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: giovanni.soldi@cmre.nato.int, paolo.braca@cmre.nato.int). F.Meyer and Moe Z. Win are with the Laboratory for Information andDecision Systems, Massachusetts Institute of Technology, Cambridge, MA,USA (e-mail: fmeyer@mit.edu, moewin@mit.edu). F. Hlawatsch is withthe Institute of Telecommunications, TU Wien, Vienna, Austria (e-mail:franz.hlawatsch@tuwien.ac.at).

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 CPU@3.10 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.

REFERENCES

[1] Y. Bar-Shalom and X.-R. Li, Multitarget-Multisensor Tracking: Princi-ples and Techniques. Storrs, CT, USA: Yaakov Bar-Shalom, 1995.

[2] S. Maresca, P. Braca, J. Horstmann, and R. Grasso, “Maritime surveil-lance using multiple high-frequency surface-wave radars,” IEEE Trans.Geosci. Remote Sens., vol. 52, no. 8, pp. 5056–5071, Aug. 2014.

[3] Y. Bar-Shalom, T. Kirubarajan, and X.-R. Li, Estimation with Applica-tions to Tracking and Navigation. New York, NY, USA: Wiley, 2002.

[4] E. Mazor, A. Averbuch, Y. Bar-Shalom, and J. Dayan, “Interactingmultiple model methods in target tracking: A survey,” IEEE Trans.Aerosp. Electron. Syst., vol. 34, no. 1, pp. 103–123, Jan. 1998.

[5] K. G. Jamieson, M. R. Gupta, and D. W. Krout, “Sequential Bayesianestimation of the probability of detection for tracking,” in Proc. FUSION,Seattle, WA, USA, July 2009, pp. 641–648.

[6] W. R. Blanding, P. K. Willett, Y. Bar-Shalom, and S. Coraluppi,“Multisensor track management for targets with fluctuating SNR,” IEEETrans. Aerosp. Electron. Syst., vol. 45, no. 4, pp. 1275–1292, Oct. 2009.

[7] R. P. S. Mahler, B. T. Vo, and B. N. Vo, “CPHD filtering with unknownclutter rate and detection profile,” in Proc. FUSION, Chicago, IL, USA,July 2011.

[8] B. T. Vo, B. N. Vo, R. Hoseinnezhad, and R. P. S. Mahler, “Robustmulti-Bernoulli filtering,” IEEE J. Sel. Topics Signal Process., vol. 7,no. 3, pp. 399–409, June 2013.

[9] B. Ristic, D. Clark, B. N. Vo, and B. T. Vo, “Adaptive target birthintensity for PHD and CPHD filters,” IEEE Trans. Aerosp. Electron.Syst., vol. 48, no. 2, pp. 1656–1668, April 2012.

[10] G. Papa, P. Braca, S. Horn, S. Marano, V. Matta, and P. Willett, “Mul-tisensor adaptive Bayesian tracking under time-varying target detectionprobability,” IEEE Trans. Aerosp. Electron. Syst., vol. 52, no. 5, pp.2193–2209, Oct. 2016.

[11] E. Leitinger, S. Grebien, X. Li, F. Tufvesson, and K. Witrisal, “On theuse of MPC amplitude information in radio signal based SLAM,” inIEEE Stat. Signal Process. Workshop, Freiburg, Germany, Jun. 2018,pp. 633–637.

[12] F. Meyer, P. Braca, P. Willett, and F. Hlawatsch, “A scalable algorithmfor tracking an unknown number of targets using multiple sensors,” IEEETrans. Signal Process., vol. 65, no. 13, pp. 3478–3493, Jul. 2017.

[13] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs andthe sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp.498–519, Feb. 2001.

[14] H.-A. Loeliger, J. Dauwels, J. Hu, S. Korl, L. Ping, and F. R. Kschis-chang, “The factor graph approach to model-based signal processing,”Proc. IEEE, vol. 95, no. 6, pp. 1295–1322, Jun. 2007.

[15] F. Meyer, P. Braca, F. Hlawatsch, M. Micheli, and K. D. LePage,“Scalable adaptive multitarget tracking using multiple sensors,” in Proc.IEEE GLOBECOM, Washington D.C., USA, Dec. 2016.

[16] G. Soldi and P. Braca, “Online estimation of unknown parametersin multisensor-multitarget tracking: A belief propagation approach,” inProc. FUSION-18, Cambridge, UK, Jul. 2018, pp. 2155–2161.

[17] F. Meyer, T. Kropfreiter, J. L. Williams, R. A. Lau, F. Hlawatsch,P. Braca, and M. Z. Win, “Message passing algorithms for scalablemultitarget tracking,” Proc. IEEE, vol. 106, no. 2, pp. 221–259, Feb.2018.

[18] R. P. S. Mahler, Statistical Multisource-Multitarget Information Fusion.Norwood, MA, USA: Artech House, 2007.

[19] J. L. Williams and R. Lau, “Approximate evaluation of marginal as-sociation probabilities with belief propagation,” IEEE Trans. Aerosp.Electron. Syst., vol. 50, no. 4, pp. 2942–2959, Oct. 2014.

[20] H. V. Poor, An Introduction to Signal Detection and Estimation. NewYork, NY, USA: Springer, 1994.

[21] A. Doucet, N. de Freitas, and N. Gordon, Sequential Monte CarloMethods in Practice. New York, NY, USA: Springer, 2001.

[22] D. Schuhmacher, B. T. Vo, and B. N. Vo, “A consistent metric forperformance evaluation of multi-object filters,” IEEE Trans. SignalProcess., vol. 56, no. 8, pp. 3447–3457, Aug. 2008.

[23] M. A. Richards, W. M. Melvin, J. A. Scheer, and W. A. Holm, Principlesof Modern Radar: Basic Principles. Raleigh, NC, USA: SciTechPublishing, 2010.

[24] B. J. Tetreault, “Use of the automatic identification system (AIS) formaritime domain awareness (MDA),” in Proc. MTS/IEEE OCEANS-05,vol. 2, Washington, D.C., USA, Sep. 2005, pp. 1590–1594.

[25] L. Bruno, P. Braca, J. Horstmann, and M. Vespe, “Experimental evalu-ation of the range-Doppler coupling on HF surface wave radars,” IEEEGeosci. Remote Sens. Letters, vol. 10, no. 4, pp. 850–854, Jul. 2013.

[26] M. Z. Win, F. Meyer, Z. Liu, W. Dai, S. Bartoletti, and A. Conti,“Efficient multi-sensor localization for the Internet-of-Things,” IEEESignal Process. Mag., vol. 35, no. 5, pp. 153–167, Sep. 2018.

[27] S. Safavi, U. A. Khan, S. Kar, and J. M. F. Moura, “Distributedlocalization: A linear theory,” Proc. IEEE, vol. 106, no. 7, pp. 1204–1223, Jul. 2018.

[28] S. Mazuelas, A. Conti, J. C. Allen, and M. Z. Win, “Soft rangeinformation for network localization,” IEEE Trans. Signal Process.,vol. 66, no. 12, pp. 3155–3168, Jun. 2018.

[29] W. Dai, Y. Shen, and M. Z. Win, “A computational geometry frameworkfor efficient network localization,” IEEE Trans. Inf. Theory, vol. 64,no. 2, pp. 1317–1339, Feb. 2018.

[30] R. Niu, A. Vempaty, and P. K. Varshney, “Received signal strength basedlocalization in wireless sensor networks,” Proc. IEEE, vol. 106, no. 6,pp. 1166 – 1182, Jun. 2018.

[31] T. Wang, Y. Shen, A. Conti, and M. Z. Win, “Network navigation withscheduling: Error evolution,” IEEE Trans. Inf. Theory, vol. 63, no. 11,pp. 7509–7534, Nov. 2017.

[32] M. Z. Win, Y. Shen, and W. Dai, “A theoretical foundation of networklocalization and navigation,” Proc. IEEE, vol. 106, no. 7, pp. 1136–1165, Jul. 2018, special issue on Foundations and Trends in LocalizationTechnologies.

[33] J. W. Koch, “Bayesian approach to extended object and cluster trackingusing random matrices,” IEEE Trans. Aerosp. Electron. Syst., vol. 44,no. 3, pp. 1042–1059, Jul. 2008.

[34] G. Vivone, P. Braca, K. Granstrom, and P. Willett, “Multistatic Bayesianextended target tracking,” IEEE Trans. Aerosp. Electron. Syst., vol. 52,no. 6, pp. 2626–2643, Dec. 2016.

[35] F. Meyer, Z. Liu, and M. Z. Win, “Scalable probabilistic data associationwith extended objects,” in Proc. IEEE ICC-19, Shanghai, China, May2019.