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Random Set/Point Random Set/Point Process in Multi-Target Process in Multi-Target TrackingTracking Ba-Ngu VoBa-Ngu Vo
EEE Department EEE Department University of University of MelbourneMelbourneAustraliaAustraliahttp://www.ee.unimelb.edu.au/staff/bv/
SAMSI, RTP, NC, USA, 8 September 2008
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.
The Bayes (single-target) filterThe Bayes (single-target) filter
Multi-target trackingMulti-target tracking
System representationSystem representation
Random finite set & Bayesian Multi-target filteringRandom finite set & Bayesian Multi-target filtering
Tractable multi-target filtersTractable multi-target filters
Probability Hypothesis Density (PHD) filterProbability Hypothesis Density (PHD) filter
Cardinalized PHD filterCardinalized PHD filter
Multi-Bernoulli filterMulti-Bernoulli filter
ConclusionsConclusions
OutlineOutline
The Bayes (single-target) Filter The Bayes (single-target) Filter
state-vector
target motion
state space
observation space
xk
xk-1
zk-1
zk
fk|k-1(xk| xk-1)
Markov Transition Density Measurement Likelihood
gk(zk| xk)
Objective
measurement history (z1,…, zk)posterior (filtering) pdf of the state
pk(xk | z1:k)
System Model
state-vector
target motion
state space
observation space
xk
xk-1
zk-1
zk
Bayes filter
pk-1(xk-1 |z1:k-1) pk|k-1(xk| z1:k-1) pk(xk| z1:k)prediction data-update
pk-1(xk-1| z1:k-1) dxk-1
fk|k-1(xk| xk-1) gk(zk| xk)K-1 pk|k-1(xk| z1:k-1)
The Bayes (single-target) Filter The Bayes (single-target) Filter
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=1
N(i) (i) {wk, xk } i=1 N(i) (i) {wk-1, xk-1}
(i) (i)Particle filter
state-vector
target motion
state space
observation space
xk
xk-1
zk-1
zk
fk|k-1(xk| xk-1)
gk(zk| xk)
The Bayes (single-target) Filter The Bayes (single-target) Filter
Multi-target trackingMulti-target tracking
observation produced by targets
target motion
state space
observation space
5 targets 3 targetsXk-1
Xk
Objective: Jointly estimate the number and states of targets
Challenges:
Random number of targets and measurements
Detection uncertainty, clutter, association uncertainty
Multi-target trackingMulti-target tracking
System RepresentationSystem Representation
0
0
1
1
X
0
0
1
1
X
1
1'
0
0
X
1
1'
0
0
X
Estimate is correct but estimation error ???
TrueMulti-target state
EstimatedMulti-target state
|| ' || 2X X
How can we mathematically represent the multi-target state?
2 targets 2 targets
Usual practice: stack individual states into a large vector!
Problem:
Remedy: use( ')
min || ' || 0perm X
X X
1
1'
0
0
X
1
1'
0
0
X
True
Multi-target state
?X ?X
EstimatedMulti-target State
2 targetsno target
1
1'
0
0
X
1
1'
0
0
X
True
Multi-target state
0
0X
0
0X
EstimatedMulti-target State
2 targets1 target
System RepresentationSystem Representation
What are the estimation errors?
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
( ')min || ' || 0
perm XX X ( ')
min || ' || 0perm X
X X
System RepresentationSystem Representation
observation produced by targets
target motion
state space
observation space
5 targets 3 targetsXk-1
Xk
Number of measurements and their values are (random) variables
Ordering of measurements not relevant!
Multi-target measurement is represented by a finite set
System RepresentationSystem Representation
RFS & Bayesian Multi-target FilteringRFS & Bayesian Multi-target Filtering
targets target set
observed set
observations
X
Z
Need suitable notions of density & integration
pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1) prediction data-update
1| 1 1: 1( | ) ( | )k k k k k k kK g Z X p X Z
1| 1 1: 1( | ) ( | )k k k k k k kK g Z X p X Z
Reconceptualize as a generalized single-target problem [Mahler 94]
Bayesian: Model state & observation as Random Finite Sets [Mahler 94]
RFS & Bayesian Multi-target FilteringRFS & Bayesian Multi-target Filtering
S
N(S) = | S|
point process or random counting measure
random finite set or random point pattern
state space E
state space E
Belief “density” of
f : F(E) [0,)
(T ) = T f (X)X
Belief “distribution” of (T ) = P(T ) , T E
E
Probability density of p : F(E) [0,)
P (T ) = T p (X)(dX)
Probability distribution of P (T ) = P(T ) , T F(E)
F(E)
Collection of finite
subsets of E State space
Mahler’s Finite Set Statistics (1994)
Choquet (1968)
T T
Conventional integral Set integral
Vo et. al. (2005)
Point Process Theory (1950-1960’s)
RFS & Bayesian Multi-target FilteringRFS & Bayesian Multi-target Filtering
x x’
X’
xdeath
creation
X’
x
spawn
motion
Multi-target Motion ModelMulti-target Motion Model
fk|k-1(Xk|Xk-1 )Multi-object transition
density
Xk = Sk|k-1(Xk-1)Bk|k-1(Xk-1)k
Evolution of each element x of a given multi-object state Xk-1
Multi-target Observation ModelMulti-target Observation Model
gk(Zk|Xk)
Multi-object likelihood
Zk = k(Xk)Kk(Xk)
x z
x
likelihood
misdetection
clutter
state space observation space
Observation process for each element x of a given multi-object state Xk
pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1) prediction data-update
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
)()|()|( 1:111| dXZXpXXf skkkkk )()|()|( 1:111| dXZXpXXf skkkkk
)()|()|(
)|()|(
1:11|
1:11|
dXZXpXZg
ZXpXZg
skkkkk
kkkkkkk
)()|()|(
)|()|(
1:11|
1:11|
dXZXpXZg
ZXpXZg
skkkkk
kkkkkkk
Multi-target Bayes FilterMulti-target Bayes Filter
Multi-target Bayes filter
Particle Multi-target Bayes FilterParticle Multi-target Bayes Filter
AlgorithmAlgorithm
for i =1:N, % Initialise => Sample: Compute:
end;normalise weights;for k =1: kmax ,
for i =1:N, % Update => Sample:
Update:end;normalise weights;resample;MCMC step;
end;
( )
( )1:
1
( | ) ( ),ik
Ni
k k k k kXi
p X Z w X
( )
( )1:
1
( | ) ( ),ik
Ni
k k k k kXi
p X Z w X
( )0 0 0~ ( )iX q X
( )0
( )0 0 0 0
1
( ) ( ),i
Ni
Xi
p X w X
( )0
( )0 0 0 0
1
( ) ( ),i
Ni
Xi
p X w X
( ) ( ) ( )0 0 0 0 0( ) ( )i i iw p X q X
( ) ( ) ( ) ( ) ( ) ( ) ( )1 | 1 1 1 1:( | ) ( | ) ( | , )i i i i i i i
k k k k k k k k k k k k kw w g Z X f X X q X X Z
( ) ( )1 1:~ ( | , )i i
k k k k kX q X X Z
pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1) prediction data-update
Multi-target Bayes filter: very expensive!
single-object Bayes filter
multi-object Bayes filter
state of system: random vector
first-moment filter(e.g. -- filter)
state of system: random set
first-moment filter(“PHD” filter)
Single-object
Multi-object
)()|()|( 1:111| dXZXpXXf skkkkk )()|()|( 1:111| dXZXpXXf skkkkk
)()|()|(
)|()|(
1:11|
1:11|
dXZXpXZg
ZXpXZg
skkkkk
kkkkkkk
)()|()|(
)|()|(
1:11|
1:11|
dXZXpXZg
ZXpXZg
skkkkk
kkkkkkk
The PHD FilterThe PHD Filter
x0 state space
v PHD (intensity function) of a RFS
S
v(x0) = density of expected
number of objects
at x0
The Probability Hypothesis DensityThe Probability Hypothesis Density
v(x)dx = expected number
of objects in SS
= mean of, N(S), the
random counting
measure at S
The PHD FilterThe PHD Filter
state space
vk vk-1
PHD filter
vk-1(xk-1|Z1:k-1) vk(xk|Z1:k) vk|k-1(xk|Z1:k-1) PHD
prediction
PHD
update
Multi-object Bayes filter
pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1)prediction update
Avoids data association!
PHD PredictionPHD Prediction
vk|k-1(xk |Z1:k-1) = k|k-1(xk, xk-1) vk-1(xk-1|Z1:k-1)dxk-1 k(xk) intensity from
previoustime-step
term for spontaneousobject births
= intensity of k
k|k-1(xk, xk-1) = ek|k-1(xk-1) fk|k-1(xk|xk-1) + k|k-1(xk|xk-1)
Markovtransitionintensity
probabilityof objectsurvival
term for objectsspawned by
existing objects= intensity of Bk(xk-1)
Markov transition density
predictedintensity
Nk|k-1 = vk|k-1 (x|Z1:k-1)dxpredicted expected number of objects
(k|k-1)(xk) k|k-1(xk, x)(x)dx k(xk)
vk|k-1 k|k-1vk-1
PHD UpdatePHD Update
vk(xk|Z1:k) zZk Dk(z) + k(z)
pD,k(xk)gk(z|xk) + 1 pD,k(xk)]vk|k-1(xk|Z1:k-1)
Dk(z) = pD,k(x)gk(z|x)vk|k-1(x|Z1:k-1)dx Nk= vk(x|Z1:k)dx
Bayes-updated intensity
predicted intensity (from previous time)
intensity offalse alarms
sensor likelihood function
probabilityof detection
expected number of objects
measurement
vk kvk|k-1
(k)(x) =zZk
<k,z,> + k(z) k,z(x)
+ 1 pD,k(x)](x) [
Particle PHD filterParticle PHD filter
| 1 1( )k k k k kv v
Particle approximation of vk-1 Particle approximation of vk
state space
[Vo, Singh & Doucet 03, 05], [Sidenbladh 03], [Mahler & Zajic 03]
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
Need to cluster the particles to obtain multi-target estimates
Particle PHD filterParticle PHD filter
AlgorithmAlgorithm
Initialise;for k =1: kmax ,
for i =1: Jk , Sample: ; compute: ;
end;
for i = Jk +1: Jk +Lk-1 , Sample: ; compute: ;
end;for i =1: Jk +Lk-1 ,
Update: ;end;
Redistribute total mass among Lk resampled particles;end;
ki
k p ~ )(x)(
)(1)(
)()(
1| ikk
ikk
k
ikk pJ
wx
x
ki
k q~)(x)(
),()(
)(1
)(1
)(1|)(
1| ikk
ik
ik
ikkki
kk q
ww
x
xx
)(1|
1
)(1|
)(,
)(,)()(
1 )()(
)()(1 i
kkZz
LJ
j
jkk
jzkk
izki
Di
k wwz
pwk
kk
x
xx
Convergence: [Vo, Singh & Doucet 05], [Clark & Bell 06], [Johansen et. al. 06]
Gaussian Mixture PHD filterGaussian Mixture PHD filter
Closed-form solution to the PHD recursion exists for linear Gaussian multi-target model
vk-1( . |Z1:k-1) vk(. |Z1:k) vk|k-1(. |Z1:k-1)
1| kk k
1| kk k{wk-1, mk-1, Pk-1} i=1
Jk-1(i) (i) (i) {wk|k-1, mk|k-1, Pk|k-1} i=1Jk|k-1(i) (i) (i) {wk, mk, Pk } i=1
Jk(i) (i) (i)
PHD filter
Gaussian Mixture (GM) PHD filter [Vo & Ma 05, 06]
Gaussian mixture prior intensity Gaussian mixture posterior intensities at all subsequent times
Extended & Unscented Kalman PHD filter [Vo & Ma 06]
Jump Markov PHD filter [Pasha et. al. 06]
Track continuity [Clark et. al. 06]
Cardinalised PHD FilterCardinalised 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.
More complex PHD update step (higher computational costs)
CPHD filter [Mahler 06,07]
vk-1(xk-1|Z1:k-1) vk(xk|Z1:k) vk|k-1(xk|Z1:k-1) intensity
prediction
intensity
update
pk-1(n|Z1:k-1) pk(n|Z1:k) pk|k-1(n|Z1:k-1) cardinality
prediction
cardinality
update
Gaussian Mixture CPHD FilterGaussian Mixture CPHD Filter
{wk-1, xk-1} i=1Jk-1(i) (i) {wk|k-1, xk|k-1} i=1
Jk|k-1(i) (i) {wk, xk } i=1 Jk(i) (i)intensity
prediction
intensity
update
cardinality
prediction
cardinality
update {pk-1(n)}n=0
{pk|k-1(n)} n=0
{pk(n)}n=0
Particle CPHD filter [Vo 08]
Closed-form solution to the CPHD recursion exists for linear Gaussian multi-target model
Gaussian mixture prior intensity Gaussian mixture posterior intensities at all subsequent times [Vo et. al. 06, 07]
Particle-PHD filter can be extended to the CPHD filter
CPHD filter DemonstrationCPHD filter Demonstration
10 20 30 40 50 60 70 80 90 1000
5
10
Time
Car
dina
lity
Sta
tistic
s
True
Mean
StDev
10 20 30 40 50 60 70 80 90 1000
5
10
Time
Car
dina
lity
Sta
tistic
s
True
Mean
StDev
1000 MC trial average1000 MC trial average
GMCPHD filter
GMPHD filter
CPHD filter DemonstrationCPHD filter Demonstration
1000 MC trial average1000 MC trial average
Comparison with JPDA: linear dynamics, Comparison with JPDA: linear dynamics, vv = 5, = 5, = 10, = 10, 4 targets,4 targets,
Sonar imagesSonar images
CPHD filter DemonstrationCPHD filter Demonstration
MeMBer FilterMeMBer Filter
{(rk-1, pk-1)} i=1
Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1
Mk|k-1(i) (i)
{(rk, pk )} i=1
Mk(i) (i)
prediction
update
Valid for low clutter rate & high probability of detection
Multi-object Bayes filter
pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1)prediction update
(Multi-target Multi-Bernoulli ) MeMBer filter [Mahler 07], biased
Approximate predicted/posterior RFSs by Multi-Bernoulli RFSs
Cardinality-Balanced MeMBer filter [Vo et. al. 07], unbiased
Cardinality-Balanced MeMBer FilterCardinality-Balanced MeMBer Filter
{(rk-1, pk-1)} i=1
Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1
Mk|k-1(i) (i)
{(rk, pk )} i=1
Mk(i) (i)
prediction
update
{(rP,k|k-1, pP,k|k-1)} {(r,k, p,k)} (i) (i) (i) (i)
i=1
Mk-1
i=1
M,k
rk-1pk-1, pS,k(i) (i)
fk|k-1(|), pk-1 pS,k(i)
pk-1, pS,k(i)
term for object births
Cardinality-Balanced MeMBer filter [Vo et. al. 07]
{(rk-1, pk-1)} i=1
Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1
Mk|k-1(i) (i)
{(rk, pk )} i=1
Mk(i) (i)
prediction
update
{(rL,k, pL,k)} {(rU,k,(z), pU,k(z))} (i) (i)
zZki=1
Mk|k-1
1 pk|k-1, pD,k(i)
pk|k-1(1 pD,k)(i)
1 rk|k-1 pk|k-1, pD,k(i) (i)
rk|k-1(1 pk|k-1, pD,k)(i)(i)
Cardinality-Balanced MeMBer FilterCardinality-Balanced MeMBer Filter
rk|k-1(1 rk|k-1) pk|k-1, pD,kgk(z|)
1 rk|k-1 pk|k-1, pD,k(i) (i)
rk|k-1 pk|k-1, pD,kgk(z|)(i)(i)
i=1
Mk|k-1
(1 rk|k-1pk|k-1, pD,k)2(i) (i)
(i)(i) (i)
i=1
Mk|k-1
(z)
1 rk|k-1(i)
rk|k-1 pk|k-1(i)(i)
i=1
Mk|k-1
pD,kgk(z|)
rk|k-1pk|k-1, pD,kgk(z|)
1 rk|k-1(i)
(i)(i)
i=1
Mk|k-1
Cardinality-Balanced MeMBer filter [Vo et. al. 07]
Cardinality-Balanced MeMBer FilterCardinality-Balanced MeMBer Filter
Closed-form (Gaussian mixture) solution [Vo et. al. 07],
Particle implementation [Vo et. al. 07],
{(rk-1, pk-1)} i=1
Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1
Mk|k-1(i) (i)
{(rk, pk )} i=1
Mk(i) (i)
prediction
update
{wk-1, xk-1} j=1
Jk-1(i,j) (i,j)j=1Jk|k-1(i,j) (i,j){wk|k-1, xk|k-1 } {wk, xk } j=1
Jk(i,j) (i,j)
{wk-1, mk-1, Pk-1} j=1
Jk-1(i,j) (i,j) (i,j) {wk|k-1, mk|k-1, Pk|k-1} j=1Jk|k-1(i,j) (i,j) (i,j) {wk, mk, Pk } j=1
Jk(i,j) (i,j) (i,j)
More useful than PHD filters in highly non-linear problems
Performance comparisonPerformance comparison
Example:Example: 10 targets max on scene, with births/deaths 4D states: x-y position/velocity, linear Gaussian
observations: x-y position, linear Gaussian
-1000 -800 -600 -400 -200 0 200 400 600 800 1000-1000
-800
-600
-400
-200
0
200
400
600
800
1000
x-coordinate (m)
y-co
ord
inat
e (m
)
/start/end positions
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
10 20 30 40 50 60 70 80 90 1000
5
10
15
20
Time
Car
dina
lity
Sta
tistic
s
True
Mean
StDev
10 20 30 40 50 60 70 80 90 1000
5
10
15
20
Time
Car
dina
lity
Sta
tistic
s
Cardinality-BalancedRecursion
Mahler’sMeMBerRecursion
1000 MC trial average1000 MC trial average
Gaussian implementationGaussian implementation
Gaussian implementationGaussian implementation
1000 MC trial average1000 MC trial average
CPHD Filter has better performance
10 20 30 40 50 60 70 80 90 1000
10
20
30
Time
OS
PA
Loc
(m
)(c=
300,
p=
1)
10 20 30 40 50 60 70 80 90 1000
100
200
300
Time
OS
PA
Car
d (m
)(c=
300,
p=
1)
10 20 30 40 50 60 70 80 90 1000
100
200
300
Time
OS
PA
(m
)(c=
300,
p=
1)
GM-CBMeMBer
GM-PHDGM-CPHD
GM-MeMBer
Particle implementationParticle implementation
1000 MC trial average1000 MC trial average
CB-MeMBerFilter has better performance
10 20 30 40 50 60 70 80 90 1000
100
200
300
Time
OS
PA
(m
)(c=
300,
p=
1)
SMC-CBMeMBer
SMC-PHDSMC-CPHD
SMC-MeMBer
10 20 30 40 50 60 70 80 90 1000
50
100
Time
OS
PA
Loc
(m
)(c=
300,
p=
1)
10 20 30 40 50 60 70 80 90 1000
100
200
300
Time
OS
PA
Car
d (m
)(c=
300,
p=
1)
Concluding RemarksConcluding Remarks
Thank You!
Random Finite Set frameworkRandom Finite Set framework
Rigorous formulation of Bayesian multi-target filteringRigorous formulation of Bayesian multi-target filtering
Leads to efficient algorithmsLeads to efficient algorithms
Future research directions Future research directions
Track before detectTrack before detect
Performance measure for multi-object systemsPerformance measure for multi-object systems
Numerical techniques for estimation of trajectoriesNumerical techniques for estimation of trajectories
For more info please see http://randomsets.ee.unimelb.edu.au/
ReferencesReferences
• 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. 4091-4104, 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. 3553-3567, 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: http://www.ee.unimelb.edu.au/staff/bv/publications.html
1 1
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( , ),
m n
pm
c p pi i
i
X x x Y y y
m n
d X Y d x y c n m m nn
d Y X
( )
( , ) min(|| ||, )c
m n
d x y x y c
permutation
1 1
1/
( )( )
1
{ ,..., }, { ,..., }
0, 0
1( , ) min ( , ) ( ) ,
( , ),
m n
pm
c p pi i
i
X x x Y y y
m n
d X Y d x y c n m m nn
d Y X
( )
( , ) min(|| ||, )c
m n
d x y x y c
permutation
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
Representation of Multi-target stateRepresentation of Multi-target state
Gaussian Mixture PHD PredictionGaussian Mixture PHD Prediction
vk-1(x) = wk-1N(x; mk-1, Pk-1)
i=1
Jk-1
(i) (i) (i)
vk|k-1(x) = [pS,kwk-1N(x; mS,k|k-1, PS,k|k-1) +i=1
Jk-1
(i) (i) (i)wk-1w,kN(x; m,k|k-1, P,k|k-1)] + k(x) (i) (i,l) (i,l)
l=1
J,k
(l)
Gaussian mixture posterior intensity at time k-1:
Gaussian mixture predicted intensity to time k:
k|k-1vk-1 mS,k|k-1 = Fk-1mk-1
PS,k|k-1 = Fk-1 Pk-1 Fk-1 + Qk-1
(i) (i)
T(i)(i)
(i,l)
(i,l) (l)
m,k|k-1 = F,k-1mk-1 + d,k-1
P,k|k-1 = F,k-1 Pk-1 (F,k-1 )T + Q,k-1
(l)
(l)
(l) (i)
(i) (l)
Gaussian Mixture PHD UpdateGaussian Mixture PHD Update
vk|k-1(x) = wk|k-1N(x; mk|k-1, Pk|k-1)i=1
Jk|k-1
(i) (i) (i)
Gaussian mixture predicted intensity to time k:
Gaussian mixture updated intensity at time k:
vk(x) =i=1
Jk|k-1
(i) (i) N(x; mk|k(z), Pk|k) + (1 pD,k)vk|k-1(x)
zZk
(i)
(j)
(i)
j=1
Jk|k-1pD,k wk|k-1qk (z) + k(z)
pD,kwk|k-1qk (z)
(j)
Pk|k = (IKk Hk )Pk|k-1
(i) (i)(i)
Kk = Pk|k-1Hk (Hk Pk|k-1Hk + Rk )1(i) (i) (i)T T
mk|k(z) = mk|k-1 + Kk (zHk mk|k-1 )(i) (i)(i)(i)
qk(z) = N(z; Hkmk|k-1, HkPk|k-1Hk + Rk )T(i)(i)(i)
kvk|k-1
vk|k-1(xk) = pS,k(xk-1) fk|k-1(xk|xk-1) vk-1(xk-1)dxk-1 k(xk) intensity from
previoustime-step
intensity of spontaneous
object births k
probabilityof survival
Markov transition density
predictedintensity
pk|k-1(n) = p,k(n - j) k|k-1[vk-1,pk-1](j)
probability of n - j spontaneous births
predictedcardinality
j=0
n
probability of j surviving targets
Cardinalised PHD PredictionCardinalised PHD Prediction
Cjl <pS,k ,vk-1> j <1 pS,k ,vk-1> l-j
l=j
<1,vk-1>lpk-1 (l)
vk(xk) = vk|k-1(xk)k, Zk(xk)
predicted intensity
updated intensity
zZk
k,z(xk)<k[vk|k-1, Zk], pk|k-1>
<k[vk|k-1, Zk\{z}], pk|k-1>1
0<k[vk|k-1, Zk], pk|k-1>
<k[vk|k-1, Zk], pk|k-1>0
1
(1pD,k(xk))
predicted cardinality distribution
k[vk|k-1, Zk](n)pk|k-1(n)
updated cardinality distribution
0
<k[vk|k-1, Zk], pk|k-1>pk(n) = 0
Cardinalised PHD UpdateCardinalised PHD Update
k[v, Z](n) = pK,k(|Z|–j) (|Z|–j)! Pj+u
esfj({<v,k,z>: zZk})
<1 pD,k ,v >n-(j+u)
<1,v >n
n
j=0
min(|Z|,n)u
SS Z,|S|=jesfj(Z) = likelihood
functionprob. of
detectionclutter intensity
pD,k(xk)gk(z|xk)<1,k>/k(z)clutter cardinality distribution
Mahler’s MeMBer FilterMahler’s MeMBer Filter
{(rk-1, pk-1)} i=1
Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1
Mk|k-1(i) (i)
{(rk, pk )} i=1
Mk(i) (i)
prediction
update
Valid for low clutter rate & high probability of detection
Multi-object Bayes filter
pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1)prediction update
(Multi-target Multi-Bernoulli ) MeMBer filter [Mahler 07]
Approximate predicted/posterior RFSs by Multi-Bernoulli RFSs
Biased in Cardinality (except when probability of detection = 1)
{(rk-1, pk-1)} i=1
Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1
Mk|k-1(i) (i)
{(rk, pk )} i=1
Mk(i) (i)
prediction
update
1 rk|k-1 pk|k-1, pD,k(i) (i)
rk|k-1 pk|k-1(i)(i)
i=1
Mk|k-1
vk|k-1 =~
(1 rk|k-1pk|k-1, pD,k)2(i) (i)
rk|k-1(1 rk|k-1) pk|k-1(i)(i) (i)
i=1
Mk|k-1
vk|k-1 = (1)
1 rk|k-1(i)
rk|k-1 pk|k-1(i)(i)
i=1
Mk|k-1
vk|k-1 =~*
{(rL,k, pL,k)} {(rU,k,(z), pU,k(z))} (i) (i)
zZki=1
Mk|k-1
(z) vk|k-1, pD,kgk(z|)
vk|k-1, pD,kgk(z|)(1)
~
1 pk|k-1, pD,k(i)
pk|k-1(1 pD,k)(i)
vk|k-1, pD,kgk(z|)
vk|k-1 pD,kgk(z|)
~*
~*
1 rk|k-1 pk|k-1, pD,k(i) (i)
rk|k-1(1 pk|k-1, pD,k)(i)(i)
Cardinality-Balanced MeMBer FilterCardinality-Balanced MeMBer Filter
Cardinality-Balanced MeMBer filter [Vo et. al. 07]
Linear Jump Markov PHD filter [Pasha et. al. 06]
-6 -4 -2 0 2 4 6
x 104
-6
-4
-2
0
2
4
6x 10
4
x coordinate (in m)
y co
ordi
nate
(in
m)
Aircraft 1 start of flight at k= 1;end of flight at k=90
Aircraft 2 start of flight at k= 3;end of flight at k=95
Aircraft 3 start of flight at k= 12;end of flight at k=100
Payload 1 separates from Aircraft 1at k= 31; end of flight at k=100
Payload 2 separates from Aircraft 2at k= 44; end of flight at k=88
Extensions of the PHD filterExtensions of the PHD filter
10 20 30 40 50 60 70 80 90 100
-5
0
5
x 104
time step
x co
ordi
nate
(in
m) PHD filter estimates
True tracks
10 20 30 40 50 60 70 80 90 100
-5
0
5
x 104
time step
y co
ordi
nate
(in
m)
Example: 4-D, Linear JM target dynamics with 3 modelsExample: 4-D, Linear JM target dynamics with 3 models
4 targets, birth rate= 3x0.05, death prob. = 0.01, 4 targets, birth rate= 3x0.05, death prob. = 0.01, clutter rate = clutter rate = 4040
Extensions of the PHD filterExtensions of the PHD filter
What is a Random Finite Set (RFS)?What is a Random Finite Set (RFS)?
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
Pine saplings in a Finish forest [Kelomaki & Penttinen]
Childhood leukaemia & lymphoma in North Humberland [Cuzich & Edwards]
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