Sequential Monte-Carlo Method

-Introduction, implementation and application

Fan, Xin2005.3.28

Probabilistic state estimation for a

dynamic system

• Dynamic system, a system with changes over time

What can SMC do

-Economics, weather

-Moving object, image

-Generally speaking, anything in the world

Extracting relevant information of the system through investigating the

observations

• State , hidden information to describe the system

What can SMC do--State-Space Modeling

-Kinematic characteristics in tracking

• Measurements , made on the system

—observed noisy data-Image data available up to current time

kx

kz

—Evolving over time (Dynamic model):

—What we are interested

-Intensities of pixels in image estimation

),( 11 kkkk vxfx

-Intensities of the degraded image—Associated with states (Measurement Model):

),( 11 kkkk nxhz

• State evolution is described in terms of transition probability

What can SMC do--Probabilistic formulation

• How the given fits the available measurement is described in terms of likelihood probability

kx kz

)(1kkp xx

1111

111111

)()),((

)(),()(

kkkkkk

kkkkkkkk

dpf

dppp

vvvxx

vxvvxxxx

)(:1 kkp zx

kx

kkkkkkkk dpp nnnxhxxz )()),(()( 11

Determining the belief in the state taking deferent values, given the

measurements k:1z

kx

• Prediction:

What can SMC do—Recursive Estimation

• Update with the innovative measurement

)()()()( 11:1111:1 kkkkkkk dppp xzxxxzx

)(

)()()(

1:1

1:1:1

kk

kkkk

kk p

ppp

zz

zxxzzx

Starting from , at time is estimated with available :

)()( 000 xzx pp )(:1 kkp zx k

)(1:11 kkp zx

kz

Why use SMC

)( 0xp

)(1kkp xx

)(kkp xz

Only when all of the distributions are Gaussian, the posterior distribution is Gaussia

n and analytical solution exists-- Kalman filter

Non-Gaussian process noise

Nonlinear Dynamics --sudden and jerky motion

Multiple targets tracking

Partial occlusion

Implementation—Basic idea

Use SAMPLES with associated weights to approximate posterior densitysN

jjk

jk w 1:0 },{ x

sN

i

ikk

ikkk xxwzxp

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

• Examples:

--discrete probability:

coin, galloping dominoes

--continuous density

sampling Gaussion density

Implementation—Basic assumptions

No explicit assumptions on the forms of both transition and likelihood probabilities, SMC is

applicable for nonlinear and non-Gaussian estimation

• Measurements are independent, both mutually and with respect to dynamical process:

)()( 11:1 kkkk xxpxxp

• Markov Chain:

1

11:11:11 )()(),(k

iiikkkkk ppp xzxxxxz

Implementation—A 1D nonlinear example

We need to infer the state at the time with available measurements

• Measurements Model:

kkkkk k vxxxx ))1(2.1cos(8)1/(255.0 2111

• Dynamic Model:

kkk nxz 20/2

kx

Implementation—Results

Implementation—Algorithm

Initialization

))20/)ˆ((

exp()ˆ(2

22

n

ikki

kkik

xxpw

z

z

-Draw samples from i0x )( 0xp

-Set weights

si Nw /10

1. Prediction:

ik

ik

ik

ik

ik

ikk

ik

k

f

vxxx

vxx

))1(2.1cos(8))(1/(255.0

),(ˆ2

111

11

2. Update:

-Normalization -

3. Resample

sNi

ii w 100 },{ x

),},({},{ 1111 kNi

ik

ik

Ni

ik

ik

ss wSMCw zxx

Implementation—Results

Implementation—Results

Implementation—Results

Implementation—Discussion

• Relaxes :- Linearity of dynamic and measurement models

- The forms of the distributions of process and measurement noise.

• Requires :- Initial prior density )( 0xp

- The likelihood can be evaluated

- State samples can be generated easily Do not make use of any knowledge of the measurements inefficient and sensitive to outliers

)( kkp xz

)( 1kkp xx

Implementation—Generic SIS Algorithm

),(

)()(

1:11:0

1

1

kik

ik

ik

ik

ikki

kik

q

ppww

zxx

xxxz

- Draw

1. Prediction:

2. Update:

-Normalization

-

3. Resample

Introducing an Importance density to facilitate sampling and using observations

),( :11:0 kkkq zxx

),(~ :11:0 kkkik q zxxx

Application—Contour extraction

• Probabilistic state estimation formulation:

• Problem Definition:

- Grouping edge points into continuous cures, represented by a series of control points.

- The positions of the control points are the states , then a contour turns out to be a state sequence.

),...,( 0:0 cc NN xxx

- Edge points are those pixels with larger intensity gradients, which are used as measurements

)( kk I xy

Application—Contour extraction

Definitions of the probabilities

• Likelihood:

N

jjjjk IIp

1

)()())(1(exp())(( uhunux

• Dynamics:))(( 11 kkkkk xxRxx ),0;()1()( 2

kk Np

• Importance density:))(( 11 kkkkk xxRxx ),0;())(1(

)()( 2

kkk

k Nxcxc

p

Perform the standard procedure to estimate the states

Application—Some results

Summary of using SMC

• Define the probability densities

• Modeling problems as probabilistic estimation

-States / what we want, but cannot observe directly-Measurements / observations

- Likelihood / the relationship between states and measurements / functional form that can be evaluated

- Transition / determine the evolution of the states over time / the prior knowledge of the system under investigation

- Importance / employ the observations / easy for sampling

Future work

• Apply SMC to various problems

- Vision tracking

- Constrain the state space by using better dynamic model / incorporate more prior knowledge

- Elaborate techniques for efficiently sampling / SA / move samples to density peaks

- Data fusion

- Image restoration/super-resolution

- Digital communication• High computational expense

- Decompose a high dimensional problem to several lower dimensional ones…

Reference

[4] P. Pérez, A. Blake, and M. Gangnet. JetStream: Probabilistic contour extraction with particles. Proc. Int. Conf. on Computer Vision (ICCV), II:524-531, 2001. --- Contour extraction

[3] Gordon, N., Salmond, D., and Smith, A. ." Novel approach to nonlinear/non-Gaussian Bayesian state estimation". IEE Proc. F, 140, 2, 107-113. --- the simple 1D example

[1] Proceedings of the IEEE, vol. 92, no. 3, Mar. 2004. Special issue

[2] IEEE Trans On Signal Processing, Vol. 50, no. 2. Special issue

[5] M. Isard and A. Blake, "Contour tracking by stochastic propagation of conditional density", ECCV96,pp. 343-356,1996. – Application to vision tracking, in which significant performance was achieved.[6] Jun S. Liu and Rong Chen, "Sequential Monte Carlo Methods for Dynamic Systems", Journal of the American Statistical Association, Vol. 93, No. 443, pp.1032--1044, 1998. – SMC from the point of statisticians