1 Poisson multi-Bernoulli mixture filter with general target-generated measurements and arbitrary clutter

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Poisson multi-Bernoulli mixture ﬁlter with general

target-generated measurements and arbitrary clutter

Ángel F. García-Fernández, Yuxuan Xia, Lennart Svensson

Abstract—This paper shows that the Poisson multi-Bernoulli

mixture (PMBM) density is a multi-target conjugate prior for

general target-generated measurement distributions and arbi-

trary clutter distributions. That is, for this multi-target mea-

surement model and the standard multi-target dynamic model

with Poisson birth model, the predicted and ﬁltering densities are

PMBMs. We derive the corresponding PMBM ﬁltering recursion.

Based on this result, we implement a PMBM ﬁlter for point-

target measurement models and negative binomial clutter density

in which data association hypotheses with high weights are

chosen via Gibbs sampling. We also implement an extended

target PMBM ﬁlter with clutter that is the union of Poisson-

distributed clutter and a ﬁnite number of independent clutter

sources. Simulation results show the beneﬁts of the proposed

ﬁlters to deal with non-standard clutter.

Index Terms—Multi-target ﬁltering, Poisson multi-Bernoulli

mixtures, Gibbs sampling, arbitrary clutter.

I. INTRODUCTION

Multi-target ﬁltering consists of estimating the current states

of an unknown and variable number of targets based on noisy

sensor measurements up to the current time step. It is a key

component of numerous applications, for example, defense [1],

automotive systems [2] and air trafﬁc control [3]. Multi-target

ﬁltering is usually addressed using probabilistic modelling,

with the main approaches being multiple hypothesis tracking

[4], joint probabilistic data association [5] and random ﬁnite

sets [6].

In detection-based multi-target ﬁltering, sensors collect

scans of data that may contain target-generated measurements

as well as clutter, which refers to undesired detections. For

example, in radar, clutter can be caused by reﬂections from the

environment, such as terrain, sea and rain, or other undesired

objects [7], [8]. Multi-target ﬁlters applied to real data must

then account for these clutter measurements for suitable per-

formance, for instance, sea clutter and land clutter in radar data

[9], false detections in image data [10], [11], false detections

in audio visual data [12], and underwater clutter in active sonar

data [13], [14].

In the standard detection model, clutter is modelled as

a Poisson point process (PPP) [6]. The PPP is motivated

by the fact that, for sufﬁciently high sensor resolution and

independent clutter reﬂections in each sensor cell, the detection

A. F. García-Fernández is with the Department of Electrical Engineer-

ing and Electronics, University of Liverpool, Liverpool L69 3GJ, United

Kingdom (angel.garcia-fernandez@liverpool.ac.uk). He is also with the

ARIES Research Centre, Universidad Antonio de Nebrija, Madrid, Spain.

Y. Xia and L. Svensson are with the Department of Electrical Engineering,

Chalmers University of Technology, SE-412 96 Gothenburg, Sweden (ﬁrst-

name.lastname@chalmers.se).

process can be accurately approximated as a PPP [15]. The

PPP is also convenient mathematically and it can be char-

acterised by its intensity function on the single-measurement

space.

For the point-target measurement model, PPP clutter and

the standard multi-target dynamic model with PPP birth, the

posterior is a Poisson multi-Bernoulli mixture (PMBM) [16],

[17]. The PMBM consists of the union of a PPP, representing

undetected targets, and an independent multi-Bernoulli mix-

ture (MBM) representing targets that have been detected at

some point and their data association hypotheses. The posterior

density is also a PMBM for the extended target model [18] and

for a general target-generated measurement model [19], both

with PPP clutter and the standard dynamic model. Extended

target modelling is required when, due to the sensor resolution

and the target extent, a target may generate more than one de-

tection at each time step [20]. This makes the data association

problem considerably more challenging than for point targets.

The general target generated-measurement model includes the

point and extended target cases as particular cases, and can for

example be used when there can be simultaneous point and

extended targets in a scenario [19].

If the birth model is multi-Bernoulli instead of PPP, the

posterior density in the above cases is an MBM, which is

obtained by setting the Poisson intensity of the PMBM ﬁlter

to zero and by adding the Bernoulli components of the birth

process in the prediction step. The MBM ﬁlter can also be

written in terms of Bernoulli components with deterministic

target existence, which results in the MBM01 ﬁlter [17, Sec.

IV]. It is also possible to add unique labels to the target

states in the MBM and MBM01 ﬁlters. The (labelled) MBM01

ﬁltering recursions are analogous to the δ-generalised labelled

multi-Bernoulli (δ-GLMB) ﬁltering recursions [21], [22].

All the above recursions to compute the posterior assume

PPP clutter, but other clutter distributions are also important

in various contexts, for instance, to model bursts of radar sea

clutter [23]. For general target-generated measurements and

arbitrary clutter density, the Bernoulli ﬁlter was proposed in

[24], and the probabilistic hypothesis density (PHD) ﬁlter,

which provides a PPP approximation to the posterior, was

derived in [25]. The corresponding PHD ﬁlter update depends

on the set derivative of the logarithm of the probability

generating functional (PGFL) of the clutter process. A version

of this PHD ﬁlter update, written in terms of densities, which

is more suitable for implementation than [25], is provided

in [26]. There are also other PHD ﬁlter variants with non-

PPP clutter for point targets, for example, a PHD ﬁlter with

negative binomial clutter [27], a second order PHD ﬁlter with

arXiv:2210.12983v2 [stat.AP] 24 May 2023

Panjer clutter [28], and a linear-complexity cumulant-based

ﬁlter for clutter described by its intensity and second-order

cumulant [29], [30]. A cardinalised PHD ﬁlter with general

target-generated measurements and arbitrary clutter is derived

in [31]. Another approach is to estimate the states of Bernoulli

clutter generators by including them in the multi-target state

[6], [32].

This paper shows that for general target-generated mea-

surements and arbitrary clutter density, the posterior is also a

PMBM and we derive the ﬁltering recursion. This contribution

extends the family of closed-form recursions to calculate

the posterior to a more general detection-based measurement

model. As a direct result, this paper also derives the corre-

sponding (labelled or not) MBM and MBM01 ﬁlters, including

δ-GLMB ﬁlters. We also directly obtain the Poisson multi-

Bernoulli (PMB) ﬁlter, which can be derived by Kullback-

Leibler divergence minimisation on a target space augmented

with auxiliary variables [16], [33]. We also propose a Gibbs

sampling algorithm for selecting global hypotheses with high

weights [21], [34]–[36] for the PMBM ﬁlter for point targets

and clutter that is an independent and an identically distributed

(IID) cluster process [6] with arbitrary cardinality distribution.

We show via simulation results the advantages of the proposed

ﬁlter in two scenarios: one scenario for point targets and

IID clutter with negative binomial cardinality distribution, and

another scenario for extended targets where there are a ﬁnite

number of clutter sources.

The rest of this paper is organised as follows. Section

II introduces the models and an overview of the PMBM

posterior. The PMBM ﬁlter update is presented in Section III.

The Gibbs sampling data association algorithm for point-target

PMBM ﬁltering with arbitrary clutter is proposed in Section

IV. Finally, simulation results and conclusions are provided in

Sections V and VI.

II. MODELS AND OVERVIEW OF THE PMBM POSTERIOR

This section presents the dynamic and measurement models

in Section II-A, and an overview of the PMBM posterior in

Section II-B, with its global hypotheses in Section II-C.

A. Models

A target state is denoted by xand it contains the variables

that describe its current dynamics, such as position and veloc-

ity, and maybe other attributes, such as orientation and extent.

The state x∈ X , where Xis a locally compact, Hausdorff

and second-countable (LCHS) space [6]. The set of targets at

time kis Xk∈ F (X), where F(X)represents the set of

ﬁnite subsets of X.

Targets move according to the standard multi-target dynamic

model [6]. Given Xk, each target x∈Xksurvives with

probability pS(x)and moves with a transition density g(· |x),

or disappears with probability 1−pS(x). New targets are

born at time step kaccording to an independent Poisson point

process (PPP) with intensity λB

k(·).

A measurement state z∈ Z contains a sensor measurement.

The set Zk∈ F (Z)of measurements at time kis the union of

target-generated measurements and independent clutter mea-

surements, modelled by

•Each target x∈Xkgenerates a set of measurements with

density f(·|x).

•The set of clutter measurements at each time step has

density c(·).

•Given Xk, the measurements generated by different tar-

gets are independent of each other and of the clutter

measurements.

It should be noted that f(·|x)is a general density for

target-generated measurements and we can accommodate any

probabilistic model for target-generated measurements. The

probability that at least one measurement is generated from the

target (effective probability of detection [37]) is 1−f(∅|x).

For example, the standard point target model in which a

target xis detected with probability pD(x)and generates a

measurement with density l(·|x)[6], is obtained by

f(Z|x) = 









1−pD(x)Z=∅

pD(x)l(z|x)Z={z}

0|Z|>1.

(1)

In the standard extended target model, we can receive more

than one measurement from a target. Speciﬁcally, a target xis

detected with probability pD(x)and, if detected, it generates a

PPP measurement with intensity γ(x)l(·|x), where l(·|x)is a

single-measurement density and γ(x)is the expected number

of measurements [20], [38]. It is obtained by

f(Z|x) = (1−pD(x) + pD(x)e−γ(x)Z=∅

pD(x)γ|Z|(x)e−γ(x)Qz∈Zl(z|x)|Z|>0.

(2)

We can also combine both (1) and (2) to model coexisting

point and extended targets, or use any of the probabilistic

extended target models in [20]. In addition, the choice of

f(·|x)can also take into account reﬂectivity models and the

propagation conditions in the environment [8], [39]–[41].

B. PMBM posterior

We will show in this paper that, for the dynamic and

measurement models in Section II-A, the predicted and pos-

terior densities are PMBMs. That is, given the sequence of

measurements (Z1, ..., Zk′), the density fk|k′(·)of Xkwith

k′∈ {k−1, k}is

fk|k′(Xk) = X

Y⊎W=Xk

k|k′(Y)fmbm

k|k′(W),(3)

k|k′(Xk) = e−Rλk|k′(x)dx Y

x∈Xk

λk|k′(x),(4)

fmbm

k|k′(Xk) = X

a∈Ak|k′

k|k′X

⊎nk|k′

l=1 Xl=Xk

nk|k′

i=1

fi,ai

k|k′Xi,

(5)

where λk|k′(·)is the intensity of the PPP fp

k|k′(·), representing

undetected targets, and fmbm

k|k′(·)is a multi-Bernoulli mixture

representing targets that have been detected at some point

up to time step k′. The sum in (3) is the convolution sum,

which implies that the PPP and MBM are independent [6].

The symbol ⊎denotes the disjoint union and the sum is taken

over all mutually disjoint (and possibly empty) sets Yand W

whose union is Xk.

The MBM in (5) has nk|k′Bernoulli components (potential

targets), each with hi

k|k′local hypotheses. There is a local

hypothesis ai∈n1, ..., hi

k|k′ofor each Bernoulli i > 0. To

handle arbitrary clutter, we also introduce a local hypothesis

a0∈n1, ..., h0

k|k′ofor clutter. Each aiis an index that

associates the i-th Bernoulli, for i≥0, or the clutter, for

i= 0, to a speciﬁc sequence of subsets of the measurement

set, see Section II-C. A global hypothesis is denoted by

a=a0, a1, ..., ank|k′∈ Ak|k′, where Ak|k′is the set of

global hypotheses, see Section II-C. The weight of global

hypothesis ais wa

k|k′and meets

k|k′∝

nk|k′

i=0

wi,ai

k|k′(6)

where wi,ai

k|k′is the weight of the i-th Bernoulli compo-

nent, or clutter if i= 0, with local hypothesis ai, and

Pa∈Ak|k′wa

k|k′= 1. A difference with PMBM ﬁlters with

PPP clutter [16], [18], [19] is that global hypotheses and

weights explicitly consider clutter, with index i= 0.

The i-th Bernoulli component with local hypothesis aihas

a density

fi,ai

k|k′(X) = 









1−ri,ai

k|k′X=∅

ri,ai

k|k′pi,ai

k|k′(x)X={x}

0 otherwise

(7)

where ri,ai

k|k′is the probability of existence and pi,ai

k|k′(·)is the

single-target density. It should be noted that in this paper

we use the following nomenclature for Bernoulli components,

densities and local hypotheses: Each Bernoulli component,

which is indexed by iand is initiated by a non-empty subset of

measurements at a given time step (see Section III), has hi

k|k′

local hypotheses, each with an associated Bernoulli density,

indexed by i, ai. Then, the total number of Bernoulli densities

in (5) across all global hypotheses is Pnk|k′

i=1 hi

k|k′, and the

number of multi-Bernoulli densities is |Ak|k′|, which denotes

the cardinality of Ak|k′.

C. Set of global hypotheses

We proceed to describe the set Ak|k′of global hypotheses.

We denote the measurement set at time step kas Zk=

z1

k, ..., zmk

k. We refer to measurement zj

kusing the pair

(k, j)and the set of all such measurement pairs up to and

including time step kis denoted by Mk. Then, a local

hypothesis aifor i=0,1, ..., nk|k′has an associated set

of measurement pairs denoted as Mi,ai

k⊆ Mk. The set Ak|k′

of all global hypotheses meets

Ak|k′=na0, a1, ..., ank|k′:ai∈n1, ..., hi

k|k′o∀i,

nk|k′

[

i=0

Mi,ai

k′=Mk′,Mi,ai

k′∩ Mj,aj

k′=∅,∀i̸=j).

That is, each measurement must be assigned to a local hypoth-

esis, and there cannot be more than one local hypothesis with

the same measurement. In this paper, we construct the (trees

of) local hypotheses recursively, and we allow for more than

one measurement to be associated to the same local hypothesis

at the same time step, i.e., Mi,ai

kmay contain zero, one or

more measurements. At time step zero, the ﬁlter is initiated

with n0|0= 0,w0,1

0|0= 1,h0

0|0= 1, and M0,1

0=∅.

III. GENERAL PMBM FILTER

This section presents the PMBM ﬁlter for general target-

generated measurements and arbitrary clutter, with models

described in Section II-A. We consider the standard dynamic

model so the prediction step is the standard PMBM prediction

step [16], [17]. The update is presented in Section III-A. We

provide a discussion and extension of the result to other ﬁlter

variants in Sections III-B and III-C. The PMBM for point

targets and arbitrary clutter is explained in Section III-D.

Finally, an analysis of the number of global hypotheses for

different PMBM ﬁlters is provided in Section III-E.

A. General PMBM update

Given two real-valued functions a(·)and b(·)on the target

space, we denote their inner product as

⟨a, b⟩=Za(x)b(x)dx. (8)

Then, the update of the predicted PMBM fk|k−1(·)with Zk

is given in the following theorem.

Theorem 1. Assume the predicted density fk|k−1(·)is a

PMBM of the form (3). Then, the updated density fk|k(·)

with set Zk=z1

k, ..., zmk

kis a PMBM with the following

parameters. The number of Bernoulli components is nk|k=

nk|k−1+ 2mk−1. The intensity of the PPP is

λk|k(x) = f(∅|x)λk|k−1(x).(9)

Let Z1

k, ..., Z2mk−1

kbe the non-empty subsets of Zk. The up-

dated number of local clutter hypotheses is h0

k|k= 2mkh0

k|k−1

such that a new local clutter hypothesis is included for each

previous local clutter hypothesis and either a misdetection

or an update with a non-empty subset of Zk. The updated

local clutter hypotheses with no clutter at time step k,a0∈

n1, ..., h0

k|k−1o, have parameters M0,a0

k=M0,a0

k−1,

w0,a0

k|k=w0,a0

k|k−1c(∅).(10)

For a previous local clutter hypothesis ea0∈n1, ..., h0

k|k−1o

in the predicted density, the new local clutter hypothesis

generated by a set Zj

khas a0=ea0+h0

k|k−1j,

M0,a0

k=M0,ea0

k−1∪n(k, p) : zp

k∈Zj

ko,(11)

w0,a0

k|k=w0,a0

k|k−1cZj

k.(12)

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1Poissonmulti-Bernoullimixturefilterwithgeneraltarget-generatedmeasurementsandarbitraryclutterÁngelF.García-Fernández,YuxuanXia,LennartSvenssonAbstract—ThispapershowsthatthePoissonmulti-Bernoullimixture(PMBM)densityisamulti-targetconjugatepriorforgeneraltarget-generatedmeasurementdistributionsandarb...

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1 Poisson multi-Bernoulli mixture filter with general target-generated measurements and arbitrary clutter

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