IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4, APRIL 2008 1313
Bayesian Filtering With Random
Finite Set Observations
Ba-Tuong Vo, Ba-Ngu Vo, and Antonio Cantoni, Fellow, IEEE
Abstract—This paper presents a novel and mathematically
rigorous Bayes’ recursion for tracking a target that generates
multiple measurements with state dependent sensor field of
view and clutter. Our Bayesian formulation is mathematically
well-founded due to our use of a consistent likelihood function
derived from random finite set theory. It is established that under
certain assumptions, the proposed Bayes’ recursion reduces to
the cardinalized probability hypothesis density (CPHD) recursion
for a single target. A particle implementation of the proposed
recursion is given. Under linear Gaussian and constant sensor field
of view assumptions, an exact closed-form solution to the proposed
recursion is derived, and efficient implementations are given.
Extensions of the closed-form recursion to accommodate mild
nonlinearities are also given using linearization and unscented
transforms.
Index Terms—Bayesian filtering, CPHD filter, Gaussian sum
filter, Kalman filter, particle filter, PHD filter, point processes,
random finite sets, target tracking.
I. INTRODUCTION
T
HE objective of target tracking is to estimate the state of
the target from measurement sets collected by a sensor at
each time step. This is a challenging problem since the target can
generate multiple measurements which are not always detected
by the sensor, and the sensor receives a set of spurious measure-
ments (clutter) not generated by the target. Many existing tech-
niques for handling this problem rest on the simplifying assump-
tions that the target generates at most one measurement and that
the sensor field of view is constant. Such assumptions are not
realistic, for example, in extended object tracking or tracking in
the presence of electronic counter measures, which are increas-
ingly becoming important due to high-resolution capabilities of
modern sensors. Nonetheless, these assumptions have formed
the basis of a plethora of works e.g., the multiple hypothesis
tracker (MHT) [1], [2], the probabilistic data association (PDA)
filter [3], the Gaussian mixture filter [4], the integrated PDA
(IPDA) filter [5], and their variants. However, such techniques
are not easily adapted to accommodate multiple measurements
Manuscript received February 28, 2007; revised July 31, 2007. The associate
editor coordinating the review of this manuscript and approving it for publica-
tion was Dr. Subhrakanti Dey.
B.-T. Vo and A. Cantoni are with the Western Australian Telecommunica-
tions Research Institute, University of Western Australia, Crawley, WA 6009,
Australia (e-mail: vob@watri.org.au; cantoni@watri.org.au).
B.-N. Vo is with the Electrical Engineering Department, University of Mel-
bourne, Parkville, VIC 3010, Australia (e-mail: bv@ee.unimelb.edu.au).
Color versions of one or more of the figures in this paper are available online
at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TSP.2007.908968
generated by the target and state dependent field of view. More-
over, it is not clear how such techniques are mathematically con-
sistent with the Bayesian paradigm.
Tracking in the presence of detection uncertainty, data as-
sociation uncertainty and clutter can be elegantly cast in the
Bayesian filtering paradigm using random finite set (RFS)
modeling. While this idea has been advocated by various re-
searchers [6]–[8], Mahler’s random finite set approach (coined
as finite set statistics or FISST) is the first systematic Bayesian
framework for the study of multisensor multitarget tracking
[9]–[11]. Exciting advances and developments in the random
finite set approach have attracted substantial interest in re-
cent years, especially moment approximations for multitarget
tracking [10]–[15]. In the single-target realm, however, the RFS
approach has not been utilized to further advance single-target
tracking techniques, though connections with existing tech-
niques such as PDA and IPDA have been addressed [16].
To the best of the authors’ knowledge, this paper is the first
to use the RFS formalism to solve the problem of tracking a
target that can generate multiple measurements, in the presence
of detection uncertainty and clutter. In our Bayesian filtering
formulation, the collection of observations at any time is treated
as a set-valued observation which encapsulates the underlying
models of multiple target-generated measurements, state depen-
dent sensor field of view, and clutter. Central to the Bayes’ re-
cursion is the concept of a probability density. Since the obser-
vation space is now the space of finite sets, the usual Euclidean
notion of a density is not suitable. An elegant and rigorous no-
tion of a probability density needed for the Bayes’ recursion is
provided by RFS or point process theory [9], [10], [12], [13].
The contributions of this paper are as follows.
• A novel and mathematically rigorous Bayes’ recursion to-
gether with a particle implementation that accommodates
multiple measurements generated by the target, state de-
pendent field of view and clutter using RFS theory,
• A closed-form solution to the proposed recursion for linear
Gaussian single-target models with constant sensor field of
view, and extensions to accommodate mild nonlinearities
using linearization and unscented transforms.
• Under certain assumptions the proposed Bayes’ recursion
is shown to reduce to Mahler’s cardinalized probability
hypothesis density (CPHD) recursion [14] restricted to a
single target.
• Our approach is compared with conventional techniques
and is shown to be significantly better in terms of track loss
and localization performance.
In contrast to the traditional approaches [1]–[5], our proposed
recursion formally accommodates multiple measurements gen-
erated by the target, detection uncertainty and clutter, thereby
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1314 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4, APRIL 2008
providing a formal mechanism for handling effects such as
electronic counter measures and multipath reflections. Indeed,
assuming no clutter and that the target generates exactly one
measurement, the proposed recursion reduces to the usual
Bayes’ recursion and the particle filter implementation reduces
to the standard particle filter. Under additional linear Gaussian
assumptions, our closed-form recursion reduces to the cele-
brated Kalman filter (hence our extensions via linearization and
unscented transforms reduce to the extended and unscented
Kalman filters, respectively). In the case of a linear Gaussian
model with at most one target-generated measurement, constant
field of view, and uniform clutter, the proposed closed-form
recursion reduces to the Gaussian mixture filter given in [4].
Moreover, if at each time step, gating is performed and the
Gaussian mixture posterior density is collapsed to a single
Gaussian component, then the proposed recursion reduces to
the PDA filter [3].
We have announced some of the results of the current work
in the conference paper [17].
The paper is structured as follows. Section II presents back-
ground information on Bayesian filtering and random finite
sets. Section III then formulates the single-target tracking
problem in the Bayes framework that accommodates multiple
measurements generated by the target, state dependent sensor
field of view, and clutter; this section also establishes the con-
nection between the proposed recursion and Mahler’s CPHD
recursion restricted to a single target. A particle implementa-
tion of the proposed recursion is presented in Section IV along
with a nonlinear demonstration and numerical studies. An
exact closed-form solution to the proposed recursion is derived
for linear Gaussian single-target models in Section V along
with demonstrations and numerical studies. Extensions of the
closed-form recursion to accommodate nonlinear Gaussian
models are described in Section VI. Concluding remarks are
given in Section VII.
II. B
ACKGROUND
A. The Bayes’ Recursion
In the classical Bayes’filter [18], [19] the hidden state
is assumed to follow a first-order Markov process on
the state space
according to a transition density
, which is the probability density that the
target with state
at time moves to state at time
. The observation is assumed conditionally
independent given the states
and is characterized by a like-
lihood
, which is the probability density that, at time
, the target with state produces a measurement . Under
these assumptions, the classical Bayes’ recursion propagates
the posterior density
in time according to
(1)
(2)
where
. All inference on the target state at
time
is derived from the posterior density . Common esti-
mators for the target state are the expected a posteriori (EAP)
and maximum a posteriori (MAP) estimators.
The Bayes’ recursion (1), (2) is formulated for single-target
single-measurement systems. In practice due to multipath
reflections, electronic counter measures, etc., the target may
generate multiple measurements, in addition to spurious mea-
surements not generated by the target. Note that at any given
time step, the order of appearance of measurements received
by sensor has no physical significance. Hence, at time
the
sensor effectively receives an unordered set of measurements
denoted by
, and the observation space is now the space
of finite subsets of
, denoted by . Consequently, the
Bayes’ update (2) is not directly applicable.
To accommodate set-valued measurements, we require a
mathematically consistent generalization of the likelihood
to the set-valued case. In other words, we need a
mathematically rigorous notion of the probability density of the
set
given . However, the notion of such densities is not
straightforward because the space
does not inherit the
usual Euclidean notions of volume and integration on
.We
review in the next subsection how RFS theory or point process
theory provides rigorous notions of volume and integration on
needed to define a mathematically consistent likelihood.
B. Random Finite Sets
We describe in this subsection the bare minimum background
on RFS theory needed to develop the results in this paper. For
a classical treatment of the mathematical theory of RFSs (or
point processes), the reader is referred to [20] and [21], while
a comprehensive treatment of multitarget tracking using RFSs
can be found in [9]–[11].
Let
be a probability space, where is the
sample space,
is a -algebra on , and is a probability
measure on
.Arandom finite set on a complete sepa-
rable metric space
(e.g., )isdefined as a measurable
mapping
(3)
with respect to the Borel sets of
[20]–[22].
1
The proba-
bility distribution of the RFS
is given in terms of the proba-
bility measure
by
(4)
where
is any Borel subset of . The probability distribu-
tion of the RFS
can be equivalently characterized by a discrete
probability distribution and a family of joint probability distri-
butions. The discrete distribution characterizes the cardinality
(the number of elements) of the RFS, whilst for a given cardi-
nality, an appropriate distribution characterizes the joint distri-
bution of the elements of the RFS [20]–[22].
The probability density
of is given by the
Radon–Nikodým derivative of its probability distribution
with respect to an appropriate dominating measure
. i.e.,
1
Technically,
F
(
Z
)
is embedded in the (complete separable metric) space of
counting measures on
Z
, and inherits the Borel sets of this space.
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VO et al.: BAYESIAN FILTERING WITH RANDOM FINITE SET OBSERVATIONS 1315
. The conventional choice of
dominating measure is [23]
(5)
where
is the th product (unitless) Lebesque measure, is a
mapping of vectors to sets defined by
, and is the th Cartesian product of with the
convention
. The integral of a measurable function
with respect to is defined as follows:
(6)
The first-order moment of a random finite set
on , also
called the intensity function, is a non-negative function
on
with the property that for any closed subset
where denotes the cardinality of . In other words, for a
given point
, the intensity is the density of expected
number of targets per unit volume at
.
An important class of RFSs are the Poisson RFSs (or Poisson
point processes) [20], which are completely characterized by
their intensity functions. The cardinality of a Poisson RFS
is Poisson distributed with mean , and for a
given cardinality the elements of
are each independent and
identically distributed (i.i.d.) with probability density
.
More generally, an RFS whose elements are i.i.d. according to
, but has arbitrary cardinality distribution is called an
i.i.d. cluster process [20].
For simplicity in notation, we shall use the same symbol for
an RFS and its realizations hereon.
III. RFS S
INGLE-TARGET BAYES
’ RECURSION
The classical Bayes’ filter was formulated for the case where
the target generates exactly one measurement and there is no
clutter. Hence, in the classical Bayes’ filter, the measurement
is vector-valued and modeled as a random variable given by a
likelihood function defined on
. As previously argued, in the
presence of multiple measurements generated by the target, de-
tection uncertainty and clutter, the measurement is set-valued.
In this section, we describe a RFS measurement model and de-
rive the corresponding likelihood function on
.
A. RFS Measurement Model
The collection of measurements obtained at time
is
represented as a finite subset
of the original observa-
tion space
. More concisely, if observations
are received at time , then
(7)
Suppose at time
that the target is in state . The measure-
ment process is given by the RFS measurement equation
(8)
where
is the RFS of the primary target-generated mea-
surement,
is the RFS of extraneous target-generated
measurements, and
is the RFS of clutter. For example,
may represent a single direct path measurement,
may represent measurements generated by multipath
effects or counter measures, and
may represent state inde-
pendent spurious measurements. It is assumed that conditional
on
, , and are independent RFSs.
We model
as a binary RFS
with probability
with probability density
where is the probability of detection for the primary
measurement, and
is the likelihood for the primary mea-
surement. Hence, the probability of not obtaining the primary
measurement from a state
is , and conversely,
given that there is a primary measurement the probability den-
sity of obtaining the primary measurement
from a state is
.
We model
and in (8) as Poisson RFSs with in-
tensities
and , respectively. For convenience
we group these RFSs together as
(9)
Since
is a union of statistically independent Poisson
RFSs, it is also a Poisson RFS with intensity
(10)
The cardinality distribution
of is Poisson
with mean
. Hence, if the target is in state
at time , the probability of having exactly mea-
surements is
, whilst each measurement is inde-
pendent and identically distributed according to the probability
density
(11)
The following proposition establishes the likelihood cor-
responding to the above RFS measurement model. See
Appendix A for the proof.
Proposition 1: If the measurements follow the RFS model
in (8), then the probability density that the state
at time
produces the measurement set is given by
(12)
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1316 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 56, NO. 4, APRIL 2008
where denotes the unit of volume on , in the sense that
is the Radon–Nikodým derivative of the probability
distribution of
given with respect to the dominating mea-
sure (6), i.e.,
(13)
Remark: The expression in Proposition 1 is a probability den-
sity, and is derived from first principles using only measure the-
oretic probability concepts. A similar expression has been inde-
pendently derived by Mahler using finite set statistics (FISST)
in [11].
2
However, Mahler stresses that the FISST derivative is
not a Radon–Nikodým derivative [11, p. 716] and hence is not a
probability density. We refer the reader to [10] for more details
on the relationship between the FISST set derivative and prob-
ability density of RFS.
The likelihood (12) has
terms each of which ad-
mits an intuitive interpretation. The first term relates to a missed
primary measurement detection, whilst each of the remaining
terms relates to a primary measurement detection. To ex-
plain the first term, notice that when there is a missed primary
measurement detection,
. Hence, the likelihood
of
comprises the following: , the probability
of a missed primary measurement detection;
,
the probability that
has exactly measurements;
, the joint density of the measurements; and
a factorial term to account for all possible permutations of
.
To explain each of the
remaining terms, notice that when
there is a primary measurement detection,
and
. Hence, the likelihood of comprises the
following:
, the probability of a primary measurement
detection;
, the probability that has
exactly
measurements; ,
the joint density of the measurements and a factorial term to ac-
count for all possible permutations of
.
As a check for consistency, if there is always a primary target-
generated measurement, no extraneous target-generated mea-
surements and no clutter, i.e.,
and
( if and zero otherwise), it can be seen
that
and . In other words,
the measurement set is always a singleton containing the pri-
mary measurement, and the likelihood (12) reduces to the usual
single measurement likelihood.
Remark: If
(hence and
), and is an i.i.d. cluster process in (8),
(9), then the likelihood (12) still holds. However, if
and in (8), (9) are both i.i.d cluster processes, the RFS
is, in general, no longer an i.i.d.
cluster process. Nonetheless if
can be approximated by
an i.i.d. cluster process with matching intensity and cardinality
distribution
(14)
(15)
2
The book [11] appeared around the same time that we submitted our prelim-
inary result [17]
where denotes convolution, and are the
cardinality distributions of
and , then the likelihood
(12) is still valid.
B. RFS Single-Target Bayes’ Recursion
The Bayes’ recursion (1), (2) can be generalized to accom-
modate multiple measurements generated by the target, detec-
tion uncertainty, and clutter, by replacing the standard likelihood
with the RFS measurement likelihood in
(12). Hence, the posterior density
can be propagated
as follows:
(16)
(17)
where
.
In general, this recursion does not admit an analytic solution.
However, the problem can be solved using sequential Monte
Carlo techniques as shown in Section IV. Furthermore, a
closed-form solution to this recursion can be derived under
linear Gaussian assumptions as shown in Section V.
Remark: If there is always a primary target-generated mea-
surement, no extraneous target-generated measurements and no
clutter, then
and the recursion (16),
(17) reduces to the classical Bayes’ recursion (1), (2).
Remark: The recursion (16), (17) can be easily extended to
accommodate multiple sensors. Suppose that there are
mu-
tually independent sensors, i.e., the product of the individual
likelihoods for each sensor is the joint likelihood for all sen-
sors. More concisely, if each sensor is modeled by a likelihood
and receives a measurement set at time where
, then the combined likelihood accounting for all
sensors is
(18)
C. Connection With Mahler’s CPHD Filter
In this section, we show how the proposed RFS single-target
Bayes’ recursion is related to Mahler’s CPHD recursion [14],
which is a moment approximation of Mahler’s multitarget
Bayes’ recursion.
The following is a brief review of the relevant results con-
cerning the CPHD recursion. Denote by
the permutation
coefficient
, the inner product defined between
two real valued functions
and by
(or when and are real sequences), and
the elementary symmetric function of order defined for a finite
set
of real numbers by with
by convention.
It was established in [15] (Section V-A) that the CPHD recur-
sion for tracking an unknown and time-varying number of tar-
gets can be simplified when the number of targets is constant.
Let
and be the predicted and posterior intensities re-
spectively at time
. If there are no target births nor deaths and
is the fixed and known number of targets, then the CPHD
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VO et al.: BAYESIAN FILTERING WITH RANDOM FINITE SET OBSERVATIONS 1317
cardinality recursion reduces to
, and the CPHD intensity recursion reduces to
(19)
(20)
where
(21)
(22)
(23)
Assuming no extraneous target-generated measurements, the
proposed RFS single-target Bayes’ recursion reduces to the
above mentioned CPHD recursion restricted to a single target.
Note that for the purposes of comparison, these particular con-
ditions ensure that both recursions assume the same underlying
dynamic and measurement model for the target. The restriction
in the CPHD recursion for no target births nor deaths and ex-
actly one target present ensures consistency with the dynamic
model in the RFS single-target Bayes’ recursion. Indeed, the
CPHD recursion for a single target coincides with the multiple
hypothesis tracker (MHT) for a single target (see [24]). The
restriction in the RFS single-target Bayes’ recursion for no
extraneous target-generated measurements ensures consistency
with the measurement model in the CPHD recursion which
accommodates at most one target-generated measurement.
The agreement between these two recursions can be expected
for the following reason. Consider the above CPHD recursion
and recall its assumption that the target state RFS at any time
is an i.i.d. cluster process [14], [15] (see Section II-B for the
meaning of an i.i.d. cluster process). Observe that if the cardi-
nality distribution of an i.i.d. cluster process is
(i.e., the
value of the process is always a singleton set), then the inten-
sity is the same as the normalized intensity, and hence the in-
tensity is actually the probability density (i.e., the intensity of
the RFS is the probability density of the single constituent point
of the RFS). If
in the above CPHD recursion, (i.e., ex-
actly one target is present at all times), then the CPHD cardi-
nality recursion states that
(i.e., the target state
RFS is always a singleton), and hence the CPHD intensity recur-
sion for
actually propagates the probability density (i.e.,
the propagated intensity is actually the probability density of
the single-target state). Thus, since both the RFS single-target
Bayes’ recursion and the above CPHD recursion assume the
same underlying model, and both recursions propagate the pos-
terior density of the target state, it would be expected that their
recursions are consistent. This is indeed true and is stated in the
following proposition (See Appendix B for the proof).
Proposition 2: The special case of the proposed RFS single-
target Bayes’ recursion (16), (17) with no extraneous target-gen-
erated measurements (i.e., (16), (17) with
hence
and
) is identical to the special case of the CPHD
recursion (19), (20) with no target births nor deaths and exactly
one target present (i.e., (19), (20) with
).
This result establishes that under the common dynamic and
measurement model stated above, the proposed derivation of
the RFS single-target Bayes’ recursion from first principles
using point process theory agrees with Mahler’s derivation
of the CPHD recursion using FISST. This agreement further
consolidates the utility and power of FISST.
IV. S
EQUENTIAL
MONTE CARLO
IMPLEMENTATION
In this section, we describe a generic sequential Monte Carlo
(SMC) (see also [19], [25]) implementation of the RFS single-
target Bayes’ recursion (16), (17) and demonstrate the proposed
filter on a nonlinear tracking example. Note that the proposed
SMC implementation inherits the usual convergence properties
[26], [27] since the recursion (16), (17) propagates the true pos-
terior density of the target state.
A. Recursion
Suppose at time
that the posterior density is
represented by set of weighted particles
, i.e.,
(24)
Then, for a given proposal density
satisfying
support
support , the particle filter approximates
the posterior density
by a new set of weighted particles
, i.e.
(25)
where
(26)
(27)
(28)
It can be seen that the proposed algorithm has the same
computational complexity as the standard single-target particle
filter.
The recursion is initialized by generating a set of weighted
particles
representing . Equations (26)–(28)
then provide a recursion for computing the set of weighted par-
ticles representing
from those representing when a new
measurement arrives.
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