Tracking in High Target Densities Using a First-Order Multitarget Moment Density
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Transcript of Tracking in High Target Densities Using a First-Order Multitarget Moment Density
Tracking in High Target Densities Using aFirst-Order Multitarget Moment Density
Ronald Mahler, Ph.D.Lockheed Martin NE&SS Tactical Systems
Eagan, Minnesota, USA651-456-4819 / [email protected]
IMA Industrial Seminar Series, University of MinnesotaOctober 4, 2002
Problem
• It is not always feasible—or necessary—to detect, track, and identify individual targets with accuracy
– large-formation tracking: track density is so large that only knowledge of overall geometrical target distribution is feasible
– group tracking: detection and tracking of force-level objects (brigades, battalions, etc.) is of greater interest than detection and tracking of their individual targets
– cluster tracking: a few Targets of Interest (ToI's) are obscured by a “multi-target background” of low-priority targets, which is of interest only because it might contain a ToI
Approach: “Bulk Tracking”
• Conventional: detect & track all targets
– most feasible: where density is lowest (outskirts of a formation)
– least feasible: where inter-track confusion is greatest because density is highest (i.e., where ToI's are most likely to be found)
• Bulk tracking: detect & track bulk target groupings first, then sort out individual targets as data permits
– estimate what is knowable given current data quantity / quality (at
first, only bulk multitarget behavior) instead of attempting to estimate what cannot be known until sufficient, and sufficiently good, data has been collected (i.e., individual target behavior)
– resolve individual targets out of the "multitarget background" as
separate identifiable tracks only as additional information about them is accumulated over time
Approach (Ctd.): 1st-Order Multitarget Moment Filter
single-sensor, single-target Bayes filter
Bayes-optimalapproach
Bayes problemformulation
computationalstrategy
multi-sensor, multi-target Bayes filter
state of system = random
state vector
state of system = random state-set(point process)
first-moment filter(e.g. -- filter)
first-moment filter(“PHD” filter)
multi-sensor,single-target
multi-sensor,multi-target
Topics
1. First-order moment filtering2. Multitarget first-order moments: the “PHD”3. Multitarget first-order moment filtering4. Simulations
xk|k xk+1|k+1^^-- filter
fk|k(xk|Zk) fk+1|k(xk+1|Zk)optimal Bayes filter
randomobservations z
produced by target
randomstate-
vector, x
target motion
state space
observation space
1st- and 2nd-Order Moment Filters
fk+1|k+1(xk+1|Zk+1)
xk+1|k^
time-updatestep
data-updatestep
Xk+1k+1
Xk|k
zk
zk+1
how can we extend this reasoning to multitarget systems?Pk|k Pk+1|k Pk+1|k+1
^ ^ ^Kalman filter
Xk+1|k
Multi-Sensor/Target Problem: Point Process Formulation
sensors
targets
random
obser-vation-
set
diverse observations
“meta-sensor”
“meta- target”
reformulate multi-object problem as generalized single-object problem
“meta-observation”
random state-set
multitarget state
X = {x1,…,xn}
multisensor state
X*k = {x*1,…,x*s}
multisensor-multitargetobservation
Z = {z 1,…,z m}
random object-set
random density random countingmeasure
S
(x) = y(x) N(S) = | S|
Geometric Point Processes (= Random Finite Sets)
three equivalent formulations of a (multidimensional) simple point process
preferred by mathematicians
preferred byphysicists
“engineering-friendly”(multi-object systems are modeled
as visualizable random images)
y
sum the Dirac deltas concentrated at the
elements of
Integral and Derivative for Simple Point Processes
][][lim][
0
hFghFh
g
F
nn
n ddffXXf xxxx 11
1 }),...,({)()(
Set integral:
Functional (Gateaux) derivative:
][][lim][
0
hFhFh
F
x
x
Dirac delta function
physics:“functionalderivative”
Probability Law of a Geometric Point Process
][E][ |1
|1kkhhG kk
)Pr(][)( |1|1|1 SGS kkSkkkk 1
)Pr()(1)( |1|1|1 SSS kkc
kkkk
probability generatingfunctional (p.g.fl.)
beliefmeasure
plausibilitymeasure
(= Choquet functional)
X
X hhx
x)(
)"Pr(" ]0[)( |1|1
|1
1
XG
Xf kkkk
n
kk
n
xx multitarget
posteriordistribution
(= Janossy densities)
discrete-space notation used only for claritication
X = {x1,…,xn}
probability that all targets are in S
probability that some target is in S
Frechét functionalderivative
h = bounded real-valued test function
xx
gg
GG
fk|k(X|Z(k))multitarget posterior
multisensor-multitarget measurements: Zk = { zzm(k)}
individual measurementscollected at time k
multitarget state
Z(k)Z ,...,Zk
fk|k(X|Z(k))X = 1normality condition
fk|k(|Z(k)) (no targets)
fk|k(x|Z(k) (one target)
fk|k(xx2|Z(k)) ( two targets)
…fk|k(xxn|Z(k)) (n targets)
measurement-stream
Multitarget Posterior Density Functions
fk|k(X|Z(k),U(k-1)) fk+1|k+1(X|Z(k+1),U(k))
randomobservation-
sets producedby targets
multi-target motion
state space
observation space
fk+1|k(X|Z(k),U(k-1))
The Multi-Sensor/Target Recursive Bayes Filter
time-update data-update
k|k k+1|k+1k+1|kevolving random state-set
Zk+1
X*k+1
future observation-set (unknowable)
new sensor state-set (to be determined)
fk+1|k(Y|X) fk+1(Zk+1|X, X*k+1)multitarget Markov
transition density multisensor-multitarget likelihood function
(target-generated observations & clutter)
p1D(x,x*),…, ps
D(x,x*)sensor FoVs
recursive Bayes filter
What is a Multitarget First-Order Moment?
Naïve concepts of multitarget expected value fail
So: we must resort to “indirect” multitarget moments
X = {x1 ,…, xn} (X) = ({x1 ,…, xn})
vector spacesubset space well-behaved function
(X Y) = (X) + (Y) if X Y = (disjoint unions aretransformed into sums)
“Indirect” multitarget expectation:
E[()] expected value of random vector ()corresponding to random set
() = y(x) = (x) random state-set
random density
random counting measure
S
Two Possible Choices for
() = | S|y
sum the Dirac deltas concentrated at the
points of
three different notations for a simple point process
first-moment density
“probability hypothesisdensity” (PHD)
first-momentmeasure
S
Indirect Expected Values of a Random State-Set
D(x) = E[(x)] M(S) = E[| S|]
PHD for a Discrete State Space (Picture)
state space (discrete)x0
three-state instantiations
{x, x2, x3} of
two-state instantiations
{x, x2} of
one-state instantiations
{x1} of
four-state instantiations
{x, x2, x3, x4} of
D(x0) = Pr(x0 ) = p1 + p6 + p9 + p11 + p16
p1
p2
p3
p4
p5
p6
p7
p8
p9
p11
p12
p13
p14
p15
p17
p10
p16
probability of the hypothesis: “the multitarget system
contains a target with state x0“
The PHD (Ctd.)
x0state space
D(x0) = expected target
density at x0
PHD magnitude
S
D(x)dx = expected number
of targets in SS
If state space is discrete then the PHD is a fuzzy membership function (fuzzy subset of target states)
five peaks (largest target densities)correspond to locations of seven
partially resolved, closely spaced targets
X
target density,
D(x)
Example of a PHD on 2-D Euclidean Space
cluster
fk|k(X|Z(k)) fk+1|k+1(X|Z(k+1))
randomobservation-
sets Z producedby targets
randomstate-set multi-target motion
state space
observation space
fk+1|k(X|Z(k))
five targets three targets
First-Order Multarget Bayes Filtering
time-updatestep
data-updatestep
k|k k+1|k+1k+1|k
random
state-set
Dk+1|k(x|Z(k)) Dk+1|k+1(x|Z(k+1))Dk|k(x|Z(k))
multitargetBayes filter
1st-moment“PHD” filter
PHD Functional Derivative Formula
]1[)(x
x
G
D
]1[)( |1|1
x
x
kk
kk
GD
]1[)( 1|11|1
x
x
kk
kk
GD
PHD Filter Assumptions: Motion Model
time-step k time-step k+1
fk+1|k(y|x)probability that target will have statey if it had state x
Xbk+1|k(X|x)
x y
1 dk+1|k(x)
x
probability that target will vanish if it had state x
xprobability that target will spawn atarget-set X if it had state x
X
probability that a target-set X will appear in scene
bk+1|k(X)
all target motions are assumed statistically independent
death
creation
spawn
motion
PHD Filter Assumptions: Sensor Model
fk(z|x)
x z
probability that target will generateobservation z if it has state x
1. observations and clutter are statistically independent2. multitarget posteriors are approximately Poisson (need high SNR):
x
pD probability that target will not generate an observation (assumed state-independent)
ck+1|k(Z) probability that a set Z = {zzm}
of clutter observations will be generated; Poisson false alarms:
ck|k(Z) = e- k|kn ck|k(z1) ck|k(zm)
likelihood
misdetection
clutter
fk|k(X|Z(k)) e-N Nk|k sk|k(x1) sk|k(xn)
state space observation space
X = {x1,…,xn}
Dk+1|k(y|Z(k)) = bk+1|k(y) +Dk+1|k(y|x) Dk|k(x|Z(k))dx
PHD Time-Update Step
PHD from previous
time-step
term for spontaneoustarget births
= PHD of bk+1|k(X)
time-updated
PHD
Nk+1|k = dk+1|k(x) + nk+1|k (x) Dk|k(x|Z(k)) dx
expected number of targets spawned by x
Dk+1|k(y|x) = dk+1|k(x) fk+1|k(y|x) + bk+1|k(y|x)
Markovtransition
PHD
probabilityof targetsurvival
term for targets spawned
by existing targets= PHD of bk+1|k(X|x)
Markov transi-tion density
Nk+1|k = Dk+1|k(y|Z(k))dy
predicted expected number of targets
nk+1|k (x) = bk+1|k(y|x)dy
Dk+1|k+1(x|Z(k+1)) zZk+1
Given new scan of data, Zk+1 = {zzm}
k+1ck+1(z)+pDDk+1(z)pDDk+1(z)
Dk+1(x|z) + (1-pD) Dk+1|k(x|Z(k+1))
Dk+1(z) = fk+1(z|x) Dk+1|k(x|Z(k+1))dx
Dk+1(x|z) = f(z|x) Dk+1|k(x|Z(k+1))
Dk+1(z) Nk+1|k+1= Dk+1|k+1(x|Z(k+1))dx
expected number of targets after new scan
Nk+1|k+1 zZk+1
k+1ck+1(z)+pDDk+1(z)pDDk+1(z)
+ (1-pD)Nk+1|k
Bayes-updated PHD
single-observation Bayes update of PHD
predicted PHD (fromprevious time-step)
averageno. of
false alarms
distributionof false alarms
predicted expected number of targets(from previous time-step)
sensor likelihood function
PHD Filter Bayes Update Step
Proof, I: Transform PHD into p.g.fl. Form
1. Data-updated multitarget posterior:
YZYfYZf
ZXfXZfZXf
kkkkk
kkkkk
kk )|()|(
)|()|()|(
)(|111
)(|11)1(
1|1
YZYfYZf
XZXfXZfhZhG
kkkkk
kkkk
X
kkk
)|()|(
)|()|(]|[
)(|111
)(|11)1(
1|1
YZYfYZf
XZXfXZfh
ZDk
kkkk
kkkk
X
kkk
)|()|(
)|()|(
)|()(
|111
)(|11
)1(1|1
xx
2. p.g.fl. of multitarget posterior:
3. PHD of multitarget posterior:
Proof, II: Transform PHD into p.g.fl. Form
4. Define Bayes characterizing functional (B.c.fl.):
]1,0[
]1,0[
)|(
1
1)1(1|1
m
m
gg
mg
hgg
hmg
kkk F
F
ZD
zz
xzzx
5. PHD in B.c.fl. form:
XZXfXgGh
ZXZXfXZfghhgF
kkkk
X
kkkk
ZX
)|(]|[
)|()|(],[
)(|11
)(|11
p.g.fl. of multitarget measurement density
Proof, III: Choice of Likelihood Function
6. Multitarget likelihood function:
11 ][1 ))()()(1(]|[
kk g
XgDDk epppXgG
x
xxx
7. Simplified B.c.fl.:
)]1([
)|()1(
)|(]|[],[
|1][
)(|1
][
)(|11
11
11
gDDkkg
kkk
XgDD
Xg
kkkk
X
ppphGe
XZXfppphe
XZXfXgGhhgF
kk
kk
zxzzx dfgpg )|()()( zzz dcgg )()(][
each target generates at most one observation with probability pD
Poisson false alarm process w/ spatialdistribution c(z)
Proof, IV: Simplification
8. Assume predicted multitarget posterior is Poisson:
9. Final simplified B.c.fl.:
][
|1 ][ gpkk
sehG
][)]1([][exp],[ 11 gDsDskk phppphpghgF
10. Inductively determine denominator and numerator of B.c.fl. form of PHD using functional derivatives
xxx dshhps )()(][
Dk+1|k(y|Z(k)) = fk+1|k(y|x) Dk|k(x|Z(k))dx
Special Case
Nk+1|k = Nk|k
Dk+1|k+1(x|Z(k+1)) zZk+1
Dk+1(x|z)
Nk+1|k+1 = |Zk+1|
time-prediction
data update
assuming:no target deaths or births
assuming:no misdetections or false alarms
Following example is based on above assumptions:Example: bulk tracking of two clusters of three more separated targets
Example: Six Targets, Two Clusters
= actual target locations = noisy observations = estimated target locations
x
y
Targets are further apart. Increased data quality (relativeto filter resolution) allows filter to resolve targets as wellas track the two formations.
Two Clusters: PHD at 3rd Observation
direction ofmotion
PHDvalue
x y
Since the PHD filter estimatesthe number N of targets, a peakextraction algorithm is used tolook for the N tallest peaks
Two Clusters : PHD at 9th Observation
Two Clusters: PHD at 17th Observation
Two Clusters: PHD at 27th Observation
Two Clusters: PHD at 31st Observation
Implementation Based on Particle-System Filters
- Non-restrictive with respect to measurement models
- Very general continuous-state Markov models– e.g. heavy-tail models, non-smooth models
- Very strong, general guaranteed-convergence properties
for every observation sequence, particle distribution converges a.s. to posterior
- Computational order: O(pd) (low-SNR detection), O(p) (low-SNR tracking)p = no. particles, d = dimensionality, N = pd = no. of unknowns
- LMTS is co-developing these filters with U. Alberta (Prof. M. Kouritzin)
posterior, time k posterior, time k+1
“particles”= samples
Deltafunctions
Branching Particle-System Filtersfast and rapidly convergent
least-probable particles“die” in next time-step
most-probable particles “spawn” new particles
in next time-step
Simple Simulation: Multitarget Tracking in Clutter
May Jun Jul Aug Sept Oct Nov Dec
2002Jan Feb Mar Apr May
2003
200 scans collected
average of 120 clutter observationsper scan
targets cross at 40th scan
26% increase in RMS localizationaccuracy overtarget-generatedobservations
velocity alsotracked successfully
first target second target
Summary / Conclusions
• First-order moment filtering provides a potential means of tracking clusters of emitters until enough information has been accumulated to begin extracting RVs
• Computational power can be shifted from low-interest regions of the PHD to regions that may contain targets of interest, to allow “peaking up” of those targets
• Efficient computational implementation requires particle-systems methods