Stanford CS223B Computer Vision, Winter 2007
Lecture 12
Tracking Motion
Professors Sebastian Thrun and Jana Košecká
CAs: Vaibhav Vaish and David Stavens
Overview





The Tracking Problem
Bayes Filters
Particle Filters
Kalman Filters
Using Kalman Filters
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
The Tracking Problem
Image 1



Image 2
Image 3
Image 4
Can we estimate the position of the object?
Can we estimate its velocity?
Can we predict future positions?
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
The Tracking Problem



Given Sequence of Images
Find center of moving object
Camera might be moving or stationary

We assume: We can find object in individual
images.
The Problem: Track across multiple images.

Is a fundamental problem in computer vision

Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Methods
Bayes Filter
Kalman Filter
Uncented
Kalman Filter
Particle Filter
Extended
Kalman Filter
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Further Reading…
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Example: Moving Object
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Tracking
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particle Filter Tracking
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Mixture of KF / PF (Unscented PF)
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Overview





The Tracking Problem
Bayes Filters
Particle Filters
Kalman Filters
Using Kalman Filters
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Example of Bayesian Inference
p(staircase)
Slow
= 0.28
Down!
Cost
Sensor
model
model
cost(fast
p(imagewalk
| staircase)
| staircase)
= 0.7= $1,000
cost(fast
p(imagewalk
| no staircase)
| no staircase)
= 0.2= $0
cost(slow+sense) = $1
Environment prior
p(staircase) = 0.1
Decision
BayesianTheory
inference
p(staircase walk)]
| image) = $1,000 • 0.28 = $280
E[cost(fast
p(image | staircasse)
p(staircase)
E[cost(slow+sense)]
= $1
=
p(im | stair) p(stair) + p(im | no stair) p(no stair)
= 0.7 • 0.1 / (0.7 • 0.1 + 0.2 • 0.9) = 0.28
?
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Bayes Filter Definition

Environment state xt

Measurement zt

Can we calculate p(xt | z1, z2, …, zt) ?
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Bayes Filters Illustrated
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Bayes Filters: Essential Steps

Belief:
Bel(xt)
∝

Measurement update: Bel(xt) ← Bel(xt) p(zt|xt)

Time update: Bel(xt+1) ← Bel(xt) ⊗ p(xt+1|ut,xt)
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
x = state
t = time
z = observation
u = action
η = constant
Bayes Filters
Bel ( xt ) =
=
p( xt | z0t , u0t )
p( xt | zt , ut − 1 , zt − 1 , , u0 )
Bayes
= η p( zt | xt , ut − 1 , zt − 1 , , z0 ) p( xt | ut − 1 , zt − 1 , , z0 )
Markov
= η p( zt | xt ) p( xt | ut − 1 , zt − 1 , , z0 )
= η p( zt | xt ) ∫ p( xt | xt − 1 , ut − 1 , , z0 ) p( xt − 1 | ut − 1 , , z0 ) dxt − 1
Markov
= η p ( zt | xt )
∫ p( x
t
| xt −1 , ut −1 ) p ( xt −1 | zt −1 , ut − 2  , u0 ) dxt −1
= η p( zt | xt ) ∫ p( xt | xt − 1 , ut − 1 ) Bel ( xt − 1 ) dxt − 1
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Bayes Filters
x
t
z
u
= state
= time
= observation
= action
Bel ( xt ) = η p( zt | xt ) ∫ p( xt | xt − 1 , ut − 1 ) Bel ( xt − 1 ) dxt − 1
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Bayes Filters Illustrated
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Bayes Filters


Initial Estimate of State
Iterate
– Receive measurement, update your belief (uncertainty
shrinks)
– Predict, update your belief (uncertainty grows)
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Methods
Bayes Filter
Kalman Filter
Uncented
Kalman Filter
Particle Filter
Extended
Kalman Filter
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Overview





The Tracking Problem
Bayes Filters
Particle Filters
Kalman Filters
Using Kalman Filters
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particle Filters: Basic Idea
p(x )
xt
set of n particles Xt
p( xt ∈ X t ) ≈ p( xt | z1...t )
Sebastian Thrun and Jana Košecká
(equality for n ↑ ∞ )
CS223B Computer Vision, Winter 2007
Particle Filter Explained
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Basic Particle Filter Algorithm
Initialization:
X0 ← n particles x0 [i] ~ p(x0)
particleFilters(Xt −1 ){
for i=1 to n
xt [i] ~ p(xt | xt −1[i])
(prediction)
wt[i] = p(zt | xt[i])
(importance weights)
endfor
for i=1 to n
include xt [i] in Xt with probability ∝ wt[i]
(resampling)
}
p( xt | z1...t , u1...t ) = η p( zt | xt ) ∫ p( xt | ut , xt − 1 ) p( xt − 1 | z1...t − 1 , u1...t − 1 ) dxt − 1
p( xt ∈ X t ) ≈ p( xt | z1...t , u1...t )
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Importance Sampling
Weight samples: w = f / g
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particle Filter
By Frank Dellaert
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Case Study:
Track moving objects from Helicopter
1.
2.
3.
4.
5.
Harris Corners
Optical Flow (with clustering)
Motion likelihood function
Particle Filter
Centroid Extraction
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
1. Harris Corner Extraction
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
2. Optical Flow + Motion Detection
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
3. Motion Likelihood Function
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
4. Particle Filters
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
5. Extract Centroid
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
More Particle Filter Tracking
David Stavens, Andrew Lookingbill, David Lieb, CS223b Winter 2004
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Some Robotics Examples





Tracking Hands, People
Mobile Robot localization
People localization
Car localization
Mapping
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Examples Particle Filter
Siu Chi Chan McGill University
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Another Example
Mike Isard and Andrew Blake
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Tracking Fast moving Objects
Sebastian Thrun and Jana Košecká
K. Toyama, A.Blake
CS223B Computer Vision, Winter 2007
Particle Filters: Illustration
With: Wolfram Burgard, Dieter Fox, Frank Dellaert
p( xt ∈ X t ) ≈ p( xt | z1...t , u1...t )
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particle Filters (1)
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particle Filters (2)
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particles = Robustness
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Tracking People from Moving Platform
• robot location (particles)
• people location (particles)
• laser measurements (wall)
Sebastian Thrun and Jana Košecká
With Michael Montemerlo
CS223B Computer Vision, Winter 2007
Tracking People from Moving Platform
• robot location (particles)
• people location (particles)
• laser measurements (wall)
Sebastian Thrun and Jana Košecká
With Michael Montemerlo
CS223B Computer Vision, Winter 2007
Particle Filters for Tracking Cars
With Anya Petrovskaya
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Mapping Environments (SFM)
With Dirk Haehnel
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Overview





The Tracking Problem
Bayes Filters
Particle Filters
Kalman Filters
Using Kalman Filters
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Tracking with KFs: Gaussians!
initial estimte
prediction
y
y
x
Sebastian Thrun and Jana Košecká
measurement
update
y
x
y
x
CS223B Computer Vision, Winter 2007
x
Kalman Filters
p( x ) ~ N ( µ , Σ )
 1

T −1
p( x ) ∝ exp  − ( x − µ ) Σ ( x − µ ) 
 2

Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filters
p( x | z ) ∝ p( z | x ) p( x )
posterior
Measurement
evidence
prior
p( x ' ) = ∫ p( x ' | x ) p( x ) dx
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
A Quiz
p( x | z ) ∝ p( z | x ) p( x )
Measurement
evidence
prior
posterior?
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Gaussians
Univariate
µ
p( x) ~ N ( µ , σ 2 ) :
p( x) =
1
e
2π σ
−
1 ( x −µ )2
2 σ2
σ
-σ
Multivariate
p (x) ~ Ν (μ,Σ) :
µ
p ( x) =
1
(2π )
d /2
Σ
1/ 2
e
1
− ( x −μ ) t Σ −1 ( x −μ )
2
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Properties of Univariate Gaussians
X ~ N ( µ , σ 2 )
2 2
⇒
Y
~
N
(
a
µ
+
b
,
a
σ )

Y = aX + b 
2
2
 σ 22

X 1 ~ N ( µ1 , σ 1 ) 
σ
1
1


⇒ p( X 1 ) ⋅ p( X 2 ) ~ N  2
µ1 + 2
µ2 ,
2
2
−2
−2 
2 
σ1 + σ 2
σ1 + σ 2 
X 2 ~ N ( µ 2 , σ 2 )
 σ1 + σ 2
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Measurement Update Derived
2
 1 ( x − µ1 ) 2 1 ( x − µ 2 ) 2 
X 1 ~ N ( µ1 , σ 1 ) 
⇒ p( X 1 ) ⋅ p ( X 2 ) = const. ⋅ exp−
−

2
2
2 
2
2
σ
σ
X 2 ~ N ( µ 2 , σ 2 )
1
2


2
2
∂  1 ( x − µ1 )
1 ( x − µ2 ) 
−
−
 =
2
2
∂x  2 σ 1
2 σ2

x − µ1 x − µ 2
+
2
2
σ1
σ2
( µ − µ1 )σ 2 2 + ( µ − µ 2 )σ 12
= 0
µ (σ 1 + σ 2 ) = µ1 σ 2 + µ 2σ 1
2
µ
∂2
∂x 2
2
2
µ σ + µ 2σ 1
= 1 22
2
σ1 + σ 2
2
= 0 (for new µ )
2
2
 1 ( x − µ1 ) 2 1 ( x − µ 2 ) 2 
−2
−2
−
=
σ
+
σ
−

1
2
2
2
2
2
σ
σ
1
2


σ =
Sebastian Thrun and Jana Košecká
1
−2
−2
σ1 +σ 2
CS223B Computer Vision, Winter 2007
Properties Multivariate Gaussians
Essentially the same as in the 1-D case, but with more general notation
X ~ N ( µ , Σ) 
T
⇒
Y
~
N
(
A
µ
+
B
,
A
Σ
A
)

Y = AX + B 
X 1 ~ N ( µ1 , Σ1 ) 
−1
−1
−1
−1 −1
 ⇒ p ( X 1 ) ⋅ p ( X 2 ) ~ N (Σ1 + Σ 2 ) Σ 2 µ1 + (Σ1 + Σ 2 ) Σ1µ 2 , (Σ1 + Σ 2 )
X 2 ~ N ( µ 2 , Σ 2 )
(

We stay in the “Gaussian world” as long as we start with Gaussians
and perform only linear transformations.
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
)
Linear Kalman Filter
Estimates the state x of a discrete-time
controlled process that is governed by the
linear stochastic difference equation
xt = At xt −1 + Bt ut + ε t
with a measurement
zt = Ct xt + δ t
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Components of a Kalman Filter
At
Matrix (n × n) that describes how the state
evolves from t to t+1 without controls or
noise.
Bt
Matrix (n × i) that describes how the control
ut changes the state from t to t+1.
Ct
Matrix (k × n) that describes how to map the
state xt to an observation zt.
εt
Random variables representing the process
and measurement noise that are assumed to
be independent and normally distributed
with covariance Rt and Qt respectively.
δt
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Algorithm
1.
Algorithm Kalman_filter( µt-1, Σt-1, ut, zt):
2.
Prediction:
3.
µ t = At µt −1 + Bt ut
4.
Σ t = At Σ t −1 AtT + Rt
5.
Correction:
6.
7.
K t = Σ t CtT (Ct Σ t CtT + Qt ) −1
µ t = µ t + K t ( z t − Ct µ t )
8.
Σ t = ( I − K t Ct ) Σ t
9.
Return µt, Σt
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Updates in 1D
measurement
belief
belief
measu
remen
t
new belief
old belief
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Updates in 1D
 µ t = µ t + K t ( zt − µ t )
bel ( xt ) = 
2
2
σ
=
(
1
−
K
)
σ
t
t
t

 µ t = µ t + K t ( z t − Ct µ t )
bel ( xt ) = 
 Σ t = ( I − K t Ct ) Σ t
σ t2
with K t = 2
2
σ t + σ obs
,t
with
K t = Σ t CtT (Ct Σ t CtT + Qt ) −1
measu
remen
t
new belief
old belief
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Updates in 1D
 µt = at µt −1 + bt ut
bel ( xt ) =  2
2 2
2
σ
=
a
σ
+
σ
t
t
act ,t
 t
 µt = At µt −1 + Bt ut
bel ( xt ) = 
T
Σ
=
A
Σ
A
t
t t −1 t + Rt

remen
t
new belief
new belief
measu
newest belief
old belief
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Updates
belief
latest belief
measurement
belief
Sebastian Thrun and Jana Košecká
belief
CS223B Computer Vision, Winter 2007
The Prediction-Correction-Cycle
Prediction
 µt =at µt −1 +bt ut
bel ( xt ) = 2
2
2
2
σt =at σt +σact ,t
µ = At µt −1 +Bt ut
bel ( xt ) = t
T
Σt = At Σt −1 At +Rt
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
The Prediction-Correction-Cycle
µ = µ + K t ( zt − µt )
σt2
bel ( xt ) =  t 2 t
,
K
=
t
2
2
σt2 +σobs
 σt = (1 − K t )σt
,t
µ = µt + K t ( zt −Ct µt )
bel ( xt ) =  t
, K t = Σt CtT (Ct Σt CtT +Qt ) −1
Σ
=
(
I
−
K
C
)
Σ
t
t
t
t

Correction
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
The Prediction-Correction-Cycle
Prediction
µ = µ + K t ( zt − µt )
σt2
bel ( xt ) =  t 2 t
,
K
=
t
2
2
σt2 +σobs
 σt = (1 − K t )σt
,t
 µt =at µt −1 +bt ut
bel ( xt ) = 2
2
2
2
σt =at σt +σact ,t
µ = µt + K t ( zt −Ct µt )
bel ( xt ) =  t
, K t = Σt CtT (Ct Σt CtT +Qt ) −1
Σ
=
(
I
−
K
C
)
Σ
t
t
t
t

µ = At µt −1 +Bt ut
bel ( xt ) = t
T
Σt = At Σt −1 At +Rt
Correction
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter Summary

Highly efficient: Polynomial in measurement
dimensionality k and state dimensionality n:
O(k2.376 + n2)

Optimal for linear Gaussian systems!

Most robotics systems are nonlinear!
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Overview





The Tracking Problem
Bayes Filters
Particle Filters
Kalman Filters
Using Kalman Filters
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Let’s Apply KFs to Tracking Problem
Image 1
Image 2
Sebastian Thrun and Jana Košecká
Image 3
Image 4
CS223B Computer Vision, Winter 2007
Kalman Filter with 2-Dim Linear Model

Linear Change (Motion)
x' = Ax + B + N (0, R )
1 0

A = 
0 1

t is
a
h
W
0
B =  
0
R?
,
D
C,
ρ2
R = 
 0
0

2
ρ 
Linear Measurement Model
z = Cx + D + N (0, Q)
1 0

C = 
0 1
Sebastian Thrun and Jana Košecká
s
i
t
a
Wh
?
Q
,
A, B
σ 2 0 
0

D =   Q = 
2
0
 0 σ 
CS223B Computer Vision, Winter 2007
Kalman Filter Algorithm
1.
Algorithm Kalman_filter( µt-1, Σt-1, ut, zt):
2.
Prediction:
4.
µ t = At µt −1 + Bt ut
Σ t = At Σ t −1 AtT + Rt
5.
Correction:
3.
6.
7.
8.
9.
1 0

A = 
0 1
K t = Σ t CtT (Ct Σ t CtT + Qt ) −1
µ t = µ t + K t ( z t − Ct µ t − D )
Σ t = ( I − K t Ct ) Σ t
1 0

C = 
Return µt, Σt
0 1
Sebastian Thrun and Jana Košecká
0
B =  
0
ρ2
R = 
 0
0

2
ρ 
σ 2 0 
0

D =   Q = 
2
0
 0 σ 
CS223B Computer Vision, Winter 2007
Kalman Filter in Detail




Measurements z = x + N (0, R )
x ' = x + N (0, Q )
Change
Σ' = Σ+Q
µ' = µ
Prediction
−1
−1 − 1
Measurement Update Σ ' ' = ( Σ ' + R )
µ ' ' = µ '+ Σ ' ' R − 1 ( z − µ ' )
initial position prediction
y
measurement
y
x
Sebastian Thrun and Jana Košecká
update
y
x
y
x
CS223B Computer Vision, Winter 2007
x
Can We Do Better?
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman, Better!
initial position prediction
x
measurement
x
update
x
x
x
x
Sebastian Thrun and Jana Košecká
x
ne
xt
x
pr
e
dic
it
on
x
x
CS223B Computer Vision, Winter 2007
We Can Estimate Velocity!
prediction
past measurements
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Kalman Filter For 2D Tracking

Linear Measurement model (now with 4 state variables)
 xobs   1 0 0 0 

 = 

 yobs   0 1 0 0 

 x
 
 y
 x  + N (0, R )
 
 y 
 
r2
R = 
0
0

2
r 
Linear Change
 x'   1
  
 y'  0
 x '  =  0
  
 y '   0
  
0 ∆t
1
0
0
0
1
0
0

∆t 
0

1 
 x
 
 y
 x  + N (0, Q)
 
 y 
 
Sebastian Thrun and Jana Košecká
0

0
Q=
0

0

0
0
0 0
0 q2
0 0
0

0
0

2
q 
CS223B Computer Vision, Winter 2007
Putting It Together Again


Change

Prediction

  1 0 0 0 
z = 
 x + N (0, R )
 0 1 0 0
Measurements
1

 0
x' = 
0

0
Σ ' = Σ + CQC
µ ' = Cµ
Measurement Update
Sebastian Thrun and Jana Košecká
0 ∆t 0 

1 0 ∆t  
x + N (0, Q )

0 1 0

0 0 1
T
Σ ' = ( Σ + A R A)
−1
T
−1
−1
µ ' = µ + Σ ' AT R − 1 ( z − Aµ )
CS223B Computer Vision, Winter 2007
Summary Kalman Filter

Estimates state of a system
– Position
– Velocity
– Many other continuous state variables possible

KF maintains
– Mean vector for the state
– Covariance matrix of state uncertainty

Implements
– Time update = prediction
– Measurement update

Standard Kalman filter is linear-Gaussian
– Linear system dynamics, linear sensor model
– Additive Gaussian noise (independent)
– Nonlinear extensions: extended KF, unscented KF: linearize

More info:
– CS226
– Probabilistic Robotics (Thrun/Burgard/Fox, MIT Press)
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007
Particle
Summary Kalman Filter

Estimates state of a system
– Position
and discrete
– Velocity
– Many other continuous state variables possible

KF maintains
– Mean vector for the state
– Covariance matrix of state uncertainty

set of particles
(example states)
Implements
– Time update = prediction = predictive sampling
– Measurement update
= resampling, importance weights

Standard Kalman filter is linear-Gaussian
– Linear system dynamics, linear sensor model
fully nonlinear
– Additive Gaussian noise (independent)
– Nonlinear extensions: extended KF, unscented KF: linearize

More info:
easy t
o
– CS226
– Probabilistic Robotics (Thrun/Burgard/Fox, MIT Press)
Sebastian Thrun and Jana Košecká
implem
ent
CS223B Computer Vision, Winter 2007
Summary





The Tracking Problem
Bayes Filters
Particle Filters
Kalman Filters
Using Kalman Filters
Sebastian Thrun and Jana Košecká
CS223B Computer Vision, Winter 2007