http://www.erp.ac.uk
Bayesian Filtering, Smoothing and Distributed Data Fusion for Joint Detection and Tracking
Daniel Clark, RAEng/ EPSRC Research Fellow
Joint Research Institute in Signal and Image Processing
Joint Target-Detection and Tracking (JoTT
(JoTT))
Gaussian Mixture JoTT Filter
A closed-form solution to the JoTT filter recursion exists
for linear Gaussian multi-target model
observation space
observation set
produced by sensor
zk
Zk-1
Probability of target existence recursion has analytic solution
Zk
zk-1
Gaussian mixture prior density results in Gaussian mixture
posterior density:
state space
target motion
state-set
state-set
Xk-1
Gaussian mixture JoTT filter
Xk
Bayes filter
prediction
fk-1 (Xk-1 |Z1:k-1 )
⋅⋅⋅
fk|k-1(Xk | Z1:k-1)data-update fk (Xk | Z1:k )
new observation set
Bernoulli Markov Transition
Random Finite Set Likelihood
JoTT Markov Motion Model
Xk-1
Xk
⋅⋅⋅
⋅⋅⋅
(i)
(i)
(i)
J
(i)
k-1
{wk-1, mk-1, Pk-1}i=1
(i)
J
(i)
(i)
(i) J
k
{wk, mk,Pk } i=1
⋅⋅⋅
JoTT Forward-Backward Smoother
Bayes smoothed
Bernoulli density
Forward
updated
posterior at
time k
Bernoulli
Markov
transition
Smoothed
density at time
k+1
Predicted density
to time k+1
x
death
(i)
k|k-1
{wk|k-1, mk|k-1,Pk|k-1} i=1
fk|k-1(Xk |Xk-1 )
Joint Target-Detection
and Tracking transition
density
motion
x’
⋅⋅⋅
⋅⋅⋅
⋅⋅⋅
K-1 fk|k-1(Xk| Z1:k-1) gk(Z k| X k)
∫ fk|k-1(Xk| Xk-1) fk-1(Xk-1| Z1:k-1)δXk-1
⋅⋅⋅
∅
The smoothed probability of target existence and track density are:
x’
creation
∅
no target
∅
∅
Random Finite Set Observation Model
likelihood
x
Random set of
observations
z
misdetection
Bernoulli
object state
gk(Zk|Xk )
Random-set observation
likelihood
∅
x
JoTT Two-Filter Smoother
The backward-prediction, backward-update and smoothed estimate are
Backward
predicted density
Backward
posterior update
at time k+1
Markov
transition
Artificial
distribution
clutter
∅
state space
Clutter
intensity
observation space
Missed
detection term probability of
Single object
likelihood
object detection
Smoothed
density at time k
Forward
prediction to time k
Backward
update at time k
Distributed Data Fusion
Two multi-object posteriors can be combined to form a
fused estimate with information from both
Joint Target-Detection and Tracking Filter
Bernoulli State Bayes Prediction
Bernoulli Markov transition
For Bernoulli posteriors, this can be computed explicitly
Predicted
Bernoulli density
posterior at
time k-1
Bernoulli State Bayes Update
Bayes updated
Bernoulli density
Likelihood for
Random Set
Observations
Predicted
Bernoulli
density
Bernoulli process
integral
The optimal value of ‘w’ can be found with the Chernoff
Information