Random finite set Markov Chain Monte Carlo predetection fusion

Abstract

Predetection fusion is an efficient (and, depending on what underlies it, indispensable) way to process high volume data from large networks of low quality sensors and thus, an aid to multisensor multitarget tracking. In previous work we derived both the GLRT (presumably “optimal”) technique and a more practicable contact-sifting variant. Unfortunately, the gaps between the two in terms of computation time and performance are not inconsiderable. Hence in this paper we propose a new approach based on random finite sets (RFS) and implemented by Monte Carlo (MCMC) simulation. We trust that it is found interesting; but even if not, we show that it offers improved results, in the sense of RMSE and number of declared targets.

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Random Finite Set
Markov Chain Monte Carlo
Predetection Fusion
Ramona Georgescu
Electrical and Computer Engineering Department
University of Connecticut
Storrs, CT
Email: ramona@engr.uconn.edu
Peter Willett
Electrical and Computer Engineering Department
University of Connecticut
Storrs, CT
Email: willett@engr.uconn.edu
Abstract—Predetection fusion is an efficient (and, depending
on what underlies it, indispensable) way to process high volume
data from large networks of low quality sensors and thus, an
aid to multisensor multitarget tracking. In previous work we
derived both the GLRT (presumably “optimal”) technique and a
more practicable contact-sifting variant. Unfortunately, the gaps
between the two in terms of computation time and performance
are not inconsiderable. Hence in this paper we propose a new
approach based on random finite sets (RFS) and implemented
by Monte Carlo (MCMC) simulation. We trust that it is found
interesting; but even if not, we show that it offers improved
results, in the sense of RMSE and number of declared targets.
Keywords: Predetection Fusion, Markov Chain Monte
Carlo, Random Finite Sets, Sensor Networks, Tracking.
I. INTRODUCTION
The optimal sensor decision rule in the case of multiple sensor
systems and known target location is known to be a likelihood
ratio test [1]. This approach, however, is not applicable to
many practical scenarios, such as sonar, in which the location
of the target is not known and hence the alternative hypothesis
becomes composite. We propose a practical implementation,
Random Finite Set Markov Chain Monte Carlo (RFS MCMC)
predetection fusion.
Additional motivation comes from the fact that in recent
years, interest has shifted towards deploying a large sensor
network that consists of many but cheap, low quality sensors.
Data fusion in large sensor networks is expected to provide
better target tracking capability in terms of increased area
coverage, expanded geometric diversity, increased target hold,
robustness to sensor loss and jamming, improved localization
and gains in probability of detection [2]. A possible drawback
is an increased false alarm rate after the fusion step.
A multistatic sonar system, which consists of multiple sonar
sources and receivers distributed over the surveillance area,
is one such sensor network. Transmissions from one or more
sources may be processed by one or more receivers to produce
a large number of sonar echo contacts [2]. The realistic
multistatic sonar Metron [3] dataset provides a motivating
example.
Figure 1 shows the setup of this Metron dataset. There are
25 stationary sensors, all of them receivers with the exception
of four sensors which are colocated source/receiver units;
four targets display rectangular trajectories. The probability
of detection is poor, on average PD = 0.12 per sensor per
scan. The high difficulty of this dataset is due to the extremely
large number of contacts per scan and the low quality of the
measurements. Figure 2 shows the first scan of data plotted in
Cartesian coordinates: out of 890 contacts only 15 originate
from a target, a major challenge to any tracking paradigm.
Predetection fusion responds to this demanding data fusion
problem by taking advantage of the benefits in both batch- and
scan-based processing. RFS MCMC predetection fusion does
so by blending multisensor measurements into a considerably
smaller set of measurements to serve as input to a tracker,
considerably reducing the number of false alarms while pre-
serving most valid target detections. The technique is therefore
beneficial to algorithms such as the Cardinalized Probability
Hypothesis Density (CPHD) tracker [4] which is of O(nm3)
complexity, where n is the number of targets and m is the
number of measurements.
II. PREVIOUS WORK
The optimal technique to solve the data fusion problem, the
multihypothesis Generalized Likelihood Ratio Test (GLRT),
has been proposed by Guerriero et al [5]. In previous work, we
have introduced 2D predetection fusion, a practical approach
that is many orders of magnitude faster than the GLRT.
Here we present a short description of each method, and
thereafter compare their performance against that of RFS
MCMC predetection fusion.
A. GLRT
The natural way to tackle the problem of data fusion in large
sensor networks is the GLRT approach, in which, for each
hypothesized target, the location estimate which maximizes
the likelihood function is found and the hypothesis with the
largest likelihood is selected. Thus, the likelihood function is

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01234567
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meters
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T2
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Figure 1.
RX26, 4 targets with square trajectories T1-T4).
Setup in the Metron dataset (4 sources S1-S4, 25 receivers RX2-
maximized with respect to both the number of targets and their
locations in Cartesian coordinates.
Let Λ0(Z) is the likelihood function given that all measure-
ments are false alarms:
Λ0(Z) =
N
∏
i=1
1
umiµF(mi)
(1)
where Z = {Zij} is the measurement set, i = 1,2,···,N
is the sensor number, j = 1,2,···,mi is the measurement
number from sensor i, u is the search volume and µF(.) is
the probability mass function of the number of false alarms
(usually Poisson).
Λ1(Z|θ1) is the likelihood function given that there is one
target:
[1−PD
Λ1(Z|θ1)=
N
∏
i=1
umiµF(mi)+PDµF(mi−1)
umi−1mi
mi
∑
j=1
p(Zij|θ1)
]
(2)
where p(Zij|θ1) = N(Zij;θ1,Σ) is the likelihood that mea-
surement j from sensor i originated from a target located at
θ1with Σ as the sensor covariance matrix. Generalization to
an arbitrary number of targets is straightforward but tedious
to repeat. A constraint is imposed on the maximum number
of targets, Tmax.
With a generalized likelihood ratio test (GLRT) approach,
one can find the target location estimates which maximize the
likelihood function for each hypothesized number of targets,
and choose the largest. The algorithm starts at:
?Λ1(Z) = max
?θ1= argmax
θ1
Λ1(Z|θ1)
(3)
The optimal target location estimate is:
θ1
Λ1(Z|θ1)
(4)
01234567
4
x 10
0
1
2
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7x 10
4
Time = 0 |Z|=890
Figure 2.
(target originated measurements are emphasized).
Raw measurements collected from all sensors in a single scan
The algorithm proceeds with joint estimation of greater num-
bers of targets. More details on our implementation of the
GLRT can be found in [6].
B. 2D Predetection Fusion
In sonar surveillance systems, measurements consist of range,
bearing, possibly Doppler and maybe feature data such as
amplitude. Range and bearing can be converted into Cartesian
measurements. In this version of the algorithm, we consider
networks in which Doppler and amplitude information are not
available. The final fused measurements are two dimensional,
in the xy-plane.
1) Collection: All measurements (from all receivers) that
arrived at the same time scan are gathered together in one
measurement set, on which the following algorithm is run.
2) Sampling: The purpose of this step is to recreate the
possible locus of a target, based on the detections hypothesized
to have arisen from that target, and use it as motivation
for the quantization decisions to be made in the next step
of the algorithm. In large networks of low quality sensors,
one expects to encounter considerable measurement errors,
as a large bearing error translates into a large and elongated
resolution cell at long ranges. In order to overcome such large
measurement errors, we generate Nmc = 100 samples via
Monte Carlo for each contact1, according to the contact’s
measurement error covariance matrix. Without this step, a
large covariance measurement would still only be “seen” in
the grid cell containing the measurement’s nominal value.
A similar implementation of this step would be to calculate
which cells have edges that intersect the error ellipse and count
the corresponding contact in those cells. However, an analytic
and efficient way to find all rectanguloid cells that intersect
1The terms measurement and contact are used interchangeably.

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with a given covariance ellipsoid does not seem available; and
Monte Carlo sampling is easy.
3) Sifting: We then “sift” these measurement samples ac-
cording to a grid in the xy-plane. When a contact yields at least
one sample that is quantized to a grid cell, then that contact is
added to the cell’s list. Additional MC samples from the same
measurement in a given grid cell have no effect.
4) Thresholding: A detection is declared in a cell if and
only if there are more than τ contacts added to that cell’s list.
The threshold τ is a tunable parameter and can be computed
according to a binomial law. We test each grid cell’s number
of hits against the calculated threshold.
5) Fusion: For each cell that passes the test, a detection is
declared. The cell’s listed contacts are then used to refine the
estimated measurement location ˆ x and to estimate the posterior
covarianceˆR.
6) Merging: We merge detections that gate with each other,
since often neighboring cells have used the same detections
from the initial Monte Carlo step. Merging also helps reduce
the number of fused measurements that predetection fusion
would feed to a tracker while preserving a good target prob-
ability of detection.
A comprehensive description of 2D predetection fusion can
be found in [6] and [7].
III. RFS MCMC PREDETECTION FUSION
RFS MCMC predetection fusion treats potential targets and
their measurements as random finite sets. A RFS is a finite-
set valued random variable which can be fully characterized
by a probability mass function (pmf) and a family of joint
probability densities. The pmf describes the cardinality of the
set; for a given cardinality, an appropriate density characterizes
the joint distribution of the elements in the set [8].
Figure 3 shows an influence diagram with two (position
only) targets xi, i = 1,2 and three measurements zj, j =
1,2,3. The nuisance random quantities (k’s) model the data
association, i.e. they reflect which measurement arose from
which target. We employ the Probabilistic Multihypothesis
Tracker (PMHT) measurement model [9] in which the mea-
surement to target association process is assumed indepen-
dent across measurements and hence, the event that a target
generated more than one measurement per scan is considered
feasible.
Our technique updates, in turn, the estimated target cardi-
nality and the estimated location of each target (using only
the measurements determined to have been generated by that
particular target) through MCMC-type sampling2. Markov
Chain Monte Carlo (MCMC) is a strategy for generating
samples while exploring the state space using a Markov chain
mechanism. The mechanism is constructed so that the samples
mimic samples drawn from the target distribution [10].
The use of a PMHT-style measurement model, as opposed
2This is essentially a Gibbs sampling operation. We are hesitant to call it
that due to what will be seen: the varying cardinality of the number of targets.
z1
x2
z3
k2
k3
z2
k1
x1
Figure 3.Influence diagram with 2 targets and 3 measurements.
to a traditional one3[11], allows for the consideration of
all possible probabilistic associations, i.e. any measurement
can come from any target (Step 1). The backbone of RFS
MCMC predetection fusion is the weighted sampling of these
soft associations with the goal of declaring hard assignments
between targets and measurements (Step 2). The algorithm
continues with what is essentially a likelihood ratio test that
decides if each considered target is an actual target or a
false alarm (Step 3) and proceeds to update target location
accordingly (Step 4). The output consists of the estimated
mean and covariance of the actual targets, calculated after
the MCMC-style sampling has reached convergence (Step 5).
Merging close estimates helps reduce the number of outputs
(Step 6). A more detailed description of our technique follows.
At initialization, a maximum number of targets Tmax is
assumed and their locations are drawn uniformly over the
surveillance area. However, we would like our algorithm to
be able to estimate target cardinality on its own. Introducing
potential dummy targets when forming all possible measure-
ment to target associations in Step 1 is necessary in order to
allow RFS MCMC predetection fusion to estimate the number
of targets, as dummy targets are not counted towards the output
number of targets.
Additional motivation for the use of dummy targets comes
from the fact that it facilitates the treatment of the set of targets
as a RFS, for which the values of the individual elements in
the set (here, target locations) and the cardinality of the set
(of targets) are random variables that are free to take different
values at each time scan and moreover at each RFS MCMC
predetection fusion iteration on a scan of data [12].
After initialization, the algorithm iterates through steps 1-4
below until convergence.
Step 1The soft associations covering every measurement
3The usual model is that each target can give rise to at most one detection
per scan, so “events” that in which more than one measurement arises from
a given target are afforded a zero prior probability. Under the “PMHT”
measurement model, however, the association process is independent across
measurements, and hence there is no such constraint.

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being generated by every target are formed. These
weights are normalized so that for each measure-
ment, the weights sum to 1. A dummy (or null) target
state is included. The probability that measurement
zj with covariance Rj comes from target xi is
calculated as:
wij= πi,j
1
√|2πRj|e−1
2(zj−xi)TR−1
j
(zj−xi)
(5)
The probability that measurement zjis a false alarm
is calculated as:
wcj= πc,j
1
V
(6)
πi,j is the prior probability that a measurement is
target originated, πc,j is the prior probability that
a measurement is clutter originated and V is the
volume under surveillance.
Step 2For each measurement zj, j = 1,···N, the hard
association variable kj that assigns measurement zj
to a target (possibly the dummy target) is sampled
according to the weights computed in Eq. 5 and
Eq. 6. At the end of step 2, a decision has been
made as to which measurement comes from which
target and respectively, which target generated which
measurements.
Step 3Each target is tested to check if it is one of the actual
M targets or a dummy Φ, with prior probabilities q
and respectively, 1−q. By Bayes’ rule, we can write:
p(xi|K,Z,X¯i) ∝ p(Z|K,xi,X¯i)p(K,xi,X¯i) (7)
where X¯istands for all the targets besides the ith
target and thus p(xi,X¯i) = p(X). The present “as-
sociations” are K = {kj}N
in the previous step) kj= i means that measurement
j arises from target i. We have ignored the p(Z) term
as it is a constant. Also,
j=1, in which (as sampled
p(K,xi,X¯i)=
=
∝
p(K|xi,X¯i)p(xi,X¯i)
p(K)p(xi,X¯i)
p(K)p(xi|X¯i)
(8)
(9)
(10)
where we have ignored the constant term p(X¯i).
Substituting Eq. 10 back into Eq. 7, we get:
p(xi|K,Z,X¯i) ∝ p(Z|K,X)p(K)p(xi|X¯i)4(11)
Using the PMHT measurement model and Eq. 11, we
calculate the probability that xi is a dummy target
p(xi = Φ) and the probability that xi is an actual
4The prior probability of a target being a dummy or an actual target does
not depend on other targets, therefore the conditioning in the third term in
Eq. 11 can be dropped.
target p(xi̸= Φ):
p(xi= Φ)
∝
[∏
×
[∏
×
kj̸=i
[∏
p(zj|xkj)
∏
kj=i
(1 − q)1
∏
q1
V
1
V
]
j
πkj,j
][
V
]
(12)
p(xi̸= Φ)
∝
kj̸=i
[∏
p(zj|xkj)
kj=i
p(zj|xkj)
]
j
πkj,j
][]
(13)
Note the common terms. We can therefore further
simplify the above equations to:
p(xi= Φ) ∝ (1 − q)
∏
∏
kj=i
1
V
(14)
p(xi̸= Φ) ∝
q
kj=i
p(zj|xi)
(15)
For each of the Tmaxtargets, we sample according to
the normalized weights given in Eq. 14 and Eq. 15.
This allows us to decide if a target should be counted
as an actual target or a false alarm. The estimated
target cardinality is a by-product of this step.
Step 4In the following, the target locations are updated. If
a target has been declared in the previous step to be
a dummy, its location is uniformly sampled from the
surveillance area. If a target has been declared in the
previous step to be an actual target, its location is
updated only with the measurements assigned to it
in step 2. Its location is sampled from5:
xi∼ N(x;µ,P)
(16)
where the mean µ and covariance P are computed
according to Eq. 15:
P =(∑
µ =(∑
Step 5After convergence, the mean and covariance of each
fused measurement (representing a declared target)
are calculated based on the last (usually 5) iterations
of this MCMC-type procedure.
Step 6The last step is to check for fused measurements that
gate with each other according to the Mahalanobis
distance. The fused measurement with the smallest
covariance (by trace) is kept and the others are
kj=i
)−1(∑
R−1
j
)−1
(17)
kj=i
R−1
j
kj=i
R−1
jzj
)
(18)
5In the case of two variables, a Gibbs sampler generates a sample from
the distribution f(x) by drawing instead from the conditional distribution
f(x|y). In this step, we generate a sample for the location of the ithtarget
by drawing from the conditional distribution of the location of the ithtarget
given the locations of all other targets (knowing the locations of the other
targets permits the soft and then hard measurement to target associations to
be formed).

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01000 20003000
x (meters)
4000500060007000
0
1000
2000
3000
4000
5000
6000
7000
RFS MCMC Results (x−y): NPMHT=1
y (meters)
Figure 4.Typical run contacts.
discarded. Motivation for the merging step is given
in [7].
IV. RESULTS
We compared the performance of the newly derived RFS
MCMC predetection fusion against the 2D predetection fu-
sion and the optimal multihypothesis GLRT approach. The
sensor grid visible in Figure 4 was used, with sensors of
initial accuracy as in Table I. Scenario difficulty was then
progressively increased. Clutter was Poisson distributed and
spatially uniform.
Figure 4 shows a typical snapshot in which the target origi-
nated contacts are displayed in magenta, clutter measurements
in blue and the sensors as black diamonds. Figure 5 zooms
in and displays the true target location as a black star and the
target estimate and its covariance found through RFS MCMC
predetection fusion also in black.
We were interested in observing how good the localization
accuracy of our method is, obtaining a close match between the
true target cardinality and the cardinality estimate produced by
RFS MCMC predetection fusion and achieving a reasonable
run time. 100 Monte Carlo runs were performed.
Number of Sensors
Sensor Probability of Detection
Average Number of False Alarms per Sensor
Delay Error (sec)
Bearing Error (◦)
Number of Iterations6
Tmax
q
25
0.99
2
0.01
1
100
4
0.5
Table I
INITIAL PARAMETERS.
6100 iterations were sufficient for convergence on this simulated dataset. An
alternate stopping criterion would be testing the difference between a target’s
locations in successive iterations against a threshold.
2480 25002520
x (meters)
254025602580
2460
2480
2500
2520
2540
2560
RFS MCMC Results (x−y): NPMHT=1
y (meters)
Figure 5.Typical run results (detail).
A. RMSE Performance Study
In our RMSE performance study, we looked at how the
algorithm reacts to a progressive increase in the difficulty
of the scenario. We started with the parameters listed in
Table I and then varied in turn the number of available
sensors, the sensor probability of detection, the quality of the
bearing measurements and the average number of false alarms
generated per sensor.
1) Number of Sensors: In each run of the 100 Monte Carlo
simulations, the sensors that would generate measurements
were selected randomly from the total of 25 sensors. A large
number of sensors is expected to provide better area coverage
in terms of probability of detection and better localization, due
in part to geometric diversity. Figure 6 confirms that increasing
the number of sensors has a noticeable beneficial effect on the
RMSE obtained with the RFS MCMC predetection fusion,
and the same can be said for the other two techniques. But
regardless of the number of available sensors, RFS MCMC
predetection fusion is able to provide a smaller error than 2D
predetection fusion and approach the optimal method (GLRT)
in performance.
2) Sensor Probability of Detection: Figure 7 looks at how
the three compared methods behave with decreasing sensor
probability of detection. All techniques appear robust, being
able to estimate the target location with acceptable RMSE for
a wide range of sensor PD. Once again, the RMSE obtained
with RFS MCMC predetection fusion further improves on the
error obtained with 2D predetection fusion and is closer to the
performance obtained with the optimal method, GLRT.
3) Bearing Measurement Quality: The effect of the wors-
ening quality of the bearing measurements, simulated as an
increasing standard deviation of the zero mean Gaussian noise
added to the true measurements, on performance can be seen
in Figure 8. All methods proved sensitive to increasing error
in the angular measurements, with RFS MCMC predetection
fusion being no exception. Because the three techniques in-