sequential monte-carlo method -introduction, implementation and application fan, xin 2005.3.28
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Sequential Monte-Carlo Method
-Introduction, implementation and application
Fan, Xin2005.3.28
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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
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• 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
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• 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
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• 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
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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
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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
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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
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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
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Implementation—Results
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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
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Implementation—Results
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Implementation—Results
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Implementation—Results
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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
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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
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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
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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
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Application—Some results
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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
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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…
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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