxzpw

zxqzxxq

zxpxxpxzp

zxq

zxpw

© Michael Rubinstein

Sequential derivation 3

• Further, if • Then the weights update rule becomes

(and need not store entire particle paths and full history of observations)

• Finally, the (filtered) posterior density is approximated by

),|(),|( 1:11:0 kkkkkk zxxqzxxq

),|(

)|()|(

1

11

kik

ik

ik

ik

ikki

kik zxxq

xxpxzpww

N

i

ikk

ikkk xxwzxp

1:1 )()|(

(3)

© Michael Rubinstein

Choice of importance density

• Choose q to minimize variance of weights• Optimal choice:

– Usually cannot sample from or solve for (in some specific cases is works)

• Most commonly used (suboptimal) alternative:

– i.e. the transitional prior

)|(),|(),|(

11

11ikk

ik

ik

kikkoptk

ikk

xzpwwzxxpzxxq

optq ikw

)|()|(),|(

1

11ikk

ik

ik

ikkoptk

ikk

xzpwwxxpzxxq

© Michael Rubinstein

Generic PF

• Resampling reduces degeneracy, but new problems arise…

1.Limits parallelization2.Sample impoverishment: particles with high

weights are selected many times which leads to loss of diversity– if process noise is small – all particles tend to

collapse to single point within few interations– Methods exist to counter this as well…

© Michael Rubinstein