self-tuning algorithms for multisensor-multitarget
TRANSCRIPT
1
Self-Tuning Algorithms for Multisensor-Multitarget
Tracking Using Belief Propagation
Giovanni Soldi, Florian Meyer, Member, IEEE, Paolo Braca, Senior Member, IEEE,
and Franz Hlawatsch, Fellow, IEEE.
Abstract—Situation-aware technologies enabled by multitargettracking algorithms will create new services and applications inemerging fields such as autonomous navigation and maritimesurveillance. The system models underlying multitarget trackingalgorithms often involve unknown parameters that are potentiallytime-varying. A manual tuning of unknown model parametersby the user is prone to errors and can thus dramatically reducetarget detection and tracking performance. We address thischallenge by proposing a framework of “self-tuning” multisensor-multitarget tracking algorithms. These algorithms adapt in anonline manner to time-varying system models by continuouslyinferring unknown model parameters along with the target states.
We describe the evolution of the parameters by a Markov chainand incorporate them in a factor graph that represents thestatistical structure of the tracking problem. We then use abelief propagation scheme to efficiently calculate the marginalposterior distributions of the targets and model parameters. As aconcrete example, we develop a self-tuning tracking algorithm formaneuvering targets with multiple dynamic models and sensorswith time-varying detection probabilities. The performance ofthe algorithm is validated for simulated scenarios and for a realscenario using measurements from two high-frequency surfacewave radars.
Index Terms—Multitarget tracking, adaptive processing, prob-abilistic data association, belief propagation, message passing,factor graph, high-frequency surface wave radar.
I. INTRODUCTION
A. Background, Motivation, State of the Art
Multitarget tracking aims at estimating the time-varying
states of an unknown number of moving objects (targets) from
measurements provided by remote sensing devices, such as
radar, sonar, and cameras [1]. This challenging inference task
is a fundamental element in a variety of applications, including
air traffic control, maritime situational awareness, autonomous
driving, biomedical analytics, remote sensing, and robotics. To
obtain satisfactory performance in conditions of low signal-to-
This work was supported in part by the NATO Allied Command Transfor-mation under the DKOE project, by the European Research Council (ERC)under grant 700478 (project RANGER) within the Horizon 2020 program, bythe by the Austrian Science Fund (FWF) under grants J3886-N31, P27370-N30, and P32055-N31, and by the Czech Science Foundation (GACR) undergrant 17-19638S. Parts of this paper were previously presented at IEEEGLOBECOM 2016, Washington D.C., USA, December 2016 and at FUSION2018, Cambridge, UK, July 2018. G. Soldi and P. Braca are with the NATOCentre for Maritime Research and Experimentation (CMRE), La Spezia,Italy (e-mail: [email protected], [email protected]). F.Meyer and Moe Z. Win are with the Laboratory for Information andDecision Systems, Massachusetts Institute of Technology, Cambridge, MA,USA (e-mail: [email protected], [email protected]). F. Hlawatsch is withthe Institute of Telecommunications, TU Wien, Vienna, Austria (e-mail:[email protected]).
noise ratio, it is often necessary to use measurements provided
by multiple sensing devices.
Multitarget tracking is complicated by the fact that the
association between measurements and targets is unknown and
that measurements are affected by noise, missed detections,
and false alarms. Furthermore, certain model parameters are
often unknown and time-varying. For example, the probability
that a sensor detects a target may change over time due to,
e.g., an evolving target-sensor geometry or Bragg scattering in
over-the-horizon radars [2]. A similar discussion applies to the
parameters of the dynamic model describing the evolution of
target states. A typical scenario in this context is that of ma-
neuvering targets, whose time-evolution cannot be described
by a single dynamic model [3], [4].
Most tracking algorithms assume that relevant model pa-
rameters are fixed and known. In practice, this means that the
user has to guess or manually tune parameters for a specific
dataset or mission. The difficulty of manual parameter tuning
can be avoided by “self-tuning” algorithms that automatically
adapt unknown and potentially time-varying model parameters
online. A Bayesian algorithm that sequentially estimates the
unknown detection probability of a single sensor is proposed
in [5]. The estimated detection probability is then used in a
probabilistic data association filter for single-target tracking. In
[6], track management routines are developed by modeling the
target detection probabilities in a multisensor sonar network
using a hidden Markov model with high and low detection
probability states. In [7] and [8], the probability hypothesis
density (PHD) filter, the cardinalized PHD (CPHD) filter, and
the multi-Bernoulli filter are extended to include estimation of
clutter intensity profile parameters and the time-varying detec-
tion probability. An extension of PHD and CPHD filters that
includes estimation of the time-varying target birth intensity
is presented in [9]. In [10], an adaptive method for single-
target tracking in a network of multiple sensors with unknown
time-varying detection probabilities is proposed. The detection
probabilities are modeled by a Markov chain and estimated
sequentially. An algorithm for simultaneous localization and
mapping (SLAM) based on a belief propagation (BP) scheme
is proposed in [11]. In this algorithm, estimates of the ampli-
tudes of multipath components are used to adapt the detection
probabilities of map features.
B. Contributions and Paper Organization
In our previous work [12], we presented a multisensor-
multitarget tracking algorithm for an unknown and time-
varying number of targets and an unknown and time-varying
association between measurements and targets. This algorithm
2
is nonadaptive in that parameters such as the detection prob-
abilities are assumed known. It was derived by representing
the statistical structure of the problem by a factor graph and
using a BP algorithm [13], [14]. The BP approach exploits
conditional statistical independencies for a drastic reduction
of complexity. This resulted in excellent scalability of our
algorithm while outperforming previously proposed methods
in terms of accuracy. More specifically, the complexity of our
algorithm scales only quadratically in the number of targets,
linearly in the number of sensors, and linearly in the number
of measurements per sensor.
The main contributions of this paper can be summarized as
follows:
• We extend [12] by proposing a BP-based framework of
self-tuning algorithms for multisensor-multitarget track-
ing. These algorithms continually infer time-varying
model parameters along with the target states. The model
parameters are assumed to take on values from parameter-
specific finite sets, and their evolution is modeled by
Markov chains. Based on this statistical model, the pa-
rameters are incorporated in the factor graph represent-
ing the statistical structure of the multisensor-multitarget
tracking problem. Then, a BP algorithm is performed
on the factor graph to calculate at each time step the
marginal posterior distributions of both the target states
and the model parameters. These distributions are finally
used to detect targets and to estimate target states and
model parameters (if required).
• We use the proposed BP framework to develop a concrete
self-tuning multisensor-multitarget tracking algorithm for
scenarios with maneuvering targets and time-varying
sensor characteristics. In the developed algorithm, the
unknown model parameters are the detection probabilities
of the sensors and the dynamic model indices of the
targets. The posterior distributions of these parameters
are recursively calculated in addition to those of the target
states.
• We formulate general rules for incorporating unknown,
time-varying model parameters in a factor graph for
self-tuning multitarget tracking problems. These rules
are based on a distinction between model parameters
related to the target dynamics and those related to the
measurement model. Examples of the first class include
driving process variance, birth intensity rate, and dynamic
model indices, whereas examples of the second class
include detection probabilities and parameters of clutter
intensity profiles.
Contrary to [5]–[10], which consider specific scenarios, we
develop a general framework for tracking multiple targets
from measurements provided by one or multiple sensors. This
framework enables a systematic incorporation of unknown
time-varying model parameters within multitarget tracking
problems. Our framework is also able to accommodate physi-
cal dependencies of these model parameters; e.g., the unknown
detection probabilities can be modeled as a function of the
amplitudes of multipath components [11].
This paper advances beyond our conference publications
[15], [16] in that it considers arbitrary model parameters,
provides a general framework for developing self-tuning algo-
rithms for multisensor-multitarget tracking, formulates general
rules for incorporating model parameters in the factor graph,
and validates the performance of the proposed algorithm in
additional simulated scenarios and in a real scenario using
measurements from two high-frequency surface wave radars.
The remainder of this paper is organized as follows. In
Section II, we describe the multisensor-multitarget tracking
problem and our stochastic model. In Section III, we establish
a factor graph and develop a self-tuning BP-based tracking
algorithm. General rules for incorporating model parameters
in a factor graph for multitarget tracking are formulated in
Section IV. Sections V and VI assess the performance of
the proposed algorithm in simulated scenarios and in a real
scenario, respectively. Section VII concludes the paper.
II. SYSTEM MODEL AND STATISTICAL FORMULATION
A. Target States and Measurements
Following [12], we account for the unknown number of
targets by considering K potential targets (PTs) indexed by
k ∈ K , {1, . . . ,K}. Thus, K is the maximum possible
number of actual targets;1 that is, the number of actual targets
may be smaller than K or equal to K but not larger than
K . Each PT may exist or not; the existence of PT k at time
n ∈ {0, 1, . . .} is indicated by the binary indicator rn,k ∈{0, 1}, i.e., PT k exists at time n if rn,k = 1. The state xn,kof PT k consists of the PT’s position and possibly further
parameters; it is formally considered also if rn,k = 0. We
define xn , [xTn,1 · · · x
Tn,K ]T and x , [xT
0 · · · xTn]
T as well
as rn , [rn,1 · · · rn,K ]T and r , [rT0 · · · r
Tn]
T. The temporal
evolution of the PTs will be discussed in Section II-C.
There are S sensors s ∈ S , {1, . . . , S}. At time n, sensor
s produces M(s)n measurements z
(s)n,m , m ∈ M
(s)n ,
{
1, . . . ,
M(s)n
}
. These measurements are the output of a detector
performing a thresholding and, possibly, some further prepro-
cessing of the raw sensor data [1]. We define z(s)n ,
[
z(s)Tn,1 · · ·
z(s)T
n,M(s)n
]T, zn ,
[
z(1)Tn · · · z
(S)Tn
]T, and z , [zT
1 · · · zTn]
T as
well as mn ,[
M(1)n · · ·M
(S)n
]Tand m , [mT
1 · · · mTn]
T. The
measurement model will be discussed in Section II-D.
There is a data association (measurement origin) uncer-
tainty: it is not known which measurement z(s)n,m originated
from which PT k, and it is possible that z(s)n,m did not originate
from any PT (false alarm, clutter) or that a PT did not lead to
any measurement of sensor s (missed detection) [1], [18]. An
existing PT can generate at most one measurement at sensor
s, and a measurement at sensor s can be generated by at most
one existing PT [1], [18]. The measurement-PT associations
at sensor s and time n can be described by the “PT-oriented”
1Scalable BP-based multisensor-multitarget tracking algorithms where thenumber of PTs (i.e., the maximum possible number of actual targets) is time-varying are presented in [17].
3
association vector a(s)n =
[
a(s)n,1 · · · a
(s)n,K
]Twith entries
a(s)n,k ,
m∈M(s)n , if at time n, PT k generates
measurement m at sensor s
0 , if at time n, PT k does not generatea measurement at sensor s.
(1)
Following [19] and [12], we also use the “measurement-orient-
ed” association vector b(s)n =
[
b(s)n,1 · · · b
(s)
n,M(s)n
]Twith entries
b(s)n,m ,
k ∈K , if at time n, measurement m at sensor sis generated by PT k
0 , if at time n, measurement m at sensor sis not generated by a PT.
We also define an ,[
a(1)Tn · · · a
(S)Tn
]T, a , [aT
1 · · · aTn]
T,
bn ,[
b(1)Tn · · · b
(S)Tn
]T, and b ,
[
bT1 · · · b
Tn
]T. Note that b
is redundant since it can be derived from a and vice versa.
B. Markov Chain Modeling of Unknown Parameters
In addition to the “primary” quantities to be tracked—
i.e., the PT states xn,k , PT existence indicators rn,k , and
association variables a(s)n,k—there are other parameters θ
(d)n ,
d = 1, . . . , D that are also unknown and time-varying. Ex-
amples will be considered in Sections II-C and II-D. To
obtain a self-tuning tracking algorithm, we propose to track
the parameters θ(d)n along with the primary quantities within
a BP-based sequential inference framework.
For computational efficiency, we discretize each parameter
unless it is already discrete-valued, i.e., we model θ(d)n as a
time-varying discrete random variable taking values from a
finite set Hd ,{
ω(d)1 , . . . , ω
(d)Nd
}
. We assume that the initial
parameters θ(d)0 are independent and distributed according to
some probability mass function (pmf) p(
θ(d)0
)
, θ(d)0 ∈ Hd,
and that the θ(d)n evolve independently according to first-order
Markov chains with transition matrices Pd ∈ [0, 1]Nd×Nd.
(Here, [0, 1]Nd×Nd denotes the set of all Nd × Nd matrices
with elements in [0, 1].) The transition pmf of θ(d)n follows as
p(
θ(d)n = ω(d)j
∣
∣θ(d)n−1 = ω
(d)i
)
= [Pd]i,j .
Note that∑Nd
j=1 [Pd]i,j =1 for all i = 1, . . . , Nd. Due to the
above assumptions, the prior pmf of the vector of all parame-
ters up to time n, θ , [θT0 · · · θ
Tn]
T with θn ,[
θ(1)n · · · θ
(D)n
]T,
factorizes as
p(θ) =D∏
d=1
p(
θ(d)0
)
n∏
n′=1
p(
θ(d)n′
∣
∣θ(d)n′−1
)
. (2)
If necessary, the unknown parameters can be estimated by
means of the minimum mean-square errror (MMSE) estimator
[20, Ch. 4]
θ(d)MMSEn ,
Nd∑
i=1
ω(d)i p
(
θ(d)n = ω(d)i
∣
∣z)
. (3)
Here, an approximation of p(
θ(d)n
∣
∣z)
is calculated by the
proposed BP-based algorithm.
We will distinguish two classes of unknown parameters θ(d)n ,
referred to as state parameters and measurement parameters.
These two classes cover most parameters that are relevant in
practice. State parameters are related to the temporal evolution
of the PTs; examples include the driving process variance in a
nearly-constant velocity model, the turn rate in a coordinated
turn model, the birth intensity rate, and dynamic model in-
dices. Measurement parameters are related to the measurement
model, and include the detection probabilities, parameters of
the clutter intensity profile, and the mean number of false
alarms. In our system model and tracking algorithm, for
concreteness, we will consider one specific type of parameters
θ(d)n for each of the two classes, namely, dynamic model
indices (see Section II-C) and detection probabilities (see Sec-
tion II-D). In Section IV, we will provide a general definition
of state parameters and measurement parameters, and we will
demonstrate how to incorporate parameters of either class in
the factor graph underlying the BP-based algorithm.
C. Target Dynamics
Following the interacting multiple model (IMM) approach
[3], [4], each PT can switch between different dynamic models
(“modes”) at any time n. Accordingly, the evolution of the
state of a PT k that exists at times n−1 and n (i.e., rn−1,k =rn,k = 1) is modeled as
xn,k = ξℓn,k
(
xn−1,k,u(ℓn,k)n,k
)
. (4)
Here, ξℓn,k(· , ·) is the state-transition function of PT k that
is in force at time n. This function is selected from a set{
ξj(· , ·)}J
j=1by the dynamic model index (IMM parameter)
ℓn,k ∈ J , {1, . . . , J}. Furthermore, u(ℓn,k)n,k is a driving
process that is assumed to be independent and identically dis-
tributed (iid) across n and k [1], [3]. We note that ξj(· , ·) and
the statistics of u(j)n,k determine the state-transition probability
density funcion (pdf) fj(xn,k|xn−1,k). In addition, existing
targets can disappear and newly born targets can appear, as
described presently.
The IMM parameters ℓn,k are modeled as random variables
that are independent across k and evolve according to the
Markov chain model of Section II-B, with a transition matrix
L ∈ [0, 1]J×J that is equal for all times and all PTs k ∈ K.
Thus, the transition pmf of ℓn,k is given by p(ℓn,k=j|ℓn−1,k=i) = [L]i,j for i, j ∈ J . We also define ℓn , [ℓn,1 · · · ℓn,K ]T
and ℓ, [ℓT0 · · · ℓ
Tn]
T.
The PT states xn,k, existence variables rn,k, and IMM
parameters ℓn,k are assumed to be statistically independent
across k and to jointly evolve according to a Markovian
dynamic model, with initial prior joint pdfs f(x0,k, r0,k, ℓ0,k)at time n=0. Thus, the joint pdf of x, r, and ℓ factorizes as
f(x, r, ℓ)
=
K∏
k=1
f(x0,k, r0,k, ℓ0,k)
×n∏
n′=1
f(xn′,k, rn′,k, ℓn′,k|xn′−1,k, rn′−1,k, ℓn′−1,k). (5)
The factors in the second product can be expressed as
f(xn,k, rn,k, ℓn,k|xn−1,k, rn−1,k, ℓn−1,k)
4
= f(xn,k, rn,k|ℓn,k,xn−1,k, rn−1,k, ℓn−1,k)
× p(ℓn,k|xn−1,k, rn−1,k, ℓn−1,k). (6)
Assuming that xn,k and rn,k are conditionally independent of
ℓn−1,k given ℓn,k, xn−1,k, and rn−1,k, and furthermore that
ℓn,k is conditionally independent of xn−1,k and rn−1,k given
ℓn−1,k, expression (6) simplifies to
f(xn,k, rn,k, ℓn,k|xn−1,k, rn−1,k, ℓn−1,k)
= f(xn,k, rn,k|ℓn,k,xn−1,k, rn−1,k) p(ℓn,k|ℓn−1,k). (7)
Here, f(xn,k, rn,k|ℓn,k,xn−1,k, rn−1,k) is obtained as follows.
If PT k did not exist at time n−1, i.e., rn−1,k = 0, then the
probability that it exists at time n, i.e., that rn,k =1, is given
by the birth probability p(b)n,k , and if it does exist at time n,
then its state xn,k is distributed according to the birth pdf
f(b)ℓn,k
(xn,k). Therefore, for rn−1,k =0,
f(xn,k, rn,k|ℓn,k,xn−1,k, rn−1,k=0)
=
(
1−p(b)n,k
)
fD(xn,k), rn,k= 0 ,
p(b)n,k f
(b)ℓn,k
(xn,k), rn,k= 1 ,(8)
where fD(xn,k) is an arbitrary “dummy pdf” [12]. On the
other hand, if PT k existed at time n−1, i.e., rn−1,k = 1,
then the probability that it still exists at time n is given by the
survival probability p(s)n,k , and if it does exist at time n, then
its state xn,k is distributed according to the state-transition pdf
fℓn,k(xn,k|xn−1,k). Therefore, for rn−1,k =1,
f(xn,k, rn,k|ℓn,k,xn−1,k, rn−1,k=1)
=
{(
1−p(s)n,k
)
fD(xn,k), rn,k= 0 ,
p(s)n,k fℓn,k
(xn,k|xn−1,k), rn,k= 1 .(9)
A strategy for choosing p(b)n,k , p
(s)n,k , and f
(b)j (xn,k) has been
presented in [12]. Note that the birth pdf f(b)j (xn,k) is allowed
to be mode-dependent so that, e.g., a broader birth pdf
(expressing a higher uncertainty of the state) can be chosen
for fast maneuvering targets.
D. Measurement Model and Likelihood Function
An existing PT k is detected by sensor s—i.e., it generates a
measurement z(s)n,m at sensor s—with an unknown probability
q(s)n,k . We define q
(s)n ,
[
q(s)n,1 · · · q
(s)n,K
]T, qn ,
[
q(1)Tn · · ·
q(S)Tn
]T, and q , [qT
1 · · · qTn]
T. If z(s)n,m is generated by PT k,
i.e., a(s)n,k=m ∈M
(s)n in (1), then its conditional distribution
given PT state xn,k is described by the pdf f(z(s)n,m|xn,k).
On the other hand, z(s)n,m may also be due to some interfering
source, e.g., sea clutter in a maritime radar application. Such
a measurement is referred to as a false alarm. The number of
false alarm measurements at sensor s is modeled by a Poisson
pmf with mean µ(s), and each false alarm measurement at
sensor s is distributed according to the pdf fFA
(
z(s)n,m
)
.
The detection probabilities q(s)n,k are assumed independent
across k and s and to take their values from a finite set Q={ω1, . . . , ωQ}, where ωi∈(0, 1]. They evolve according to the
Markov chain model of Section II-B, with a transition matrix
Q(s)∈ (0, 1]Q×Q that is equal for all times and all PTs k∈K
but generally sensor-dependent. The initial distribution of q(s)n,k
is given by the pmf p(
q(s)0,k
)
. In accordance with (2), the prior
pmf of q factorizes as
p(q) =
S∏
s=1
K∏
k=1
p(
q(s)0,k
)
n∏
n′=1
p(
q(s)n′,k
∣
∣q(s)n′−1,k
)
, (10)
where p(
q(s)n,k= ωj
∣
∣q(s)n−1,k= ωi
)
= [Q(s)]i,j . Furthermore, for
use in Section III-C, we make the assumption (to be referred
to as “A1”) that the detection probability vector q is a priori
statistically independent of x, r, and ℓ. This means that the
sensors’ capabilities of detecting targets are not influenced by
the states, existences, and dynamics of the PTs.
To find an expression of the “total likelihood function”
f(z|x, r, a,m), we make the following further commonly used
assumptions [1], [12], [19]: (A2) Given x, r, a, and m, the
z(s)n are conditionally independent across n and s. (A3) Given
xn, rn, a(s)n , and M
(s)n , the z
(s)n,m , m ∈ M
(s)n at sensor s
are conditionally independent. (A4) Given x, r, a, and m,
the measurement vector z is conditionally independent of q
(since the association vector a already contains the information
whether each PT k has been detected) and of ℓ (since given
the target states x, the measurements are not affected by the
target dynamics). With these assumptions, we obtain (cf. [12])
f(z|x, r, a,m)
= C(z,m)
n∏
n′=1
S∏
s=1
K∏
k=1
w(
xn′,k, rn′,k, a(s)n′,k; z
(s)n′
)
, (11)
where C(z,m) =∏n
n′=1
∏S
s=1
∏M(s)
n′
m=1 fFA
(
z(s)n′,m
)
is a nor-
malization factor that depends only on z and m and
w(
xn,k, rn,k, a(s)n,k; z
(s)n
)
is given by
w(
xn,k, 1, a(s)n,k; z
(s)n
)
=
f(
z(s)n,m
∣
∣xn,k)
fFA
(
z(s)n,m
)
, a(s)n,k=m∈M
(s)n
1 , a(s)n,k=0
w(
xn,k, 0, a(s)n,k; z
(s)n
)
= 1.
E. Joint Prior Distribution of Association Variables and Num-
bers of Measurements
Finally, to obtain an expression of p(a,b,m|x, r,q), i.e.,
the joint prior pmf of a, b, and m given x, r, and q, we
additionally make the following commonly used assumptions
[1], [12], [19]: (A5) Given x, r, and q, both a(s)n and
M(s)n are conditionally independent across n and s. (A6) The
measurements z(s)n,m, m ∈ M
(s)n at sensor s are randomly
ordered, with each possible order equally likely. Furthermore,
we define the indicator function [12], [19]
Ψ(
a(s)n,k , b
(s)n,m
)
,
0 , a(s)n,k=m, b
(s)n,m 6= k
or b(s)n,m= k, a
(s)n,k 6=m
1 , otherwise.
This function expresses the data association constraint as-
sumed in Section II-A, namely, that a PT can generate at most
5
one measurement at sensor s, and a measurement at sensor
s can be generated by at most one PT. Note that the for-
mulation of the data association constraint via Ψ(
a(s)n,k , b
(s)n,m
)
is redundant in that it involves both the PT-oriented associa-
tion variables a(s)n,k and the measurement-oriented association
variables b(s)n,m; however, this is key to obtaining an algorithm
that has a moderate computational complexity and an excellent
estimation performance even for a large number of targets and
a large number of measurements per sensor [12], [19].
Using the above assumptions, we obtain (cf. [12])
p(a,b,m|x, r,q)
= C(m)
n∏
n′=1
S∏
s=1
K∏
k=1
h(
xn′,k, rn′,k, a(s)n′,k, q
(s)n′,k;M
(s)n′
)
×
M(s)
n′
∏
m=1
Ψ(
a(s)n′,k , b
(s)n′,m
)
. (12)
Here, C(m) =∏n
n′=1
∏S
s=1e−µ(s)
(µ(s))M
(s)
n′
M(s)
n′!
is a normaliza-
tion factor that depends only on m, and h(
xn,k, rn,k, a(s)n,k,
q(s)n,k;M
(s)n
)
is defined as
h(
xn,k, 1, a(s)n,k, q
(s)n,k;M
(s)n
)
=
q(s)n,k
µ(s), a
(s)n,k ∈M
(s)n
1− q(s)n,k , a
(s)n,k = 0
h(
xn,k, 0, a(s)n,k, q
(s)n,k;M
(s)n
)
= 1(
a(s)n,k
)
,
where 1(a) ∈ {0, 1} is the indicator function of the event
a = 0, i.e., 1(a) = 1 if a = 0 and 0 otherwise. Finally, for use
in Section III-C, we make the following further assumption,
referred to as A7: Given x, r, and q, the vectors a, b, and
m are conditionally independent of ℓ. This means that if the
detection probabilities of the sensors as well as the states
and existences of the PTs are known, the association between
measurements and PTs is not influenced by the PT dynamics.
III. THE PROPOSED ALGORITHM
In this section, we develop the proposed self-tuning BP-
based multisensor-multitarget tracking algorithm.
A. Target Detection and State Estimation
Our ultimate goal is to determine if a PT k∈K exists (i.e.,
to detect the binary variables rn,k) and to estimate the states
xn,k of the detected PTs. This detection/estimation is based on
the past and present measurements of all the sensors, i.e., on
the total measurement vector z. In the Bayesian setting, target
detection and state estimation essentially amount to calculating
the marginal posterior existence probabilities p(rn,k = 1|z)and the marginal posterior state pdfs f(xn,k|rn,k = 1, z),respectively. PT k is detected (i.e., declared to exist) if
p(rn,k = 1|z) is larger than a suitably chosen threshold Pth
[20, Ch. 2]. Furthermore, for each detected PT k, an estimate
of xn,k is provided by the MMSE estimator [20, Ch. 4]
xMMSEn,k ,
∫
xn,k f(xn,k|rn,k=1, z)dxn,k .
The marginal statistics p(rn,k=1|z) and f(xn,k|rn,k=1, z)used for target detection and state estimation can be obtained
from the posterior pdf f(xn,k, rn,k, ℓn,k|z) according to
p(rn,k=1|z) =∑
ℓn,k∈J
∫
f(xn,k, rn,k=1, ℓn,k|z)dxn,k (13)
and
f(xn,k|rn,k=1, z) =f(xn,k, rn,k=1|z)
p(rn,k=1|z)
=
∑
ℓn,k∈J f(xn,k, rn,k=1, ℓn,k|z)
p(rn,k=1|z)..
(14)
Thus, the remaining problem is to calculate f(xn,k, rn,k=1,ℓn,k|z).
B. Review of Factor Graphs
The proposed BP-based multisensor-multitarget tracking
algorithm is based on a factor graph [13], [14] represent-
ing the factorization structure of the joint posterior pdf
f(x, r, ℓ, a,b,q|z) (to be derived later). For a brief review
of factor graphs, consider the generic problem of estimating
K parameter vectors xk, k ∈ {1, . . . ,K} from a measurement
vector z. In the Bayesian setting, these vectors are random, and
the estimation of xk is based on the posterior pdf f(xk|z).This pdf is a marginal pdf of the joint posterior pdf f(x|z),
where x =[
xT1 · · · x
TK
]T. The joint posterior pdf is assumed
to be the product of certain lower-dimensional factors, i.e.,
f(x|z) ∝∏
l
ψl(x(l); z). (15)
Here, ∝ denotes equality up to a constant factor (possibly
dependent on z), and each argument x(l) comprises certain
parameter vectors xk, where each xk can appear in several
x(l). The factorization (15) can be represented by a factor
graph, which is constructed as follows: each parameter variable
xk is represented by a variable node; each factor ψl(·) is
represented by a factor node; and variable node “xk” and
factor node “ψl” are adjacent, i.e., connected by an edge, if xkis an argument of ψl(·). Next, we will derive the factorization
(15) and the corresponding factor graph for our problem.
C. Joint Posterior Distribution and Factor Graph
The posterior pdf f(xn,k, rn,k, ℓn,k|z) in (13) and (14) is a
marginal density of the joint posterior pdf f(x, r, ℓ, a,b,q|z),which involves all the states, existence variables, IMM param-
eters, association variables, detection probabilities, and mea-
surements up to the current time n. An efficient approximate
implementation of the corresponding marginalization can be
obtained by performing BP on a factor graph representing the
factorization of f(x, r, ℓ, a,b,q|z).To derive this factorization, we first note that in the
conditional pdf f(x, r, ℓ, a,b,q|z), z is observed and thus
fixed, and thus the numbers of measurements M(s)n and the
corresponding vector m are fixed as well. We then have
f(x, r, ℓ, a,b,q|z)
= f(x, r, ℓ, a,b,q,m|z)
6
s = 1 s = 1
s = S s = S
a1
aK
a−1
a−K
f−1
f−K
p−1
p−1
p−K
p−K
q−1
q−K
y−1
y−K
f−1
f−K
b−1
b−M
υ−1
υ−K
Ψ−1,1
Ψ−K,M
Ψ−1,M
Ψ−K,1
q1
qK
y1
yK
f1
fK
f1
fK
p1
pK
b1
bM
α1α1
α1
αK
αK
αK
γ1γ1
γKγK
υ1
υKηK βK
η1 β1
χ1
χ1
χK
χK
ǫ1
ǫK
Ψ1,1
ΨK,M
Ψ1,M
ΨK,1
ν1,1 ζ1,1
νM,1 ζK,1
ν1,K
νM,K
ζ1,M
n− 1 n
ζK,M
p1
pK
p−1
p−K
Fig. 1. Factor graph describing the factorization of f(x, r, ℓ,a,b,q|z) in (19), shown for time steps n − 1 and n. For simplicity, the time indices n − 1
and n and the sensor index s are omitted, and the following short notations are used: f−
k, f(yn−1,k |yn−2,k), y
−
k, yn−1,k , p−
k, p
(
q(s)n−1,k
∣
∣q(s)n−2,k
)
,
υ−
k, υ
(
xn−1,k, rn−1,k, a(s)n−1,k , q
(s)n−1,k; z
(s)n−1
)
, q−k
, q(s)n−1,k , a−
k, a
(s)n−1,k , b−m , b
(s)n−1,m , Ψ−
k,m, Ψ
(
a(s)n−1,k , b
(s)n−1,m
)
, f−
k, f(yn−1,k),
p−k, p
(
q(s)n−1,k
)
, fk , f(yn,k |yn−1,k), yk , yn,k , qk , q(s)n,k
, pk , p(
q(s)n,k
∣
∣q(s)n−1,k
)
, ak , a(s)n,k
, bm , b(s)n,m, υk , υ
(
xn,k, rn,k, a(s)n,k
, q(s)n,k
; z(s)n
)
,
Ψk,m , Ψ(
a(s)n,k
, b(s)n,m
)
, αk , α(yn,k), βk , β(
a(s)n,k
)
, ηk , η(
a(s)n,k
)
, γk , γ(s)(xn,k , rn,k), χk , χ(
q(s)n,k
)
, ǫk , ǫ(
q(s)n,k
)
, νm,k , ν(p)m→k
(
a(s)n,k
)
,
ζk,m , ζ(p)k→m
(
b(s)n,m
)
, fk , f(yn,k), and pk , p(
q(s)n,k
)
.
∝ f(z|x, r, ℓ, a,b,q,m)f(x, r, ℓ, a,b,q,m) (16)
= f(z|x, r, a,m)f(x, r, ℓ, a,b,q,m) (17)
= f(z|x, r, a,m) p(a,b,m|x, r,q)f(x, r, ℓ) p(q), (18)
where we used Bayes’ rule in (16), assumption A4 as well
as the fact that a implies b in (17), and the chain rule and
assumptions A1 and A7 in (18). Next, inserting the expressions
(11) for f(z|x, r, a,m), (12) for p(a,b,m|x, r,q), (5) for
f(x, r, ℓ), and (10) for p(q), we obtain the final factorization
f(x, r, ℓ, a,b,q|z)
∝K∏
k=1
f(x0,k, r0,k, ℓ0,k)
(
S∏
s=1
p(q(s)0,k)
)
×n∏
n′=1
f(xn′,k, rn′,k, ℓn′,k|xn′−1,k, rn′−1,k, ℓn′−1,k)
×S∏
s=1
p(
q(s)n′,k
∣
∣q(s)n′−1,k
)
υ(
xn′,k, rn′,k, a(s)n′,k, q
(s)n′,k; z
(s)n′
)
×
M(s)
n′
∏
m=1
ψ(
a(s)n′,k, b
(s)n′,m
)
, (19)
where
υ(
xn,k, rn,k, a(s)n,k, q
(s)n,k; z
(s)n
)
, w(
xn,k, rn,k, a(s)n,k; z
(s)n
)
h(
xn,k, rn,k, a(s)n,k, q
(s)n,k;M
(s)n
)
.
It will be convenient to introduce the augmented state of PT kas yn,k , [xT
n,k rn,k ℓn,k]T. The factors in the first and third
product on the right-hand side of (19) then read as f(y0,k)and f(yn′,k, |yn′−1,k), respectively.
In Fig. 1, we show the factor graph describing the factor-
ization (19) for time steps n−1 and n. With a view toward our
discussion in Section IV, we remark that xn,k, rn,k, and ℓn,kare represented in the factor graph by the single “augmented
state” variable node labeled yk , whereas q(s)n,k is represented
by a separate variable node labeled qk.
D. Review of Belief Propagation
Approximations of the marginal posterior pdfs f(yn,k|z) =f(xn,k, rn,k, ℓn,k|z) can be efficiently calculated by running
iterative BP message passing [13], [14] on the factor graph in
Fig. 1. For a brief review of the BP algorithm, we reconsider
the generic estimation problem from Section III-B, involving
the joint posterior pdf f(x|z) and the marginal posterior pdfs
f(xk|z).The BP algorithm aims at computing the marginal posterior
pdfs f(xk|z) in an efficient way. It is based on the factor graph
representing the factorization of f(x|z) in (15), which contains
the variable nodes “xk” and the factor nodes “ψl”. For each
node in the factor graph, certain messages are calculated, each
of which is then passed to one of the adjacent nodes. Let Vldenote the set of the indices k of all variables xk that are
adjacent to factor node “ψl”. Then, factor node “ψl” passes
the following message to variable node “xk” with k∈Vl:
ζψl→xk(xk) =
∫
ψl(x(l); z)
∏
k′∈Vl\{k}
ηxk′→ψl(xk′ ) dx−k .
(20)
7
Here,∫
. . . dx−k denotes integration with respect to all xk′ ,
k′ ∈ Vl except xk, and the messages ηxk′→ψl(xk′ ) are
calculated as described later. If the factorization (15) involves
(also) discrete variables, then the respective integrations in (20)
have to be replaced with summations.
Furthermore, let Fk be the set of the indices l of all those
factors nodes “ψl” that are adjacent to variable node “xk”.
Then, variable node “xk” passes the following message to
factor node “ψl” with l∈Fk:
ηxk→ψl(xk) =
∏
l′∈Fk\{l}
ζψl′→xk(xk).
For a factor graph with loops, as in Fig. 1, the calculation
of the messages is usually repeated in an iterative manner.
There is no unique order of message calculation, and different
orders may lead to different results. Finally, for each variable
node “xk”, a belief f(xk) is calculated by multiplying all
the incoming messages (passed from all the adjacent factor
nodes) and normalizing the resulting product function such
that∫
f(xk)dxk = 1. The belief f(xk) provides the desired
approximation of the marginal posterior pdf f(xk|z).
E. BP Message Passing Algorithm
We now apply the BP algorithm to the problem
of calculating the marginal posterior pdfs f(yn,k|z) =f(xn,k, rn,k, ℓn,k|z). Since our factor graph in Fig. 1 contains
loops, we have to choose an order of calculating the individual
messages. In our algorithm, the order is defined by two
rules: first, messages are not sent backward in time, and
second, iterative message passing is only performed for data
association, and separately at each time step and at each sensor.
The second rule implies that for loops involving different
sensors, only a single message passing iteration is performed.
For the formulation of the message passing algorithm, we
recall that a state xn,k is formally defined also for a nonex-
istent PT k, i.e., if rn,k = 0. Since the states of nonexistent
PTs are irrelevant, any pdf f(yn,k) = f(xn,k, rn,k, ℓn,k) of an
augmented state yn,k is defined such that for rn,k=0,
f(xn,k, rn,k= 0, ℓn,k) = f(ℓn,k)n,k fD(xn,k). (21)
Here, fD(xn,k) is an arbitrary dummy pdf [12], as previously
used in (8) and (9). Thus, f(xn,k, rn,k=0, ℓn,k) is described
by the single number f(ℓn,k)n,k .
Combining these rules with the generic BP rules for cal-
culating messages and beliefs as reviewed in Section III-D,
one obtains that the following operations are performed at
time step n in the stated order: 1) prediction, 2) measurement
evaluation, 3) data association, 4) measurement update, and 5)
calculation of beliefs. In what follows, we provide descriptions
of these operations, including explicit expressions of the
various messages and beliefs.
1) Prediction: First, a prediction step is performed, in
which a message α(yn,k) = α(xn,k, rn,k, ℓn,k) is calculated as
α(xn,k, rn,k, ℓn,k)
=∑
rn−1,k∈{0,1}
∑
ℓn−1,k∈J
∫
f(xn−1,k, rn−1,k, ℓn−1,k)
×f(xn,k, rn,k, ℓn,k|xn−1,k, rn−1,k, ℓn−1,k)dxn−1,k .
(22)
Here, f(xn−1,k, rn−1,k, ℓn−1,k) was calculated at time n−1(see (25)). Using (7)–(9) in (22), we obtain for rn,k=1
α(xn,k, 1, ℓn,k)
=∑
ℓn−1,k∈J
p(ℓn,k|ℓn−1,k)
(
p(s)n,k
∫
fℓn,k(xn,k|xn−1,k)
× f(xn−1,k, rn−1,k=1, ℓn−1,k) dxn−1,k
+ p(b)n,kf
(b)ℓn,k
(xn,k) f(ℓn−1,k)n−1,k
)
. (23)
Here, f(ℓn−1,k)n−1,k ,
∫
f(xn−1,k, rn−1,k = 0, ℓn−1,k)dxn−1,k;
note that, consistently with (21), f(xn−1,k, rn−1,k=0, ℓn−1,k)
= f(ℓn−1,k)n−1,k fD(xn−1,k). Similarly, we obtain for rn,k= 0
α(ℓn,k)n,k ,
∫
α(xn,k, 0, ℓn,k)dxn,k (24)
=∑
ℓn−1,k∈J
p(ℓn,k|ℓn−1,k)
(
(
1−p(s)n,k
)
×
∫
f(xn−1,k, rn−1,k=1, ℓn−1,k) dxn−1,k
+(
1−p(b)n,k
)
f(ℓn−1,k)n−1,k
)
,
where∫
fD(xn,k)dxn,k = 1 was used. (Note that, consistently
with (21), α(xn,k, 0, ℓn,k) = α(ℓn,k)n,k fD(xn,k).) Marginalizing
α(ℓn,k)n,k over ℓn,k and using (24) yields
αn,k ,∑
ℓn,k∈J
α(ℓn,k)n,k =
∫
A(xn,k, 0)dxn,k ,
with A(xn,k, rn,k) ,∑
ℓn,k∈J α(xn,k, rn,k, ℓn,k). Because
f(xn−1,k, rn−1,k, ℓn−1,k) is normalized (i.e., a pdf), it follows
from (22) that also α(xn,k, rn,k, ℓn,k) is normalized, which
implies∑
rn,k∈{0,1}
∫
A(xn,k, rn,k)dxn,k =1. Thus, we have
αn,k = 1−
∫
A(xn,k, 1)dxn,k .
Finally, the prediction step also comprises the calculation of
χ(
q(s)n,k
)
=∑
q(s)n−1,k∈Q
p(
q(s)n,k
∣
∣q(s)n−1,k
)
p(
q(s)n−1,k
)
,
for all s∈S. Here, p(
q(s)n−1,k
)
was calculated at time n−1 (see
(26)).
After the prediction step, messages β(
a(s)n,k
)
, η(
a(s)n,k
)
, and
γ(s)(xn,k, rn,k) are calculated for all k ∈ K and s ∈ S in
parallel, as described next.
2) Measurement Evaluation: First, in the measurement
evaluation step, the messages β(
a(s)n,k
)
are calculated as
β(
a(s)n,k
)
=∑
q(s)n,k
∈Q
∑
rn,k∈{0,1}
∑
ℓn,k∈J
∫
υ(
xn,k, rn,k, a(s)n,k, q
(s)n,k; z
(s)n
)
8
× α(xn,k, rn,k, ℓn,k)χ(
q(s)n,k
)
dxn,k
=∑
q(s)n,k
∈Q
χ(
q(s)n,k
)
∫
υ(
xn,k, 1, a(s)n,k, q
(s)n,k; z
(s)n
)
A(xn,k, 1)dxn,k
+ 1(
a(s)n,k
)
αn,k .
3) Data Association: In the subsequent iterative data asso-
ciation step, the messages β(
a(s)n,k
)
are converted into messages
η(
a(s)n,k
)
. This step is equal to the corresponding step in [12];
it involves iterated messages ν(p)m→k
(
a(s)n,k
)
and ζ(p)k→m
(
b(s)n,m
)
,
where p denotes the iteration index. These iterated messages
are shown in Fig. 1 for time step n. The data association step
also involves the factor nodes labeled Ψk,m in Fig. 1.
4) Measurement Update: Finally, in the measurement up-
date step, the messages γ(s)(xn,k, rn,k) are calculated as
follows: for rn,k =1,
γ(s)(xn,k, 1) =∑
a(s)n,k
∈{0,...,M(s)n }
∑
q(s)n,k
∈Q
υ(
xn,k, 1, a(s)n,k, q
(s)n,k; z
(s)n
)
× χ(
q(s)n,k
)
η(
a(s)n,k
)
,
and for rn,k =0,
γ(s)n,k ,
∫
γ(s)(xn,k, 0)dxn,k = η(
a(s)n,k= 0
)
.
5) Calculation of Beliefs: The beliefs f(yn,k) =f(xn,k, rn,k, ℓn,k) approximating the posterior pdfs
f(xn,k, rn,k, ℓn,k|z) are calculated as
f(xn,k, rn,k, ℓn,k) =1
Cn,kα(xn,k, rn,k, ℓn,k) Γ(xn,k, rn,k),
(25)
with Γ(xn,k, rn,k) ,∏Ss=1 γ
(s)(xn,k, rn,k) and
Cn,k =
∫
A(xn,k, 1) Γ(xn,k, 1)dxn,k + αn,kΓn,k ,
where Γn,k ,∏S
s=1 γ(s)n,k. The beliefs f(xn,k, rn,k, ℓn,k) for
rn,k=1 are used for target detection and state estimation, by
substituting f(xn,k, rn,k=1, ℓn,k) for f(xn,k, rn,k=1, ℓn,k|z)in (13) and (14). For rn,k= 0, we note that (25) implies that
f(ℓn,k)n,k =
∫
f(xn,k, rn,k= 0, ℓn,k)dxn,k is given by
f(ℓn,k)n,k =
1
Cn,kα(ℓn,k)n,k Γn,k .
Next, the beliefs p(
q(s)n,k
)
approximating the posterior pmfs
of the detection probabilities, p(
q(s)n,k
∣
∣z)
, are calculated as
p(
q(s)n,k
)
= χ(
q(s)n,k
)
ǫ(
q(s)n,k
)
, (26)
with
ǫ(
q(s)n,k
)
=∑
a(s)n,k
∈{0,...,M(s)n }
∫
υ(
xn,k, 1, a(s)n,k, q
(s)n,k; z
(s)n
)
η(a(s)n,k)
×A(xn,k, 1)dxn,k + η(
a(s)n,k= 0
)
αn,k .
Furthermore, the beliefs g(ℓn,k) approximating the posterior
pmfs of the modes, p(ℓn,k|z), are calculated as
g(ℓn,k) =∑
rn,k∈{0,1}
∫
f(xn,k, rn,k, ℓn,k)dxn,k
=1
Cn,k
(
G(ℓn,k) + α(ℓn,k)n,k Γn,k
)
, (27)
with G(ℓn,k) ,∫
α(xn,k, 1, ℓn,k) Γ(xn,k, 1)dxn,k. Finally,
beliefs fℓn,k(xn,k, rn,k) approximating the conditional pdf
f(xn,k, rn,k|ℓn,k) are calculated as
fℓn,k(xn,k, rn,k) =
f(xn,k, rn,k, ℓn,k)
g(ℓn,k). (28)
These beliefs will be used in our particle-based implementa-
tion of the proposed method, which will be described next.
6) Particle-Based Implementation: For general nonlinear
and non-Gaussian measurement and state-evolution models,
the expressions of the various messages and beliefs presented
above cannot be evaluated in closed form and, moreover, their
computation is unfeasible. However, a feasible approximate
computation that avoids the explicit evaluation of integrals and
message products is provided by a particle-based implementa-
tion of the BP message passing algorithm that is analogous to
the implementation presented in [12]. In what follows, we only
consider the prediction step; a particle-based implementation
of the measurement evaluation, data association, measurement
update, and belief calculation steps can be performed (with
minor modifications) as described in [12].
We start by using (28) in (23), which gives
α(xn,k, 1, ℓn,k)
=∑
j∈J
p(ℓn,k|ℓn−1,k=j)
(
p(s)n,k
∫
fℓn,k(xn,k|xn−1,k)
× fj(xn−1,k, rn−1,k=1) g(j) dxn−1,k
+ p(b)n,k f
(b)ℓn,k
(xn,k) f(j)n−1,k
)
=∑
j∈J
p(ℓn,k|ℓn−1,k=j) p(s)n,k g(j)
∫
fℓn,k(xn,k|xn−1,k)
× fj(xn−1,k, rn−1,k=1) dxn−1,k
+ p(b)n,k f
(b)ℓn,k
(xn,k)∑
j∈J
p(ℓn,k|ℓn−1,k=j) f(j)n−1,k. (29)
It can be seen that for each ℓn,k, α(xn,k, 1, ℓn,k) is a
weighted mixture of J + 1 component pdfs. In particu-
lar, the first term consists of |J | = J component pdfs∫
fℓn,k(xn,k|xn−1,k) fj(xn−1,k, rn−1,k = 1) dxn−1,k, involv-
ing the previous beliefs fj(xn−1,k, rn−1,k = 1), with
component weights given by p(ℓn,k|ℓn−1,k = j)p(s)n,k g(j),
for j ∈ J . The second term consists of the sin-
gle component pdf f(b)ℓn,k
(xn,k) with component weight
p(b)n,k
∑
j∈J p(ℓn,k|ℓn−1,k=j) f(j)n−1,k.
At each time n≥ 1, for each mode j ∈ J and PT k ∈ K,
T particles and weights{(
x(j,t)n−1,k, w
(j,t)n−1,k
)}T
t=1representing
fj(xn−1,k, rn−1,k = 1) were calculated at the previous time
n−1. For a particle-based computation of α(xn,k, 1, ℓn,k) in
9
(29), we draw particles from each component pdf, weight them
using the corresponding component weight, and combine them
into a joint particle set [21]. The following operations are
performed for each mode ℓn,k ∈ J in parallel. (To simplify
the notation, we suppress the index ℓn,k when possible.)
First, the message α(xn,k, 1, ℓn,k) is represented by a set
of weighted particles{(
xm(j,t)n,k , w
m(j,t)n,k
)}
j∈J, t∈{1,...,T}, with
m(j, t) , (j−1)T + t, augmented by a set of weighted parti-
cles (“birth particles”){(
x(b)(i)n,k , w
(b)(i)n,k
)}I
i=1. These weighted
particles are obtained as follows. For each j ∈ J and for
each particle x(j,t)n−1,k, one particle x
m(j,t)n,k is drawn from
fℓn,k
(
xn,k∣
∣x(j,t)n−1,k
)
, and the corresponding weight wm(j,t)n,k is
obtained as (cf. (29))
wm(j,t)n,k = p(ℓn,k|ℓn−1,k=j) p
(s)n,k g(j)w
(j,t)n−1,k .
Furthermore, the birth particles x(b)(i)n,k , i=1, . . . , I are drawn
from the birth pdf f(b)ℓn,k
(xn,k), and the corresponding weights
w(b)(i)n,k are calculated as
w(b)(i)n,k =
p(b)n,k
I
∑
j∈J
p(ℓn,k|ℓn−1,k=j) g(j)
(
1−T∑
t=1
w(j,t)n−1,k
)
.
We note that this expression of w(b)(i)n,k is based on (29) wherein
f(j)n−1,k has been approximated by g(j)
(
1 −∑Tt=1 w
(j,t)n−1,k
)
.
For each j ∈ J , the resulting particle representation of
α(xn,k, 1, j), denoted{(
x(j,t)n,k , w
(j,t)n,k
)}TJ+I
t=1, is employed as
proposal distribution in our particle-based implementation
of the measurement evaluation, measurement update, and
belief calculation steps. This particle-based implementation
uses Monte Carlo integration and importance sampling [21],
as described in detail in [12]. After the belief calculation
step, as in [12], resampling is performed to obtain par-
ticles{(
x(j,t)n,k , w
(j,t)n,k
)}T
t=1that represent the current belief
fj(xn,k, rn,k = 1). The initialization of the above recursive
scheme, as well as the choice of the birth pdfs f(b)j (xn,k), birth
probabilities p(b)n,k, and survival probabilities p
(s)n,k are discussed
in [12, Sec. VII].
7) Remarks: The BP algorithm described above exhibits
the same scalability properties as the nonadaptive algorithm
described in [12]. Its complexity scales only quadratically in
the number of PTs, linearly in the number of sensors, and
linearly in the number of measurements per sensor. Further-
more, the complexity scales quadratically both in the number
of modes J and in the number of detection probabilities Q.
We also note that our BP algorithm conforms in spirit to the
general approach of the IMM method in [3, Sec. 11.6.6], rather
than of the generalized pseudo-Bayesian estimator of first or-
der (GPB1) in [3, Sec. 11.6.4]. Both the IMM method and the
GPB1 method use J Kalman filters in parallel, each associated
with one mode. In the IMM method, the prediction step of each
Kalman filter is based on a state estimate that depends on the
associated mode, whereas in the GPB1 method, the prediction
steps of all the Kalman filters are based on the same mode-
independent state estimate. The GPB1 method has generally
been observed to perform less well than the IMM method [3,
Sec. 11.6.9]. While our BP algorithm does not use Kalman
filters, it does introduce a mode dependence in the spirit of the
IMM method. Indeed, the messages α(xn,k, rn,k, ℓn,k) in (22)
involve the beliefs f(xn−1,k, rn−1,k, ℓn−1,k), which depend
on the IMM parameter ℓn−1,k.
IV. INCORPORATING MODEL PARAMETERS IN THE
FACTOR GRAPH
We will now generalize the development in the previous
section to arbitrary unknown time-varying model parameters
θ(d)n for multitarget tracking problems. More concretely, in this
section, we propose a systematic approach to incorporating
time-varying model parameters in a factor graph representing
the joint posterior pdf in a multitarget tracking problem. This
approach is based on the distinction between state parameters
and measurement parameters introduced in Section II-B.
We first provide a formal definition of these two types
of parameters. Let us denote by θS and θM the vectors of,
respectively, all the state parameters and all the measurement
parameters of all the PTs k, all the sensors s, and all the times
up to the current time n. Then, θS and θM are defined by the
following properties:
1) Given the PT states x, the PT existence variables r,
and the measurement parameters θM, the measurement-
related quantities z, m, a, and b are conditionally
independent of θS, i.e.,
f(z,m, a,b|x, r, θM, θS) = f(z,m, a,b|x, r, θM). (30)
2) The PT states x, PT existence variables r, and state
parameters θS are independent of θM, i.e.,
f(x, r, θS|θM) = f(x, r, θS). (31)
In our case, θS= ℓ and θM= q. Thus, equation (30) reads
f(z,m, a,b|x, r,q, ℓ) = f(z,m, a,b|x, r,q). (32)
To show that it is satisfied, we process the left-hand side as
follows:
f(z,m, a,b|x, r,q, ℓ)
= f(z|x, r,q, ℓ,m, a,b) p(m, a,b|x, r,q, ℓ)
= f(z|x, r,m, a,b) p(m, a,b|x, r,q)
= f(z,m, a,b|x, r,q). (33)
where we used assumptions A4 from Section II-D and A7
from Section II-E. Expression (33) is recognized to be equal
to the right-hand side of (32). Furthermore, equation (31) reads
f(x, r, ℓ|q) = f(x, r, ℓ), which is satisfied due to assumption
A1 from Section II-D.
We now propose the following rules for incorporating the
parameters θS and θM in the factor graph:
1) The subvector θS,n,k of θS comprising all the state
parameters at time n that are related to PT k is rep-
resented in the factor graph jointly with the PT state
xn,k and the PT existence indicator rn,k by a common
variable node. Thereby, the BP algorithm performs an
(approximate) calculation of the joint posterior pdf/pmf
of xn,k, rn,k, and θS,n,k, rather than of the individual
10
posterior pdfs/pmfs of xn,k, rn,k, and θS,n,k. In our case,
θS,n,k=ℓn,k, and the common variable node is the node
labeled yk in Fig. 1, which represents the augmented
state vector yn,k = [xTn,k rn,k ℓn,k]
T.
2) Each subvector θ(s)M,n,k of θM comprising all the mea-
surement parameters at time n that are related to PT
k and sensor s is represented in the factor graph by
a separate variable node. Thereby, the BP algorithm
performs an (approximate) calculation of the individual
posterior pdfs/pmfs of the various measurement parame-
ter subvectors θ(s)M,n,k. In our case, θ
(s)M,n,k= q
(s)n,k, which
is represented by the node labeled qk in Fig. 1.
V. SIMULATION RESULTS
We assess the performance of the proposed algorithm in
three different simulated scenarios. Results for a real scenario
will be presented in Section VI.
A. Basic Simulation Setup
In all scenarios, the number of PTs K is set to 8, and the PT
states consist of two-dimensional (2D) position and velocity,
i.e., xn,k = [x1,n,k x2,n,k x1,n,k x2,n,k]T. We consider
dynamic models (DMs) ξj of the nearly-constant velocity type,
i.e. (cf. (4))
xn,k = ξj(
xn−1,k,u(j)n,k
)
= Axn−1,k +Wu(j)n,k ,
where A ∈ R4×4 and W ∈ R
4×2 are chosen as in [3, Sec.
6.3.2] (these matrices involve the scan duration ∆T , i.e., the
duration of one time step n); furthermore, the driving process
u(j)n,k ∼ N
(
0, σ2j I2)
is a sequence of 2D Gaussian random
vectors that is iid across n and k. The DMs ξj(·) differ solely
in the driving process variance σ2j . We note that σj character-
izes the average increment of target speed in a time step of
duration ∆T . Thus, higher values of σ2j are typically used to
model targets that accelerate and/or change their course. As a
rule of thumb, to model targets that follow straight trajectories,
a low value of σ2j should be chosen if ∆T is large and a high
value if ∆T is small. To model also maneuvering or suddenly
accelerating targets, we propose to use two DMs ξ1 and ξ2,
where σ21 is chosen according to the above rule of thumb and
σ22 is at least two orders of magnitude larger than σ2
1 . In our
simulation, the number of DMs is chosen as J = 1 or J =2.
Depending on the scenario, two or three targets move in the
square region given by [−80 km, 80 km]×[−80 km, 80 km]. The
targets exist at all times. (The case of appearing/disappearing
targets will be considered in Section VI.) There are S = 3sensors that measure the target position in polar coordinates,
i.e., range and bearing, with a maximum range of 160 km. The
sensor measurements are modeled according to
z(s)n,m =
[
‖xn,k−p(s)‖
φ(xn,k,p(s))
]
+ v(s)n,m , (34)
where xn,k , [x1,n,k x2,n,k]T is the position of tar-
get k, p(s) = [p(s)1 p
(s)2 ]T is the position of sensor s,
φ(xn,k,p(s)) is the angle between xn,k and p(s), and v
(s)n,m ∼
N(
0, diag(σ2r , σ
2b ))
is a sequence of 2D Gaussian random
TABLE ICOMMON PARAMETER VALUES USED IN THE SIMULATION.
Parameter Value Description
∆T 20 s Duration of time step
σr 150 m Range standard deviation
σb 1.5◦ Bearing standard deviation
µ(s) 10 Mean number of false alarms
T 5000 Number of particles
p(s) 0.999 Survival probability
p(b) 0.001 Birth probability
Pth 0.5 Detection threshold
K 8 Number of PTs
vectors that is iid across n, m, and s. The false alarm pdf
fFA
(
z(s)n,m
)
is linearly increasing on [0 km, 160 km] and zero
outside that interval with respect to the range component, and
uniform on [0◦, 360◦) with respect to the bearing component.
Equivalently, it is uniform on the surveillance region. In
the proposed self-tuning algorithm, the detection probabilities
q(s)n,k are considered unknown with Q = 11 possible values
ω1 = 0.01, ω2 = 0.1, ω3 = 0.2, . . . , ω10 = 0.9, ω11 = 1. The
corresponding transition probabilities are chosen as follows:
[Q]i,i−1 = 0.03, [Q]i,i = 0.92, and [Q]i,i+1 = 0.05 for
i = 2, 3, . . . , 10; furthermore, [Q]1,1 = 0.95, [Q]1,2 = 0.05,
[Q]11,10= 0.03, [Q]11,11= 0.97, and [Q]i,j=0 otherwise.
In the next three subsections, we describe the three sim-
ulation scenarios and present the corresponding simulation
results. The simulation parameters common to all scenarios
are listed in Table I. We note that the survival probability is
equal for all PTs k and all times n, i.e., p(s)n,k = p(s), and
similarly for the birth probability, i.e., p(b)n,k= p(b).
B. First Scenario: Two Modes
In the first scenario, three targets switch between two nearly-
constant velocity DMs ξ1 and ξ2 with σ21 = 0.012 and
σ22 = 0.52, respectively. Within 100 simulated time steps
n ∈ [1, 100], all three targets follow DM ξ2 in the time
intervals [30, 45] and [80, 95] and DM ξ1 in the remaining
time intervals. Fig. 2 shows the positions of the three sensors
and an exemplary realization of the three target trajectories.
The detection probabilities are chosen as q(s)n,k = 0.8 for all
n, k, and s. In the proposed self-tuning algorithm, the q(s)n,k
are considered unknown with discrete values and transition
probabilities as described in Section V-A; furthermore, the DM
transition probabilities are chosen as [L]1,1= [L]2,2= 0.9975and [L]1,2= [L]2,1= 0.0025.
We compare the proposed self-tuning algorithm with the
original nonadaptive BP algorithm from [12], which uses the
true detection probabilities q(s)n,k = 0.8 and constant driving
process variance σ2u = σ2
1 = 0.012, and with a “clairvoyant”
algorithm, which is the nonadaptive BP algorithm that uses
q(s)n,k= 0.8 and, in addition, knows at each time which one of
the DMs ξ1 and ξ2 is in force. All three algorithms were
simulated using particle-based implementations (cf. Section
III-E6 and [12]). Fig. 3 shows the Euclidean distance based
mean optimal sub-pattern assignment (MOSPA) error with
11
Target 1
Target 2
Target 3
x1 [km]
x2
[km
]
−80 −60 −40 −20 0 20 40 60 80−80
−60
−40
−20
0
20
40
60
80
Fig. 2. Sensor positions (marked by triangles) and an exemplary realizationof the target trajectories for the first scenario. The crosses mark the finalpositions of the targets.
10 20 30 40 50 60 70 80 90 1000
200
400
600
800
1000
Proposed
algorithm
Nonadaptive
algorithm [12]
Clairvoyant
algorithm
n
MO
SPA
erro
r[m
]
Fig. 3. MOSPA error for the first scenario. (In this figure and in Figs. 4, 9,and 10, the dashed lines indicate the times when the DM changes.)
order p = 1 and cutoff parameter c = 1000 [22], averaged
over 200 simulation runs. The MOSPA error metric takes into
account both the estimation errors for correctly detected targets
and the errors due to incorrect target detections. One can see
in Fig. 3 that initially, as long as the targets follow DM ξ1,
the MOSPA errors of the proposed algorithm, the nonadaptive
algorithm, and the clairvoyant algorithm are almost identical.
However, ever after the targets change to DM ξ2, the MOSPA
error of the proposed algorithm is smaller by a factor of about
three to four than that of the nonadaptive algorithm, and this
remains true even after the targets revert to DM ξ1. Moreover,
the MOSPA error of the proposed algorithm is almost equal
to that of the clairvoyant algorithm, which means that the DM
adaptation performed by the proposed algorithm works very
well.
Fig. 4 shows the mode beliefs g(ℓn,k) (see (27)) for the
DMs ξ1 and ξ2 calculated by the proposed algorithm, averaged
over the three targets and over the 200 simulation runs. The
averaged mode belief for DM ξ1 increases to about 0.8 in
the intervals where DM ξ1 is in force, whereas it decreases
to about 0.2 in the intervals where DM ξ2 is in force. An
analogous behavior is exhibited by the averaged mode belief
for DM ξ2. These results are consistent with the notion that
10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1ξ1ξ2
n
Aver
aged
mo
de
bel
ief
Fig. 4. Averaged mode beliefs for the DMs ξ1 and ξ2, for the first scenario.
10 20 30 40 50 60 70 80 90 1000
0.2
0.4
0.6
0.8
1
Est
imat
edd
etec
tio
np
rob
abil
ity
n
Fig. 5. Averaged estimated detection probability for the first scenario. Thedashed line indicates the true detection probability.
g(ℓn,k) approximates p(ℓn,k|z).Finally, Fig. 5 shows an estimate of the detection prob-
ability q(s)n,k that was calculated by the proposed algorithm.
Approximate MMSE estimates of the q(s)n,k were obtained as
(cf. (3)) q(s)n,k=
∑11i=1 ωi p
(
q(s)n,k=ωi
)
, with p(
q(s)n,k
)
calculated
according to (26). These estimates were then averaged over
the three targets, the three sensors, and 200 simulation runs. It
is seen that after about 20 time steps, the estimated detection
probability is very close to the true detection probability, 0.8.
We note that the initial pmf of the detection probability at
time n = 0 was chosen uniform, i.e., p(q(s)0,k = ωi) =
111 for
i ∈ {1, . . . , 11}. Therefore, the initial estimates of q(s)n,k are
given by q(s)0,k=
111
∑11i=1ωi =0.5009.
C. Second Scenario: Varying Detection Probabilities
In the second scenario, there are two targets, one heading
toward North-West and the other toward South-East, and both
following almost straight trajectories. We simulated 400 time
steps. The proposed self-tuning algorithm uses only DM ξ1,
and thus does not adapt the DM in this scenario. Fig. 6 depicts
the positions of the three sensors (labeled N, E, and S) and the
target trajectories (used in all simulation runs). The simulated
detection probabilities q(s)n,k now depend on the distances of
the targets from the sensors and thus are time-varying. They
12
Target 1
Target 2
N
E
S
x1 [km]
x2
[km
]
−80 −60 −40 −20 0 20 40 60 80−80
−60
−40
−20
0
20
40
60
80
Fig. 6. Sensor positions and target trajectories for the second scenario.
are calculated as [23, Eq. 15.25]
q(s)n,k = QM
(
√
2C/R(s)4n,k ,
√
−2 lnPFA
)
, (35)
where QM(· , ·) denotes the Marcum Q function, R(s)n,k is the
distance of target k from sensor s (in m), C = 6.3 · 1020m4,
and PFA = 10−6 is the false alarm probability.2
To show that the proposed algorithm is able to estimate the
detection probabilities q(s)n,k in an online manner, Fig. 7 depicts
the mean estimates of q(s)n,2 (i.e., for the target heading toward
South-East) and the associated standard deviations, for each
of the three sensors s. These results were obtained from 200
simulation runs, which differ in the sensor measurements and
particles; the individual estimates were calculated as explained
in Section V-B. The true detection probabilities, calculated
according to (35), are also shown. It is seen that the true detec-
tion probabilities increase or decrease depending on whether
the target moves toward or away from the respective sensor,
and the mean estimates roughly conform to this behavior. We
also calculated the time-averaged root mean squared errors
(RMSEs) of the estimates of q(s)n,k for each of the three sensors.
We obtained 0.10 for sensor N, 0.08 for sensor E, and 0.07
for sensor S.
D. Third Scenario: Two Modes and Varying Detection Prob-
abilities
In the third scenario, there are again two targets. The
nominal DM of both targets is ξ1 (with σ21 = 0.012), and
this DM is used by the nonadaptive algorithm. However,
the target trajectories conform to DM ξ1 only in the time
intervals [1, 189] and [231, 400], whereas in the intermediate
time interval [190, 230], the targets perform a coordinated turn
with nearly constant speed and constant angular rate. The
sensor positions and target trajectories are shown in Fig. 8. The
proposed algorithm switches adaptively between DM ξ1 and a
2More specifically, C = PTGTGRλ2σ/
(
(4π)3kT0FB)
, with transmitter
power PT = 31W, transmitter gain GT = 101.5, receiver gain GR = 101.5,wavelength λ=13m, target radar cross section σ=9m2, Boltzmann constantk = 1.38× 10−23 Ws/K, standard temperature T0 = 290K, receiver noisefigure F =100.5, and receiver bandwidth B=3000Hz.
50 100 150 200 250 300 350 400
0
0.2
0.4
0.6
0.8
1
True (sensor N)
True (sensor E)
True (sensor S)
Estimated (sensor N)
Estimated (sensor E)
Estimated (sensor S)
n
Est
imat
edd
etec
tio
np
rob
abil
ity
Fig. 7. Empirical mean and standard deviation of the estimated detectionprobabilities for target 2 at the three sensors, for the second scenario. Theshaded areas indicate the standard deviations.
Target 1
Target 2
N
E
S
x1 [km]
x2
[km
]
−80 −60 −40 −20 0 20 40 60 80−80
−60
−40
−20
0
20
40
60
80
Fig. 8. Sensor positions and target trajectories for the third scenario.
second nearly-constant velocity DM ξ2 with σ22 = 0.12, using
the same DM transition probabilities as in the first scenario.
As in the second scenario, the detection probabilities q(s)n,k
depend on the distance of the respective target k from the
respective sensor s according to (35). They are estimated by
the proposed algorithm, whereas the nonadaptive algorithm
assumes q(s)n,k=0.8. In this scenario, the clairvoyant algorithm
is a version of the proposed algorithm that knows at each
time which DM is in force, and thus adapts only the detection
probabilities q(s)n,k. We performed 200 simulation runs, which
differ in the sensor measurements and particles.
Fig. 9 shows that the MOSPA error of the proposed al-
gorithm is similar to or lower than that of the nonadaptive
algorithm. The time-averaged MOSPA error is 181 m for
the proposed algorithm versus 221 m for the nonadaptive
algorithm. In particular, during the turn interval [190, 230],it is 305 m for the proposed algorithm versus 504 m for the
nonadaptive algorithm. The time-averaged MOSPA error of
the clairvoyant algorithm is 166 m, and it is 285 m during the
turn interval [190, 230] in particular. As in the first scenario,
the performance of the proposed algorithm is seen to be almost
equal to that of the clairvoyant algorithm.
Fig. 10 shows the mode beliefs g(ℓn,k) for ξ1 and ξ2
13
50 100 150 200 250 300 350 4000
200
400
600
800
1000
Proposed
algorithm
Nonadaptive
algorithm [12]
Clairvoyant
algorithm
n
MO
SPA
erro
r[m
]
Fig. 9. MOSPA error for the third scenario.
50 100 150 200 250 300 350 4000
0.2
0.4
0.6
0.8
1
ξ1ξ2
n
Aver
aged
mo
de
bel
ief
Fig. 10. Averaged mode beliefs for the DMs ξ1 and ξ2, for the third scenario.
calculated by the proposed algorithm, averaged over the two
targets and over the 200 simulation runs. These beliefs are seen
to correctly picture the DM actually in force, except that they
switch between the DMs with a delay of about 20 time steps.
This delay is probably caused by the strong measurement
noise (as indicated by the high range and bearing standard
deviations, σr=150m and σb=1.5◦).
Finally, Fig. 11 shows the mean estimates of the detection
probabilities q(s)n,2 of target k=2 and the associated standard
deviations, for each of the three sensors s. The figure also
shows the true detection probabilities, which depend on the
distance between the target and the sensor according to (35).
It is seen that the mean estimates roughly approximate the
true detection probabilities. The time-averaged RMSEs were
obtained as 0.10 for sensor N, 0.07 for sensor E, and 0.07 for
sensor S.
E. Computational Complexity
In Section III-E7, we discussed the scaling of the compu-
tational complexity of the proposed self-tuning BP algorithm
with respect to various parameters. We now present a partial
experimental verification of that discussion, namely regarding
the scaling with respect to the number of modes J and the
number of detection probabilitiesQ used by the algorithm. Our
simulations using different values of J and Q always consider
the first scenario, where three targets switch between the DMs
50 100 150 200 250 300 350 400
0
0.2
0.4
0.6
0.8
1
True (sensor N)
True (sensor S)
True (sensor E)
Estimated (sensor N)
Estimated (sensor S)
Estimated (sensor E)
n
Est
imat
edd
etec
tio
np
rob
abil
ity
Fig. 11. Empirical mean and standard deviation of the estimated detectionprobabilities for target 2 at the three sensors, for the third scenario. The shadedareas indicate the standard deviations.
TABLE IIAVERAGE RUNTIMES (IN SECONDS) FOR DIFFERENT VALUES OF J AND
Q.
Q
2 10 100
J
1 7.9 15.5 103
2 27 59 406
3 55 120 838
ξ1 and ξ2 with σ21= 0.012 and σ2
2 =0.52, respectively, and the
true detection probability is q(s)n,k= 0.8. However, the number
of modes used by the algorithm, J , is 1, 2, or 3. For J =1,
we consider a clairvoyant version of the proposed algorithm
that always knows whether ξ1 or ξ2 is in force. (We note that
for J=1, this clairvoyant version has the same complexity as
the algorithm using a fixed mode.) For J = 2, the algorithm
uses DMs ξ1 and ξ2, and for J=3, it uses ξ1 and ξ2 plus an
additional DM ξ3 with σ23 = 0.12. The number of detection
probabilities used by the algorithm, Q, is 2, 10, or 100. We
performed 20 simulation runs for each combination of J and
Q. The average runtimes using a MATLAB implementation
on an Intel Core i7-8705G [email protected] GHz (single core)
are reported in Table II. The increase with J is roughly
consistent with the quadratic scaling described in Section
III-E7. However, the increase with Q is seen to be less than
linear, rather than quadratic. This is because the considered
values of Q are too low for the operations involving the
detection probabilities q(s)n,k to provide a dominant contribution
to the overall computational complexity. However, values of
Q larger than 100 are not relevant since such a fine-grained
quantization of the detection probability does not improve the
tracking performance.
VI. EXPERIMENTAL RESULTS FOR REAL DATA
Next, we assess the performance of the proposed self-tuning
BP algorithm in a real-case scenario with real measurements.
A. Experiment Setup
The measurements are part of a 25-day data set collected
between May 8 and June 4, 2009 by two high-frequency sur-
14
TABLE IIIPARAMETER VALUES USED IN THE REAL DATA EXPERIMENT.
Parameter Value Description
∆T 16 s or 33.28 s Duration of time step
σr 100 m Range standard deviation
σb 1.5◦ Bearing standard deviation
σr 0.1 m/s Range rate standard deviation
µ(s) 10 Mean number of false alarms
T 5000 Number of particles
p(s) 0.999 Survival probability
p(b) 0.08 Birth probability
Pth 0.65 Detection threshold
K 80 or 30 Number of PTs
face wave (HFSW) radars installed on the island of Palmaria
(IP) near La Spezia and in San Rossore Park (SRP) near Pisa,
on the Italian coast of the Ligurian Sea. Each measurement
z(s)n,m consists of range, bearing, and range rate. It is modeled
as (cf. (34))
z(s)n,m =
‖xn,k−p(s)‖
φ(xn,k,p(s))
(xn,k−p(s))T ˙xn,k
‖xn,k−p(s)‖
+ v(s)n,m ,
where v(s)n,m ∼ N
(
0, diag(σ2r , σ
2b , σ
2r ))
is a sequence of 3D
Gaussian random vectors that is iid across n, m, and s.The false-alarm pdf fFA
(
z(s)n,m
)
is chosen uniform on the
surveillance region R, which is the intersection of the fields-
of-view of the two radar stations. The proposed algorithm uses
two nearly-constant velocity DMs ξ1 and ξ2 with σ21= 0.0012
and σ22 = 0.012, respectively and transition probabilities
[L]1,1 = [L]2,2 = 0.985 and [L]1,2 = [L]2,1 = 0.015.
The detection probabilities q(s)n,k are considered unknown with
values and transition probabilities as described in Section V-A.
Further model parameters are listed in Table III.
As ground truth information, we use data about existing
targets (vessels) that are provided by the Automatic Identifi-
cation System (AIS) [24]. AIS reports contain both dynamic
information (latitude, longitude, course over ground, speed
over ground, and time, all with GPS accuracy) and static
information (including vessel type and dimension). However,
vessels below a certain gross tonnage and military vessels are
not reported, and thus no ground truth information is available
for them. We consider any estimated trajectory without a corre-
sponding AIS trajectory as false. Furthermore, since AIS data
and radar measurements are not temporally aligned and AIS
data are more frequent than radar measurements, we use cubic
interpolation to estimate AIS data at the time instants of the
radar measurements. We then use the following procedure to
find the associations between the AIS trajectories and the esti-
mated trajectories. Let {xAISn,j}j∈DAIS
nbe the set of AIS-reported
positions at time n, with DAISn denoting the set of vessels
reporting their position at time n. Similarly, let{
ˆxn,i}
i∈Dn
be the set of estimated positions of the vessels declared to exist
by the algorithm at time n, with Dn denoting the set of PTs
k such that p(rn,k=1|z)≥Pth. At each time n, we associate
with each AIS position xAISn,j , j ∈ DAIS
n the nearest estimated
position ˆxn,i(j) that is located in a circular search region
of radius 200 m, C(
xAISn,j
)
,{
x∈R :∥
∥x− xAISn,j
∥
∥ ≤ 200 m}
,
i.e., i(j) = argmini∈Dn: ˆxn,i∈C(xAIS
n,j)
∥
∥ˆxn,i − xAISn,j
∥
∥. Here, an
estimated position that has already been associated with an
AIS position is no longer considered for further associations
because each estimated position can only be associated with
a single AIS position.
B. Results
Fig. 12 depicts the AIS trajectories and the trajectories
estimated in the surveillance region by the proposed algorithm
from measurements produced by the two HFSW radars during
ten hours. The number of PTs was set to K=80. It can be seen
that for almost all AIS trajectories, there is a corresponding
estimated trajectory. However, some estimated trajectories are
slightly offset from the corresponding AIS trajectory, due to
a systematic bias. A potential source of this bias is range-
Doppler coupling, which is a detrimental effect in HFSW
radars resulting in a bias in the measured range that is
proportional to the radial velocity of a target [25].
Next, to compare the performance of the proposed algorithm
with that of the nonadaptive BP algorithm [12], we use a subset
of the measurements corresponding to about 5 h and a confined
part of the surveillance region. The number of PTs in this
smaller region is now chosen as K= 30 for both algorithms.
The nonadaptive algorithm uses DM ξ1 and detection proba-
bility q(s)n,k=0.5. Fig. 13 shows that the trajectories estimated
by the nonadaptive algorithm are more fragmented than those
estimated by the proposed algorithm; this is probably due to
the low density of radar measurements in the considered part
of the surveillance region. Also, the nonadaptive algorithm
produces a higher number of false detections than the proposed
algorithm. Furthermore, as highlighted by the magenta circle,
the nonadaptive algorithm does not track one of the two sharp
turns of one of the trajectories with good accuracy. This can
be explained by the low driving process variance of DM ξ1.
In Table IV, we compare the performance of the two
algorithms in terms of the normalized time on target (ToT)
and the false alarm rate (FAR) [2]. The ToT is defined as the
average ratio—averaged over all the vessels sending AIS in-
formation within the confined part of the surveillance region—
between the total number of estimated positions successfully
associated with an AIS trajectory during the recording period
and the duration (number of time steps) of that AIS trajectory.
That is, ToT , (1/MAIS)∑MAIS
v=1 Nv/Nv, where MAIS is the
total number of vessels sending AIS information within the
confined part of the surveillance region, Nv is the total number
of estimated positions associated with the vth vessel, and Nv is
the true (according to AIS information) trajectory duration for
the vth vessel. Ideally, the ToT would be 100%, which occurs
if Nv = Nv for all AIS trajectories. The FAR is defined as
FAR ,MFA/(TRA), where MFA is the number of false alarms
within the recording period, TR is the duration of the recording
period (in h), and A is the area of the surveillance region (in
km2). Here, a false alarm is defined as an estimated position
that has not been associated with any AIS position. Ideally, the
15
AIS trajectories
Estimated trajectories
IP radar
SRP radar
44.2
44.0
43.8
43.6
43.4
9.2 9.4 9.6 9.8 10.0 10.2 10.4
20 km
La Spezia
Livorno
Longitude [deg]
Lat
itu
de
[deg
]
Fig. 12. AIS trajectories (red lines) and estimated trajectories (black lines) using measurements from the IP radar (blue triangle) and the SRP radar (greentriangle). The blue and green dots represent the measurements from the IP and SRP radar, respectively. (Map courtesy of Google)
AIS trajectories
Estimated trajectories43.95
43.90
43.85
43.80
9.45 9.50 9.55 9.60 9.65 9.70 9.75 9.80
5 km
Longitude [deg]
(a)
Lat
itu
de
[deg
]
AIS trajectories
Estimated trajectories43.95
43.90
43.85
43.80
9.45 9.50 9.55 9.60 9.65 9.70 9.75 9.80
5 km
Longitude [deg]
(b)
Lat
itu
de
[deg
]
Fig. 13. AIS trajectories (red lines) and estimated trajectories (black lines) in a confined part of the surveillance region: (a) nonadaptive algorithm (using DM
ξ1 and detection probability q(s)n,k
=0.5), (b) proposed algorithm. The magenta circle highlights one of the two sharp turns of one of the trajectories.
FAR would be zero, which occurs if there are no false alarms.
Table IV shows the ToT and FAR that were obtained with the
nonadaptive algorithm using different values of the detection
probability q(s)n,k and with the proposed algorithm. These ToT
and FAR values are the result of averaging over 50 evaluation
runs, which differ in the particles. The nonadaptive algorithm
achieves the highest ToT (66.6%)—but, simultaneously, also
the highest FAR (1.7 h−1km−2)—for q(s)n,k=0.2, and the low-
est ToT (56.9%)—but also the lowest FAR (0.4 h−1km−2)—
for q(s)n,k= 0.8. In contrast, the proposed self-tuning algorithm
achieves both an acceptable ToT of 62.4% and a low FAR of
0.4 h−1km−2.
TABLE IVTOT AND FAR FOR THE NONADAPTIVE ALGORITHM (USING A FIXED
VALUE OF q(s)n,k
) AND THE PROPOSED SELF-TUNING ALGORITHM.
q(s)n,k ToT [%] FAR [h−1km−2]
0.2 66.6 1.7
0.3 66.1 1.2
0.4 65.3 0.9
0.5 59.9 0.7
0.6 58.1 0.6
0.7 58.2 0.5
0.8 56.9 0.4
self-tuning 62.4 0.4
16
VII. CONCLUSIONS
We proposed a belief propagation (BP) message passing
framework for the development of “self-tuning” algorithms for
multisensor-multitarget tracking. These algorithms are adap-
tive with respect to time-varying model parameters such as
the detection probabilities of the sensors, the clutter intensity,
or the dynamic model indices. In our approach, the evolution
of the model parameters is described by a Markov chain,
and the parameters are tracked together with the target states
using a BP-based tracking methodology. This methodology
provides a principled way to reduce complexity by exploiting
conditional statistical independencies, which results in quasi-
optimal Bayesian multisensor-multitarget tracking algorithms
with excellent scalability.
As a concrete example, we developed a self-tuning multi-
sensor-multitarget tracking algorithm for the case of unknown,
time-varying detection probabilities and dynamic model in-
dices. Simulation results showed that our algorithm is able
to track multiple targets during coordinated turns and for
range-dependent detection probabilities, and that it achieves a
significant reduction of the time-averaged mean optimal sub-
pattern assignment (MOSPA) error relative to the nonadaptive
BP-based algorithm from [12] (e.g., 181 m versus 221 m). We
also validated our algorithm with real data collected from two
high-frequency surface wave radars. Here, the algorithm was
observed to achieve a normalized time on target (ToT) of
62.4% and a false alarm rate (FAR) of 0.4 h−1km−2, which
constitutes a much better ToT–FAR compromise than that
provided by the nonadaptive algorithm.
A promising direction for future research is the extension
of our self-tuning BP framework to applications where objects
may generate more than one measurement. Such applications
include localization [26]–[32] and tracking of extended targets
[33]–[35].
ACKNOWLEDGMENT
The authors would like to thank Prof. M. Z. Win for
stimulating discussions.
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