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Particle Filters
(Sequential Monte-Carlo Methods)
Medical Imaging Team Seminar
David Lesage
05/03/07

Outline
Introduction to Bayesian Filtering / Tracking
Sequential Monte-Carlo Methods (Particle Filters)
Problems and Extensions
Applications and Illustrations
2

Bayesian Filtering / Tracking
Tracking of a dynamic process
X t = {x 0 , x 1 , ..., x t }
state vector
x t
Observation sequence
Y t = {y 1 , y 2 , ..., y t }
Dynamic System
{
Xt = F t (X t−1 , V t )
Y t = H t (X t , W t )
with
V t , W t
process and observation noises
3

Bayesian Filtering / Tracking
Hidden Markov Model (HMM)
1st order
p(x t |x 0:t−1 ) = p(x t |x t−1 )
Observations i.i.d. given the state
p(y t |x t , A) = p(y t |x t )
Notation
p(X t ) = p(x 0:t−1 )
Observed
y t−2 y t−1 y t
p(y t |x t )
Unobserved
x t−2
x t−1
x t
p(x t |x t−1 )
4

Bayesian Filtering / Tracking
Filtering = sequentially estimating the states
w.r.t. a set of observations
Y t
X t
of a system
Posterior
p(X t |Y t )
Filtering (marginal) distribution
Transition/dynamics density of the process prior
Observation likelihood
p(y t |x t )
p(x t |Y t )
p(x t |x t−1 )
Observed
y t−2 y t−1 y t
p(y t |x t )
Unobserved
x t−2
x t−1
x t
p(x t |x t−1 )
5

Bayesian Filtering / Tracking
Recursive Estimation
p(X t |Y t ) = p(X t−1 |Y t−1 ) p(y t|x t )p(x t |x t−1 )
p(y t |Y t−1 )
For the marginal distribution
prediction
p(x t |Y t−1 ) =
∫
likelihood
transition density
p(x t |x t−1 )p(x t−1 |Y t−1 )dx t−1
update
p(x t |Y t ) = p(y t|x t )p(x t |Y t−1 )
p(y t |Y t−1 )
Simple but conceptual solution !
6

Bayesian Filtering / Tracking
Different integrals intractable in the general case
(normalizing constants, some marginals of the posterior...
Exceptions: optimal filters
Kalman filter
(if F and H linear, posterior and noises Gaussian)
{
Xt = F t (X t−1 , V t )
Y t = H t (X t , W t )
Grid-based methods
(for finite discrete state spaces)
Strong limitations...
7

Bayesian Filtering / Tracking
“General” case
F and H non linear
{
Xt = F t (X t−1 , V t )
Y t = H t (X t , W t )
No assumption on the noises and the state space
Approximations
Extended Kalman filter (local linearization)
Unscented Kalman filter
Approximate grid-based Methods
Sequential Monte-Carlo Methods
8

Sequential Monte-Carlo Methods
Perfect Monte-Carlo Simulation
i.i.d. samples (“particles”)
p(X t |Y t )
{x (i)
from
0:t ; i = 1 : n} p(X t |Y t )
X t
discrete approximation
p n (dx 0:t |Y t ) = 1 n
n∑
i=1
δ x
(i)
0:t
(dx 0:t ) −→ p(X t |Y t )
n → ∞
convergence a.s. and usually in distribution
convergence rate independent of the dimensionality !
9

Sequential Monte-Carlo Methods
Problem: we cannot sample from the true posterior
p(X t |Y t )
Sequential Importance sampling (SIS)
sample from an importance/proposal distribution
substitute in the recursive equations...
q(X t |Y t )
importance weights
particle approximation
weight update
w(X t ) = p(X t|Y t )
q(X t |Y t )
n∑
p n (dx 0:t |Y t ) =
w (i)
t
∝ w (i)
t−1
i=1
p(y t |x (i)
t
w (i)
t δ (i) x
(dx 0:t )
0:t
)p(x (i)
t |x (i)
t−1 )
q(x (i)
t |x (i)
0:t−1 , Y t)
10

Sequential Monte-Carlo Methods
Fundamental properties
handle non-linear Bayesian tracking problems
non-parametric ()
theoretically able to track any (multi-modal...) distribution
map intractable integrals to sums of weighted particles
simple, recursive (efficient) and generic solution
representation of the complete posterior distribution !
p(X t |Y t )
BUT...
X t
11

Problems and Extensions
Number of particles: depends on the application
adaptation w.r.t. the complexity of the distribution:
KLD-sampling [Fox 01]
12

Problems and Extensions
Choice of the likelihood
evaluated on the data
p(y t |x t )
need for smooth likelihood landscapes
Choice of the transition model
depends on the application
classical “default” choice: Random Walk
p(x t |x t−1 )
13

Problems and Extensions
Choice of the importance distribution
critical design issue
q(x t |X t−1 , Y t )
classical choice:
w (i)
t
q(x t |X t−1 , Y t ) = p(x t |x t−1 )
∝ w (i)
t−1
p(y t |x (i)
t
)p(x (i)
t |x (i)
t−1 )
q(x (i)
t |x (i)
0:t−1 , Y t)
Bootstrap [Gordon 93], Condensation [Isard 98]...
Simple, but does not integrate last observations
14

Problems and Extensions
Multiple proposals in the literature
Anyways...
the variance of the weights increases over time ! [Kong 94]
Degeneracy Problem
15

Problems and Extensions
1st strategy: “good” importance functions
“optimal” functions in constrained cases [Zaritskii 75]
“good” generic solutions
focus on heavy-tailed importance functions
Auxiliary Particle Filter [Pitt 99]
Unscented Particle Filter [van Der Merwe 00]
16

Problems and Extensions
2nd strategy: Selection / Resampling
first introduced in the Bootstrap algorithm [Gordon 93]
idea: discard low weight samples, multiply high ones
numerous resampling schemes in the literature
most popular one: Sequential Importance Resampling (SIR)
17

Problems and Extensions
Sequential Importance Resampling (SIR)
resample particles w.r.t. their weights
T
evaluation (update)
resampling
prediction (transition)
resampling not done
at each step
T+1
18

Problems and Extensions
Animation (sequential importance resampling)
19

Problems and Extensions
SIR does not solve everything...
Problems: artificial noise and loss of diversity
In practice, cannot track multi-modal distributions...
Main issue: discrete approximation
20

Problems and Extensions
“Solutions” to the loss of diversity: not so discrete particles
Unscented particle filter [van Der Merwe 00]
Regularized particle filter [Musso 01]
Gaussian particle filter [Kotecha 03]
21

Problems and Extensions
Endless extensions and combinations
Rao-Blackwellisation
Markov chain moves (MCMC)
Markov (reversible) model jumps
...
Links with other fields
population-based optimization: genetic algorithms,
particle swarms....
Extensions to multi-target tracking
problems of (temp.) object disappearance and
appearance, collisions...
22

Applications and Illustrations
Generic tool: endless combinations with existing
techniques for parameter optimization
Model selection for N.N. [Freitas 99]
...
Illustrations of “direct” applications
23

Applications and Illustrations
Finance (volatility...)
24

Applications and Illustrations
Video tracking
[Widynksi]
25

Applications and Illustrations
Video tracking [Rensselear P.I., NY]
26

Applications and Illustrations
Video tracking [Isard, Blake]
27

Applications and Illustrations
Video tracking [Isard, Blake]
28

Applications and Illustrations
Video tracking [Isard, Blake]
29

Applications and Illustrations
Video + audio tracking (fusion) [IDIAP, MS Cambridge]
30

Applications and Illustrations
Face tracking and recognition [Mitsubishi Research]
31

Applications and Illustrations
Micro-tubule tracking [Smal, Niessen, Meijering 06]
3 3
6
2µm
2µm
tracking microusing
our teche-lapse
studies).
ithm to capture
ct object disapof
the algorithm
X Y
T
Fig. 2. Visualization of a 2D+T fluorescence microscopy image
sequence showing microtubules (bright, elongated spots) and the re-
32

Applications and Illustrations
Robotics [Fox, Washington]
33

Applications and Illustrations
Robotics [Fox, Washington]
34

Applications and Illustrations
Robotics [Fox, Washington]
35

Applications and Illustrations
“Geometric” tracking [Florin]
Key idea: time as a spatial dimension
Application to coronary tracking
36

Applications and Illustrations
“Geometric” tracking [Florin]
Dynamic stochastic shape models
learn the spatial dynamics of a shape (2D slices to 3D)
robustness to outlier and missing info (sparse info models)
application to liver segmentation
Phd Defense
Tomorrow !!
37

The Particle Family
Oxford
A. Blake (MS Cambridge), M. Isard (MS US), M. Pitt, N. Shephard
Cambridge
A. Doucet (British Columbia), N. de Freitas (B.C.), C. Andrieu (Bristol), N. Gordon
Washington
D. Fox
France
P. Del Moral (Sophia)
O. Cappe, E. Moulines (ENST)
38

Conclusion
Simple, generic, efficient (), non-parametric ()
numerous applications
numerous frameworks
numerous extensions
Fundamental difficulties
discrete approximation...
Practical difficulties
computational load (problem dimension and nb particles)
design issues !
39

Questions
Thank you for your attention
40