1. Random Set/Point Process in Multi-Target Tracking
Ba-Ngu Vo EEE Department University of Melbourne Australia
Collaborators (in no particular order):
Mahler R., Singh. S., Doucet A., Ma. W.K., Panta K., Clark D., Vo B.T., Cantoni A., Pasha A., Tuan H.D., Baddeley A., Zuyev S., Schumacher D.
SAMSI, RTP, NC, USA, 8 September 2008

2. Outline The Bayes (single-target) filter Multi-target tracking
System representation
Random finite set & Bayesian Multi-target filtering
Tractable multi-target filters
Probability Hypothesis Density (PHD) filter
Cardinalized PHD filter
Multi-Bernoulli filter
Conclusions

3. The Bayes (single-target) Filter
observation space zk
zk-1
state space
target motion xk
xk-1
state-vector
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking.
System Model
fk|k-1(xk| xk-1)
gk(zk| xk)
Markov Transition Density
Measurement Likelihood
Objective
pk(xk | z1:k)
posterior (filtering) pdf of the state
measurement history (z1,…, zk)

4. The Bayes (single-target) Filter
observation space zk
zk-1
state space
target motion xk
xk-1
state-vector
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking.
 pk-1(xk-1| z1:k-1) dxk-1
fk|k-1(xk| xk-1)
K pk|k-1(xk| z1:k-1)
gk(zk| xk)
Bayes filter
pk-1(xk-1 |z1:k-1)
prediction
pk|k-1(xk| z1:k-1)
data-update

pk(xk| z1:k)


5. The Bayes (single-target) Filter
observation space zk
zk-1
gk(zk| xk)
state space
target motion xk
fk|k-1(xk| xk-1)
xk-1
state-vector
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking.
pk-1(. |z1:k-1)
pk|k-1(. | z1:k-1)
pk(. | z1:k)
prediction
data-update

Bayes filter

N(.;mk-1, Pk-1)
N(.;mk|k-1, Pk|k-1)
N(.;(mk, Pk )
Kalman filter
i=1 N
{wk|k-1, xk|k-1}
(i)
{wk, xk }

{wk-1, xk-1}
Particle filter

6. Multi-target tracking

7. observation produced by targets
Multi-target tracking
observation space
observation produced by targets
state space
target motion
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking. Xk
Xk-1
5 targets
3 targets
Objective: Jointly estimate the number and states of targets
Challenges:
Random number of targets and measurements
Detection uncertainty, clutter, association uncertainty

8. System Representation
How can we mathematically represent the multi-target state?
Usual practice: stack individual states into a large vector!
Problem:
True
Multi-target
state
Estimated
Multi-target
state
2 targets
2 targets
Estimate is correct but estimation error ???
Remedy: use

9. System Representation
True
Multi-target
state
Estimated
Multi-target
State
1 target
2 targets
True
Multi-target
state
Estimated
Multi-target
State
no target
2 targets
What are the estimation errors?

10. System Representation
Error between estimate and true state (miss-distance)
fundamental in estimation/filtering & control
well-understood for single target: Euclidean distance, MSE, etc
in the multi-target case: depends on state representation
For multi-target state:
vector representation doesn’t admit multi-target miss-distance
finite set representation admits multi-target miss-distance: distance between 2 finite sets
In fact the “distance”
is a distance for sets not vectors

11. observation produced by targets
System Representation
observation space
observation produced by targets
state space
target motion
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking. Xk
Xk-1
5 targets
3 targets
Number of measurements and their values are (random) variables
Ordering of measurements not relevant!
Multi-target measurement is represented by a finite set

12. RFS & Bayesian Multi-target Filtering
Reconceptualize as a generalized single-target problem [Mahler 94]
observations
observed set Z 
targets
target set X X
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking.
Bayesian: Model state & observation as Random Finite Sets [Mahler 94]

prediction
pk-1(Xk-1|Z1:k-1)
pk|k-1(Xk|Z1:k-1)
data-update
pk(Xk|Z1:k)

Need suitable notions of density & integration

13. RFS & Bayesian Multi-target Filtering
state space E S
random finite set or random point pattern
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking. S
state space E
NS(S) = |S  S|
point process or random counting measure

14. RFS & Bayesian Multi-target Filtering
F(E) E
Collection of finite subsets of E
State space S S T T
Probability distribution of S
PS (T ) = P(S ÎT ) , T Í F(E)
Belief “distribution” of S
bS (T ) = P(S Í T ) , T Í E
Since Individual target probability distribution models are defined on subsets of Rn
Not tractable to derive multi-target probability distribution on the abstract Borel subsets of the finite subsets of E
However, belief distribution on the closed subsets of E can be derived
Mahler’s approach is offers a more tractable modeling alternative- Aim: for practicing engineers to write down the belief distribution using the motion models of individual targets, take set derivative to get the multi-target transition density, write down belief distribution using the sensor models, take set derivative to get the multi-target likelihood.
Choquet (1968)
Point Process Theory ( ’s)
Mahler’s Finite Set Statistics (1994)
Probability density of S
pS : F(E) ® [0,¥)
PS (T ) = òT pS (X)m(dX)
Belief “density” of S
fS : F(E) ® [0,¥)
bS (T ) = òT fS (X)dX
Vo et. al. (2005)
Conventional integral
Set integral

15. Multi-object transition
Multi-target Motion Model
Xk = Sk|k-1(Xk-1)ÈBk|k-1(Xk-1)ÈGk x x’ X’ 
death
creation
spawn
motion
fk|k-1(Xk|Xk-1 )
Multi-object transition
density
Evolution of each element x of a given multi-object state Xk-1

16. Multi-object likelihood
Multi-target Observation Model
Zk = Qk(Xk) ÈKk(Xk)
likelihood x z
gk(Zk|Xk)
misdetection 
Multi-object likelihood x
clutter 
state space
observation space
Observation process for each element x of a given multi-object state Xk

17. Multi-target Bayes Filter

pk-1(Xk-1|Z1:k-1)
prediction
pk|k-1(Xk|Z1:k-1)
data-update
pk(Xk|Z1:k)

Since Individual target probability distribution models are defined on subsets of Rn
Not tractable to derive multi-target probability distribution on the abstract Borel subsets of the finite subsets of E
However, belief distribution on the closed subsets of E can be derived
Mahler’s approach is offers a more tractable modeling alternative- Aim: for practicing engineers to write down the belief distribution using the motion models of individual targets, take set derivative to get the multi-target transition density, write down belief distribution using the sensor models, take set derivative to get the multi-target likelihood.
Computationally intractable in general
No closed form solution
Particle or SMC implementation
[Vo, Singh & Doucet 03, 05, Sidenbladh 03, Vihola 05, Ma et al. 06]
Restricted to a very small number of targets

18. Particle Multi-target Bayes Filter
Algorithm
for i =1:N, % Initialise =>
Sample:
Compute:
end;
normalise weights;
for k =1: kmax ,
for i =1:N, % Update =>
Update:
resample;
MCMC step;

19. The PHD Filter pk-1(Xk-1|Z1:k-1) pk|k-1(Xk|Z1:k-1) pk(Xk|Z1:k)  
prediction
pk|k-1(Xk|Z1:k-1)
data-update
pk(Xk|Z1:k)

Multi-target Bayes filter: very expensive!
Since Individual target probability distribution models are defined on subsets of Rn
Not tractable to derive multi-target probability distribution on the abstract Borel subsets of the finite subsets of E
However, belief distribution on the closed subsets of E can be derived
Mahler’s approach is offers a more tractable modeling alternative- Aim: for practicing engineers to write down the belief distribution using the motion models of individual targets, take set derivative to get the multi-target transition density, write down belief distribution using the sensor models, take set derivative to get the multi-target likelihood.
state of system:
random vector
single-object
Bayes filter
first-moment filter
(e.g. a-b-g filter)
Single-object
state of system:
random set
multi-object
Bayes filter
first-moment filter
(“PHD” filter)
Multi-object

20. The Probability Hypothesis Density
vS PHD (intensity function) of a RFS S
vS(x0) = density of expected
number of objects
at x0 S
vS(x)dx = expected number
of objects in S
= mean of, NS(S), the
random counting
measure at S x0
state space S

21. Multi-object Bayes filter
The PHD Filter
state space vk
vk-1
Avoids data association!
PHD filter
vk-1(xk-1|Z1:k-1)
PHD
prediction
vk|k-1(xk|Z1:k-1)
PHD
update
vk(xk|Z1:k)


pk-1(Xk-1|Z1:k-1)
prediction
pk|k-1(Xk|Z1:k-1)
update
pk(Xk|Z1:k)


Multi-object Bayes filter

22. Markov transition density
PHD Prediction 
vk|k-1(xk |Z1:k-1) = fk|k-1(xk, xk-1) vk-1(xk-1|Z1:k-1)dxk-1 + gk(xk)
predicted
intensity
Nk|k-1 = vk|k-1 (x|Z1:k-1)dx 
predicted expected
number of objects
Markov
transition
intensity
intensity from
previous
time-step
term for spontaneous
object births
= intensity of Gk
fk|k-1(xk, xk-1) = ek|k-1(xk-1) fk|k-1(xk|xk-1) + bk|k-1(xk|xk-1)
probability
of object
survival
Markov transition density
term for objects
spawned by
existing objects
= intensity of Bk(xk-1)
vk|k-1 = Fk|k-1vk-1
(Fk|k-1a)(xk) = fk|k-1(xk, x)a(x)dx + gk(xk) 

23. sensor likelihood function
PHD Update
[ S
+ 1 - pD,k(xk)]vk|k-1(xk|Z1:k-1)
pD,k(xk)gk(z|xk)
vk(xk|Z1:k) 
Dk(z) + kk(z)
zZk
Bayes-updated
intensity
measurement
intensity of
false alarms
probability
of detection
predicted intensity
(from previous time) 
Dk(z) = pD,k(x)gk(z|x)vk|k-1(x|Z1:k-1)dx
Nk= vk(x|Z1:k)dx 
sensor likelihood function
expected number
of objects
vk = Ykvk|k-1
(Yka)(x) =
zZk
<yk,z,a> + kk(z)
yk,z(x)
+ 1 - pD,k(x)]a(x)
[ S

24. Particle PHD filter
The PHD (or intensity function) vk is not a probability density
The PHD propagation equation is not a standard Bayesian recursion
Sequential MC implementation of the PHD filter
[Vo, Singh & Doucet 03, 05], [Sidenbladh 03], [Mahler & Zajic 03]
state space
Particle approximation of vk
Particle approximation of vk-1
Need to cluster the particles to obtain multi-target estimates

25. Particle PHD filter Algorithm Initialise; for k =1: kmax ,
for i =1: Jk ,
Sample: ; compute: ;
end;
for i = Jk +1: Jk +Lk-1 ,
Sample: ; compute: ;
for i =1: Jk +Lk-1 ,
Update: ;
Redistribute total mass among Lk resampled particles;
Convergence: [Vo, Singh & Doucet 05], [Clark & Bell 06], [Johansen et. al. 06]

26. Gaussian Mixture PHD filter
Closed-form solution to the PHD recursion exists for linear Gaussian multi-target model
Gaussian mixture prior intensity Þ Gaussian mixture posterior intensities at all subsequent times
vk-1( . |Z1:k-1)
vk(. |Z1:k)
vk|k-1(. |Z1:k-1)

PHD filter

{wk-1, mk-1, Pk-1}
i=1
Jk-1
(i)
{wk|k-1, mk|k-1, Pk|k-1}
Jk|k-1
{wk, mk, Pk } Jk
Gaussian Mixture (GM) PHD filter [Vo & Ma 05, 06]
Extended & Unscented Kalman PHD filter [Vo & Ma 06]
Jump Markov PHD filter [Pasha et. al. 06]
Track continuity [Clark et. al. 06]

27. Cardinalised PHD Filter
Drawback of PHD filter: High variance of cardinality estimate
Relax Poisson assumption: allows arbitrary cardinality distribution
Jointly propagate: intensity function &
probability generating function of cardinality.
pk-1(n|Z1:k-1)
pk(n|Z1:k)
pk|k-1(n|Z1:k-1)

cardinality
prediction
update
vk-1(xk-1|Z1:k-1)
intensity
prediction
vk|k-1(xk|Z1:k-1)
intensity
update
vk(xk|Z1:k)


CPHD filter [Mahler 06,07]
More complex PHD update step (higher computational costs)

28. Particle CPHD filter [Vo 08]
Gaussian Mixture CPHD Filter
Closed-form solution to the CPHD recursion exists for linear Gaussian multi-target model
Gaussian mixture prior intensity Þ Gaussian mixture posterior intensities at all subsequent times [Vo et. al. 06, 07]
Particle-PHD filter can be extended to the CPHD filter
cardinality
prediction 
cardinality
update 

{pk-1(n)} 
{pk|k-1(n)}
{pk(n)}

n=0
n=0
n=0
(i)
(i)
Jk-1
intensity
prediction
{wk-1, xk-1}
{wk|k-1, xk|k-1}
(i)
(i)
Jk|k-1
intensity
update
(i)
(i) Jk

{wk, xk }

i=1
i=1
i=1
Particle CPHD filter [Vo 08]

29. CPHD filter Demonstration
1000 MC trial average
GMCPHD filter
Well I’m getting a bit bored of equations, so I’d like to show this closed form solution in action. What I’m going to show you is a scenario where we have targets moving in a given region shown by a blue cross. The measurements received by the filter are shown by black crosses and you’ll see that we can’t distinguish which measurement came from where. The filter estimate is shown in red. On the right you’ll see a histogram of the posterior intensity showing how well the closed form Gaussian mixture implementation performs. Here we’ll actually have a total of 10 targets on the scene at any one time.
GMPHD filter

30. CPHD filter Demonstration
Comparison with JPDA: linear dynamics, sv = 5, sh = 10, 4 targets,
1000 MC trial average
Well I’m getting a bit bored of equations, so I’d like to show this closed form solution in action. What I’m going to show you is a scenario where we have targets moving in a given region shown by a blue cross. The measurements received by the filter are shown by black crosses and you’ll see that we can’t distinguish which measurement came from where. The filter estimate is shown in red. On the right you’ll see a histogram of the posterior intensity showing how well the closed form Gaussian mixture implementation performs. Here we’ll actually have a total of 10 targets on the scene at any one time.

31. CPHD filter Demonstration
Sonar images

32. Multi-object Bayes filter
MeMBer Filter
Multi-object Bayes filter
pk-1(Xk-1|Z1:k-1)
prediction
pk|k-1(Xk|Z1:k-1)
update
pk(Xk|Z1:k)


(i)
(i)
Mk-1
prediction
(i)
(i)
Mk|k-1
update
(i)
(i) Mk

{(rk-1, pk-1)}
{(rk|k-1, pk|k-1)}
{(rk, pk )}

i=1
i=1
i=1
(Multi-target Multi-Bernoulli ) MeMBer filter [Mahler 07], biased
Cardinality-Balanced MeMBer filter [Vo et. al. 07], unbiased
Approximate predicted/posterior RFSs by Multi-Bernoulli RFSs
Valid for low clutter rate & high probability of detection

33. Cardinality-Balanced
Cardinality-Balanced MeMBer Filter
(i)
(i)
Mk-1
prediction
(i)
(i)
Mk|k-1
update
{(rk-1, pk-1)}
{(rk|k-1, pk|k-1)}
{(rk, pk )}
(i)
(i) Mk


i=1
i=1
i=1
(i)
Mk-1
{(rP,k|k-1, pP,k|k-1)} È {(rG,k, pG,k)}
(i)
(i)
(i)
MG,k
i=1
i=1
rk-1á pk-1, pS,kñ
(i)
term for object births
(i)
á fk|k-1(·|·), pk-1 pS,kñ
Cardinality-Balanced
MeMBer filter
[Vo et. al. 07]
ápk-1, pS,kñ
(i)

34. Cardinality-Balanced
Cardinality-Balanced MeMBer Filter
(i)
(i)
Mk-1
prediction
(i)
(i)
Mk|k-1
update

{(rk-1, pk-1)}
{(rk|k-1, pk|k-1)}
{(rk, pk )}
(i)
(i) Mk

i=1
i=1
i=1
{(rL,k, pL,k)} È {(rU,k,(z), pU,k(z))}
(i)
(i)
Mk|k-1
i=1
zÎZk
1- rk|k-1 ápk|k-1, pD,kñ
(i)
rk|k-1(1- ápk|k-1, pD,kñ)
1- ápk|k-1, pD,kñ
(i)
pk|k-1(1- pD,k)
1- rk|k-1
(i)
rk|k-1 pk|k-1
i=1
Mk|k-1 S
pD,kgk(z|·)
rk|k-1ápk|k-1, pD,kgk(z|·)ñ
1- rk|k-1
(i)
i=1
Mk|k-1 S
Mk|k-1
rk|k-1(1- rk|k-1) ápk|k-1, pD,kgk(z|·)ñ
(i)
(i)
(i) S
(1- rk|k-1ápk|k-1, pD,kñ)2
(i)
(i)
i=1
1- rk|k-1 ápk|k-1, pD,kñ
(i)
rk|k-1 ápk|k-1, pD,kgk(z|·)ñ
i=1
Mk|k-1 S
Cardinality-Balanced
MeMBer filter
[Vo et. al. 07]
k(z) +

35. Cardinality-Balanced MeMBer Filter
Closed-form (Gaussian mixture) solution [Vo et. al. 07],
Jk-1
(i,j)
(i,j)
(i,j)
Jk|k-1
{wk-1, mk-1, Pk-1}
(i,j)
(i,j)
(i,j)
{wk|k-1, mk|k-1, Pk|k-1}
{wk, mk, Pk }
(i,j)
(i,j)
(i,j) Jk
j=1
j=1
j=1
(i)
(i)
Mk-1
prediction
(i)
(i)
Mk|k-1
update
{(rk-1, pk-1)}
{(rk|k-1, pk|k-1)}
{(rk, pk )}
(i)
(i) Mk


i=1
i=1
i=1
{wk-1, xk-1}
(i,j)
(i,j)
Jk-1
{wk|k-1, xk|k-1 }
(i,j)
(i,j)
Jk|k-1
(i,j)
(i,j) Jk
{wk, xk }
j=1
j=1
j=1
Particle implementation [Vo et. al. 07],
More useful than PHD filters in highly non-linear problems

36. Performance comparison
Example: 10 targets max on scene, with births/deaths
4D states: x-y position/velocity, linear Gaussian
observations: x-y position, linear Gaussian
Dynamics
constant velocity model:
v = 5ms-2,
survival probability:
pS,k = 0.99,
Observations
additive Gaussian noise:
 =10m,
detection probability:
pD,k = 0.98,
uniform Poisson clutter:
c = 2.5x10-6m-2
/
start/end
positions

37. Gaussian implementation
1000 MC trial average
Cardinality-Balanced
Recursion
Mahler’s
MeMBer
Recursion

38. Gaussian implementation
1000 MC trial average
CPHD Filter
has better
performance

39. Particle implementation
1000 MC trial average
CB-MeMBer
Filter has better
performance

40. Concluding Remarks Thank You! Random Finite Set framework
Rigorous formulation of Bayesian multi-target filtering
Leads to efficient algorithms
Future research directions
Track before detect
Performance measure for multi-object systems
Numerical techniques for estimation of trajectories
For more info please see
Thank You!

41. References
D. Stoyan, D. Kendall, J. Mecke, Stochastic Geometry and its Applications, John Wiley & Sons, 1995
D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer-Verlag, 1988.
I. Goodman, R. Mahler, and H. Nguyen, Mathematics of Data Fusion. Kluwer Academic Publishers, 1997.
R. Mahler, “An introduction to multisource-multitarget statistics and applications,” Lockheed Martin Technical Monograph, 2000.
R. Mahler, “Multi-target Bayes filtering via first-order multi-target moments,” IEEE Trans. AES, vol. 39, no. 4, pp. 1152–1178, 2003.
B. Vo, S. Singh, and A. Doucet, “Sequential Monte Carlo methods for multi-target filtering with random finite sets,” IEEE Trans. AES, vol. 41, no. 4, pp. 1224–1245, 2005,.
B. Vo, and W. K. Ma, “The Gaussian mixture PHD filter,” IEEE Trans. Signal Processing, IEEE Trans. Signal Processing, Vol. 54, No. 11, pp , 2006.
R. Mahler, “A theory of PHD filter of higher order in target number,” in I. Kadar (ed.), Signal Processing, Sensor Fusion, and Target Recognition XV, SPIE Defense & Security Symposium, Orlando, April 17-22, 2006
B. T. Vo, B. Vo, and A. Cantoni, "Analytic implementations of the Cardinalized Probability Hypothesis Density Filter," IEEE Trans. SP, Vol. 55,  No. 7,  Part 2,  pp , 2007.
D. Clark & J. Bell, “Convergence of the Particle-PHD filter,” IEEE Trans. SP, 2006.
A. Johansen, S. Singh, A. Doucet, and B. Vo, "Convergence of the SMC implementation of the PHD filter," Methodology and Computing in Applied Probability, 2006.
A. Pasha, B. Vo, H. D Tuan and W. K. Ma, "Closed-form solution to the PHD recursion for jump Markov linear models," FUSION, 2006.
D. Clark, K. Panta, and B. Vo, "Tracking multiple targets with the GMPHD filter," FUSION, 2006.
B. T. Vo, B. Vo, and A. Cantoni, “On Multi-Bernoulli Approximation of the Multi-target Bayes Filter," ICIF, Xi’an, 2007.
See also:

42. Representation of Multi-target state
Optimal Subpattern Assignment (OSPA) metric
[Schumacher et. al 08]
Fill up X with n - m dummy points located at a distance greater than c from any points in Y
Calculate pth order Wasserstein distance between resulting sets
Efficiently computed using the Hungarian algorithm

43. S S [pS,kwk-1N(x; mS,k|k-1, PS,k|k-1) +
Gaussian Mixture PHD Prediction
Gaussian mixture posterior intensity at time k-1:
vk-1(x) = wk-1N(x; mk-1, Pk-1) S
i=1
Jk-1
(i)
Gaussian mixture predicted intensity to time k:
vk|k-1(x) =
[pS,kwk-1N(x; mS,k|k-1, PS,k|k-1) + S
i=1
Jk-1
(i)
wk-1wb,kN(x; mb,k|k-1, Pb,k|k-1)] + gk(x)
(i,l)
l=1
Jb,k
(l)
Fk|k-1vk-1
mS,k|k-1 = Fk-1mk-1
PS,k|k-1 = Fk-1 Pk-1 Fk-1 + Qk-1
(i) T
(i,l)
(l)
mb,k|k-1 = Fb,k-1mk-1 + db,k-1
Pb,k|k-1 = Fb,k-1 Pk-1 (Fb,k-1 )T + Qb,k-1
(i)

44. S S S S Gaussian Mixture PHD Update
Gaussian mixture predicted intensity to time k:
vk|k-1(x) = wk|k-1N(x; mk|k-1, Pk|k-1) S
i=1
Jk|k-1
(i)
Gaussian mixture updated intensity at time k:
vk(x) =
i=1
Jk|k-1
(i)
N(x; mk|k(z), Pk|k) + (1- pD,k)vk|k-1(x)
S S
zÎ Zk
(j)
j=1 S
pD,k wk|k-1qk (z) + kk(z)
pD,kwk|k-1qk (z)
Ykvk|k-1
mk|k(z) = mk|k-1 + Kk (z- Hk mk|k-1 )
(i)
Pk|k = (I- Kk Hk )Pk|k-1
(i)
qk(z) = N(z; Hkmk|k-1, HkPk|k-1Hk + Rk ) T
(i)
Kk = Pk|k-1Hk (Hk Pk|k-1Hk + Rk )-1
(i) T

45. Markov transition density
Cardinalised PHD Prediction S
j=0 n
pk|k-1(n) = p,k(n - j) k|k-1[vk-1,pk-1](j)
predicted
cardinality
probability of n - j spontaneous births
probability of j surviving targets S
l=j
Cjl <pS,k ,vk-1> j <1- pS,k ,vk-1> l-j
pk-1 (l)
<1,vk-1>l 
vk|k-1(xk) = pS,k(xk-1) fk|k-1(xk|xk-1) vk-1(xk-1)dxk-1 + gk(xk)
predicted
intensity
probability
of survival
Markov transition density
intensity from
previous
time-step
intensity of spontaneous
object births Gk

46. S (P ) Cardinalised PHD Update S esfj(Z) = S pk(n) = +
¡k[vk|k-1, Zk](n)pk|k-1(n)
vk(xk) = vk|k-1(xk)Yk, Zk(xk)
predicted intensity
updated intensity
pk(n) =
<¡k[vk|k-1, Zk], pk|k-1>
updated cardinality distribution
predicted cardinality distribution
¡k[v, Z](n) =
pK,k(|Z|–j) (|Z|–j)! Pj+u S
esfj({<v,yk,z>: zZk})
<1- pD,k ,v >n-(j+u)
<1,v >n n
j=0
min(|Z|,n) u
S (P ) z
zS
S Í Z,|S|=j
esfj(Z) =
likelihood function
prob. of
detection
clutter intensity
pD,k(xk)gk(z|xk)<1,kk>/kk(z)
clutter cardinality distribution S
zZk
yk,z(xk) +
<¡k[vk|k-1, Zk], pk|k-1>
<¡k[vk|k-1, Zk\{z}], pk|k-1> 1
(1-pD,k(xk))

47. Multi-object Bayes filter
Mahler’s MeMBer Filter
Multi-object Bayes filter
pk-1(Xk-1|Z1:k-1)
prediction
pk|k-1(Xk|Z1:k-1)
update
pk(Xk|Z1:k)


(i)
(i)
Mk-1
prediction
(i)
(i)
Mk|k-1
update
(i)
(i) Mk

{(rk-1, pk-1)}
{(rk|k-1, pk|k-1)}
{(rk, pk )}

i=1
i=1
i=1
(Multi-target Multi-Bernoulli ) MeMBer filter [Mahler 07]
Approximate predicted/posterior RFSs by Multi-Bernoulli RFSs
Valid for low clutter rate & high probability of detection
Biased in Cardinality (except when probability of detection = 1)

48. Cardinality-Balanced
Cardinality-Balanced MeMBer Filter
(i)
(i)
Mk-1
prediction
(i)
(i)
Mk|k-1
update
{(rk-1, pk-1)}
{(rk|k-1, pk|k-1)}
{(rk, pk )}
(i)
(i) Mk


i=1
i=1
i=1
Cardinality-Balanced
MeMBer filter
[Vo et. al. 07]
(i)
(i)
Mk|k-1
{(rL,k, pL,k)} È {(rU,k,(z), pU,k(z))}
i=1
zÎZk
1- rk|k-1 ápk|k-1, pD,kñ
(i)
rk|k-1(1- ápk|k-1, pD,kñ)
1- ápk|k-1, pD,kñ
(i)
pk|k-1(1- pD,k)
ávk|k-1, pD,kgk(z|·)ñ
vk|k-1 pD,kgk(z|·) ~*
(1- rk|k-1ápk|k-1, pD,kñ)2
(i)
rk|k-1(1- rk|k-1) pk|k-1
i=1
Mk|k-1
vk|k-1 = S
(1)
k(z) + ávk|k-1, pD,kgk(z|·)ñ
ávk|k-1, pD,kgk(z|·)ñ
(1) ~
1- rk|k-1 ápk|k-1, pD,kñ
(i)
rk|k-1 pk|k-1
i=1
Mk|k-1
vk|k-1 = ~ S
1- rk|k-1
(i)
rk|k-1 pk|k-1
i=1
Mk|k-1
vk|k-1 = ~* S

49. Extensions of the PHD filter
Linear Jump Markov PHD filter [Pasha et. al. 06]

50. Extensions of the PHD filter
Example: 4-D, Linear JM target dynamics with 3 models
4 targets, birth rate= 3x0.05, death prob. = 0.01, clutter rate = 40

51. What is a Random Finite Set (RFS)?
I will then present finite sets stats-a stat tool derived from RS for attacking MS MT tracking.
Pine saplings in a Finish forest [Kelomaki & Penttinen]
Childhood leukaemia & lymphoma in North Humberland [Cuzich & Edwards]
The number of points is random,
The points have no ordering and are random
Loosely, an RFS is a finite set-valued random variable
Also known as: (simple finite) point process or random point pattern