# Multiobject Tracking for
Thresholded Cell Measurements

Thomas Kropfreiter1, Jason L. Williams2, and Florian Meyer1 This work was
supported by the Austrian Science Fund (FWF) under Grant J 4726-N and the
National Science Foundation (NSF) under CAREER award 2146261. 1University of
California San Diego, La Jolla, CA, USA (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS> 2Whipbird Signals, Brisbane, Australia<EMAIL_ADDRESS>
###### Abstract

In many multiobject tracking applications, including radar and sonar tracking,
after prefiltering the received signal, measurement data is typically
structured in cells. The cells, e.g., represent different range and bearing
values. However, conventional multiobject tracking methods use so-called point
measurements. Point measurements are provided by a preprocessing stage that
applies a threshold or detector and breaks up the cell’s structure by
converting cell indexes into, e.g., range and bearing measurements. We here
propose a Bayesian multiobject tracking method that processes measurements
that have been thresholded but are still cell-structured. We first derive a
likelihood function that systematically incorporates an adjustable detection
threshold which makes it possible to control the number of cell measurements.
We then propose a Poisson Multi-Bernoulli (PMB) filter based on the likelihood
function for cell measurements. Furthermore, we establish a link to the
conventional point measurement model by deriving the likelihood function for
point measurements with amplitude information (AM) and discuss the PMB filter
that uses point measurements with AM. Our numerical results demonstrate the
advantages of the proposed method that relies on thresholded cell measurements
compared to the conventional multiobject tracking based on point measurements
with and without AM.

###### Index Terms:

Multiobject tracking, multiobject tracking, Poisson Multi-Bernoulli filtering,
random finite sets.

##  I Introduction

Multiobject tracking (MOT) aims to estimate the time-dependent number and
states of multiple objects based on the data provided by one or more sensors.
MOT algorithms use either sensor data in a preprocessed form or the raw sensor
data without any preprocessing. In the first approach, known as _detect-then-
track_ (DTT) MOT, a detector discards all measurements that are below a
predefined threshold in order to reduce data flow and computational complexity
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. In the second
approach, known as _track-before-detect_ (TBD) MOT, the tracker uses all the
raw sensor data, which typically results in improved tracking performance at a
higher computational complexity.

Irrespective of the measurement model that is in force, the multiobject state
can either be modeled as a random vector [1, 2, 3, 4, 5, 6, 11, 12, 14, 18,
19] or a random set, more precisely, a random finite set (RFS) [7, 8, 9, 10,
13, 15, 16, 17]. A high-performing RFS-based MOT filter is the track-oriented
marginal multi-Bernoulli/Poisson filter, which will be simply referred to as
Poisson Multi-Bernoulli (PMB) filter in the following. The PMB filter was both
proposed for DTT MOT [10, 11, 13, 17] and recently also for TBD MOT [20]. The
PMB filter models the multiobject state as the union of a Poisson RFS and a
multi-Bernoulli (MB) RFS. Whereas the MB part of the PMB filter is used for
the tracking of already detected objects, the Poisson part models objects that
have not been are detected so far. A modified version of the PMB filter
proposed in [21] extends the use of the Poisson RFS from the modeling of
undetected objects to the tracking of objects that are unlikely to exist by
using the concept of recycling. In the recycling step, Bernoulli components
with a low existence probability are transferred from the MB part of the
multiobject state to the Poisson part of the multiobject state instead of
pruning them. The PMB filter can be implemented very efficiently using the
framework of belief propagation (BP) [11, 22] resulting in an overall linear
complexity scaling. The PMB filter has also been derived and extended to
multiple sensors using the framework of factor graphs and BP [11, 12, 18, 19].
It has been demonstrated that it can outperform existing DTT methods and that
it is highly scalable in all relevant system parameters [21, 22, 10, 11, 12,
13, 14, 18, 20].

In this paper, we introduce a measurement model for thresholded cell
measurements and propose a PMB filter based on this new model. The novel
method can be interpreted as a tradeoff between the already existing DTT PMB
and TBD PMB filters. In particular, as in DTT filtering, a threshold controls
the number of considered measurements, but, as in TBD filtering, measurements
still possess the cell structure of the original, unthresholded, measurements.
In addition, this cell structure makes it possible to avoid the Poisson
clutter approximation used by conventional DTT approaches. We also establish a
link to the conventional likelihood function for point measurements with
amplitude information (AM) and discuss the approximations applied in
conventional point measurement models. Our numerical results indicate an
improved tracking performance compared to the DTT PMB filter with and without
AM in both tracking accuracy and runtime.

We will use the following basic notation. Random variables are displayed in
sans serif, upright fonts; their realizations in serif, italic fonts. Vectors
and matrices are denoted by bold lowercase and uppercase letters,
respectively. For example, $\mathsfbr{x}$ is a random variable, and $x$ is its
realization and $\bm{\mathsfbr{x}}$ is a random vector and $\bm{x}$ is its
realization. Random sets and their realizations are denoted by upright sans
serif and calligraphic font, respectively. For example, $\mathsfbr{X}$ is a
random set and ${\cal{X}}$ is its realization. We denote probability density
functions (pdfs) by $f(\cdot)$ and probability mass functions (pmfs) by
$p(\cdot)$. Further, ${\cal{N}}(\bm{x};\bm{\mu},\bm{\varSigma})$ denotes the
Gaussian pdf (of random vector $\bm{\mathsfbr{x}}$) with mean $\bm{\mu}$ and
covariance matrix $\bm{\varSigma}$ and ${\cal{R}}(x;\sigma)$ denotes the
Rayleigh pdf (of scalar random variable $\mathsfbr{x}$) with scale parameter
$\sigma$, ${\cal{P}}(n;\mu)$ denotes the Poisson pmf (of scalar random integer
variable $\mathsfbr{n}$) with mean parameter $\mu$, and ${\cal{B}}(n;N,p)$
denotes the binomial pmf (of scalar random integer variable $\mathsfbr{n}$) of
size $N$ and probability $p$.

##  II Measurement Model

The measurement model describes the statistics of the measurements
$\bm{\mathsfbr{z}}_{k}$ conditioned on the multiobject state ${\cal{X}}_{k}$
and can be characterized by the likelihood function
$f(\bm{z}_{k}|{\cal{X}}_{k})$. We here model the multiobject state at time $k$
by the RFS
$\mathsfbr{X}_{k}=\\{\bm{\mathsfbr{x}}_{k}^{(1)},\ldots,\bm{\mathsfbr{x}}_{k}^{(n)}\\}$
and the measurements by the random vector
$\bm{\mathsfbr{z}}_{k}=[\mathsfbr{z}_{k}^{(1)}\ldots\mathsfbr{z}_{k}^{(M)}]^{\text{T}}$.
The single-object state vectors $\bm{\mathsfbr{x}}_{k}$ are defined as
$\bm{\mathsfbr{x}}_{k}\\!=[\bm{\mathsfbr{p}}_{k}^{\text{T}}\,\,\sgamma_{k}]^{\text{T}}$,
where
$\bm{\mathsfbr{p}}_{k}=[\mathsfbr{p}_{k,1}\,\,\mathsfbr{p}_{k,2}\,\,\dot{\mathsfbr{p}}_{k,1}\,\,\dot{\mathsfbr{p}}_{k,2}]^{\text{T}}$
is a two-dimensional (2D) position and velocity vector and $\sgamma_{k}$ is
the object’s intensity. The modeling of $\bm{\mathsfbr{z}}_{k}$ will be
described in the following.

###  II-A Detector-Based Association Model

Our modeling of $\bm{\mathsfbr{z}}_{k}$ is based on $M$ data cells
characterized by the random vector
$\bm{\mathsfbr{c}}_{k}=[\mathsfbr{c}_{k}^{(1)}\ldots\mathsfbr{c}_{k}^{(M)}]^{\text{T}}$.
Here, each $\mathsfbr{c}_{k}^{(m)}$,
$m\in\mathcal{M}\triangleq\\{1,\ldots,M\\}$ represents the scalar intensity or
magnitude of data cell $m$, and the cell index $m\in\mathcal{M}$ represents
the corresponding positional information. Since we consider point objects, the
influence of the object with state $\bm{x}_{k}\in{\cal{X}}_{k}$ on cell
$m\in{\cal{M}}$ is determined by an indicator function, i.e.,

$\delta^{(m)}(\bm{x}_{k})=\begin{cases}1,&\bm{x}_{k}\hskip 2.84526pt\text{is
in bin}\hskip 2.84526ptm,\\\\[2.84526pt] 0,&\bm{x}_{k}\hskip 2.84526pt\text{is
not in bin}\hskip 2.84526ptm\hskip 0.85358pt.\end{cases}\vspace{0.5mm}$ (1)

In particular, object $\bm{x}_{k}$ contributes to cell
$\mathsfbr{c}_{k}^{(m)}$ only if the object is located in that cell, i.e., by
knowing $\bm{x}_{k}$, we also know the index of the influenced cell. If cell
$\mathsfbr{c}_{k}^{(m)}$ is influenced by $\bm{x}_{k}$, it is distributed
according to $f_{1}\big{(}\mathsfbr{c}^{(m)}_{k}\big{|}\bm{x}_{k}\big{)}$. If
cell $\mathsfbr{c}_{k}^{(m)}$ is not influenced by any object, it is due to
clutter and is distributed according to
$f_{0}\big{(}\mathsfbr{c}^{(m)}_{k}\big{)}$.

Since $\mathsfbr{X}_{k}$ is random, it is unknown which object contributes to
which cell. We therefore introduce the random object-cell association vector
$\bm{\mathsfbr{a}}_{k}\triangleq\big{[}\mathsfbr{a}_{k}^{(1)}\ldots\mathsfbr{a}_{k}^{(n)}\big{]}^{\text{T}}$
with $n=|{\cal{X}}_{k}|$ and entries $\mathsfbr{a}_{k}^{(i)}\in\mathcal{M}$.
Here, $\mathsfbr{a}_{k}^{(i)}\in\mathcal{M}$ indicates that object
$\bm{x}_{k}^{(i)}$ contributes to cell $\mathsfbr{c}_{k}^{(m)}$. We assume now
in the following that an object contributes to at most one cell and,
conversely, that a cell is at most influenced by one object. These assumptions
result in a set of valid association vectors $\bm{a}_{k}$ collected in the set
${\cal{A}}_{k}$.

We assume now that a detector with threshold $\eta$ is applied to each cell
$\mathsfbr{c}^{(m)}_{k}$, $m\in\mathcal{M}$ to obtain “thresholded”
measurements
$\bm{\mathsfbr{z}}_{k}=[\mathsfbr{z}_{k}^{(1)}\ldots\mathsfbr{z}_{k}^{(M)}]^{\text{T}}$.
In particular, we set $\mathsfbr{z}_{k}^{(m)}=\mathsfbr{c}^{(m)}_{k}$ if
$\mathsfbr{c}^{(m)}_{k}>\eta$ and $\mathsfbr{z}_{k}^{(m)}=\eta$ if
$\mathsfbr{c}^{(m)}_{k}\leqslant\eta$. In what follows, we refer to
$\mathsfbr{z}_{k}^{(m)}>\eta$ as detections. Following classical detection
theory [23], we can directly obtain a statistical model for the thresholded
measurements. Each detection, $\mathsfbr{z}^{(m)}_{k}>\eta$, is either
originated by an object or by false alarm/clutter. In the former case, it is
distributed according to

$f_{1,\eta}\big{(}z^{(m)}_{k}\big{|}\bm{x}_{k}\big{)}=\frac{f_{1}\big{(}z^{(m)}_{k}\big{|}\bm{x}_{k}\big{)}}{p_{\text{D}}^{(m)}(\bm{x}_{k})}\vspace{0.5mm}$
(2)

and, in the latter case, according to

$f_{0,\eta}\big{(}z^{(m)}_{k}\big{)}=\frac{f_{0}\big{(}z^{(m)}_{k}\big{)}}{p^{(m)}_{\text{FA}}}.$
(3)

Here, the normalization constants $p_{\text{D}}^{(m)}(\bm{x}_{k})$ and
$p^{(m)}_{\text{FA}}$ are referred to as detection probability and false alarm
probability and can be computed as

$\displaystyle p_{\text{D}}^{(m)}(\bm{x}_{k})$
$\displaystyle=\int_{\eta}^{\infty}f_{1}\big{(}z_{k}^{(m)}\big{|}\bm{x}_{k}\big{)}\hskip
0.85358pt\hskip 0.85358pt\mathrm{d}z_{k}^{(m)},$ $\displaystyle
p^{(m)}_{\text{FA}}$
$\displaystyle=\int_{\eta}^{\infty}f_{0}\big{(}z_{k}^{(m)}\big{)}\hskip
0.85358pt\text{d}z_{k}^{(m)}.$

###  II-B The Joint Likelihood Function

The conditional pdf $f(\bm{z}_{k}|{\cal{X}}_{k})$ is obtained by first
computing the conditional pdf $f(\bm{z}_{k},\bm{a}_{k}|{\cal{X}}_{k})$ and
then marginalizing out the association variable $\bm{\mathsfbr{a}}_{k}$, i.e.,

$f(\bm{z}_{k}|{\cal{X}}_{k})\hskip 0.85358pt=\sum_{\bm{a}_{k}\in\hskip
0.85358pt\mathcal{A}_{k}}f(\bm{z}_{k},\bm{a}_{k}|{\cal{X}}_{k})\hskip
0.85358pt.$ (4)

Based on the aforementioned assumptions, the conditional pdf
$f(\bm{z}_{k},\bm{a}_{k}|{\cal{X}}_{k})$ is found as

$\displaystyle f(\bm{z}_{k},\bm{a}_{k}|{\cal{X}}_{k})$
$\displaystyle=\bigg{(}\prod_{i\hskip 0.85358pt\in\hskip
0.85358pt{\cal{I}}_{k}}\delta^{(a_{k}^{(i)})}\big{(}\bm{x}_{k}^{(i)}\big{)}\hskip
0.85358ptf\big{(}z_{k}^{(a_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}\bigg{)}$
$\displaystyle\hskip 2.84526pt\times\prod_{m\hskip 0.85358pt\in\hskip
0.85358pt\mathcal{M}_{\bm{a}_{k}}}f(z_{k}^{(m)})\hskip 0.85358pt$ (5)

where we introduced ${\cal{I}}_{k}\triangleq\\{1,\ldots,n\\}$ and
$\mathcal{M}_{\bm{a}_{k}}\triangleq\mathcal{M}\setminus\big{\\{}a_{k}^{(1)},\dots,a_{k}^{(n)}\big{\\}}$.
The set $\mathcal{M}_{\bm{a}_{k}}$ comprises all measurement indices that are
not associated to any object by association $\bm{a}_{k}$. In (5), the
distribution $f(z_{k}^{(m)}|\bm{x}_{k}^{(i)})$ is given by

$f\big{(}z_{k}^{(m)}|\bm{x}_{k}^{(i)}\big{)}=\begin{cases}1-p_{\text{D}}^{(m)}\big{(}\bm{x}_{k}^{(i)}\big{)},&z_{k}^{(m)}=\eta\\\\[7.11317pt]
p_{\text{D}}^{(m)}\big{(}\bm{x}_{k}^{(i)}\big{)}\hskip
0.85358ptf_{1,\eta}\big{(}z_{k}^{(m)}\big{|}\bm{x}_{k}^{(i)}\big{)},&z_{k}^{(m)}>\eta\vspace{0.5mm}\end{cases}$
(6)

Furthermore, the distribution $f(z_{k}^{(m)})$ reads

$f(z_{k}^{(m)})=\begin{cases}1-p_{\text{FA}}^{(m)},&z_{k}^{(m)}=\eta\\\\[7.11317pt]
p_{\text{FA}}^{(m)}\hskip
0.85358ptf_{0,\eta}\big{(}z_{k}^{(m)}\big{)},&z_{k}^{(m)}>\eta.\\\\[7.11317pt]
\end{cases}$ (7)

Assuming that every object is located in a different data cell and
${\cal{X}}_{k}$ fixed, the conditional pdf in (5) is nonzero only for a single
association vector $\bm{a}_{k}$. The case where multiple objects are in the
same data cell is not directly captured by our proposed model. However, this
event is rather unlikely in many applications anyway, as the data cell size is
small there. The proposed method achieves very good object detection and
tracking performance, as reported in Section V.

##  III Multiobject Tracking with Thresholded Cell Measurements

In the following, we present our PMB filtering method based on the thresholded
cell measurement model introduced in the previous section.

###  III-A Sequential Bayesian Estimation

To derive our PMB filter, we use the sequential Bayesian estimation framework
for RFS-based multiobject state models. Here, the statistics of the
multiobject state $\mathsfbr{X}_{k}$ at time $k$, conditioned on all obtained
measurements
$\bm{z}_{1:k}\triangleq[\bm{z}_{1}^{\text{T}}\ldots\bm{z}_{k}^{\text{T}}]^{\text{T}}$
up to time $k$, are described by the posterior pdf
$f({\cal{X}}_{k}|\bm{z}_{1:k})$. The estimation framework consists of a
prediction and an update step. The prediction step transforms the posterior
pdf at time $k-1$, i.e., $f({\cal{X}}_{k-1}|\bm{z}_{1:k-1})$, into the
predicted posterior pdf at time $k$, i.e., $f({\cal{X}}_{k}|\bm{z}_{1:k-1})$
using the state-transition pdf $f({\cal{X}}_{k}|{\cal{X}}_{k-1})$.
Analogously, the update step transforms the predicted posterior pdf into the
updated posterior pdf at time $k$, i.e., $f({\cal{X}}_{k}|\bm{z}_{1:k})$,
using the likelihood function $f(\bm{z}_{k}|{\cal{X}}_{k})$ presented in
Section II-B and the current measurements $\bm{z}_{k}$.

Following the PMB MOT paradigm [10, 13, 17], we here model the multiobject
state ${\cal{X}}_{k-1}$ at time $k-1$ as the union of a Poisson RFS and an MB
RFS, i.e., the posterior pdf $f({\cal{X}}_{k-1}|{\cal{X}}_{k-1})$ is of PMB
form. Using standard model assumptions for $f({\cal{X}}_{k}|{\cal{X}}_{k-1})$
[10, 11, 12, 13], it was shown in, e.g., [10, 11, 12, 13], that, after
applying the prediction step, the predicted posterior pdf
$f({\cal{X}}_{k}|\bm{z}_{1:k-1})$ is again of PMB form. In fact,
$f({\cal{X}}_{k}|\bm{z}_{1:k-1})$ is given according to

$f({\cal{X}}_{k}|\bm{z}_{1:k-1})\hskip
0.85358pt=\sum_{{\cal{X}}_{k}^{(1)}\uplus\hskip
0.85358pt{\cal{X}}_{k}^{(2)}=\hskip
0.85358pt{\cal{X}}_{k}}f_{k|k-1}^{\text{P}}\big{(}{\cal{X}}_{k}^{(1)}\big{)}\hskip
0.85358ptf_{k|k-1}^{\text{MB}}\big{(}{\cal{X}}_{k}^{(2)}\big{)}\hskip
0.85358pt.\vspace{-0.5mm}$ (8)

Here, $\sum_{{\cal{X}}_{k}^{(1)}\uplus\hskip
0.85358pt{\cal{X}}_{k}^{(2)}=\hskip 0.85358pt{\cal{X}}_{k}}$ denotes the sum
over all disjoint decompositions of ${\cal{X}}_{k}$ into ${\cal{X}}_{k}^{(1)}$
and ${\cal{X}}_{k}^{(2)}$. Furthermore,
$f_{k|k-1}^{\text{P}}\big{(}{\cal{X}}_{k}\big{)}$ is a Poisson pdf [7], which
is fully parametrized by its probability hypothesis density (PHD)
$\lambda_{k|k-1}(\bm{x}_{k})$, and
$f_{k|k-1}^{\text{MB}}\big{(}{\cal{X}}_{k}\big{)}$ is an MB pdf [7] consisting
of $J_{k-1}$ Bernoulli pdfs, where each Bernoulli pdf is itself parametrized
by an existence probability $r_{k|k-1}^{(j)}$ and a spatial pdf
$f_{k|k-1}^{(j)}(\bm{x})$,
$j\in{\cal{J}}_{k-1}\triangleq\\{1,\ldots,J_{k-1}\\}$. See, e.g., [7], for
more information on RFSs and their statistical description.

###  III-B Exact Update Step

In the following two subsections, we present the exact and approximate update
steps. Note that at this point, the current measurements $\bm{z}_{k}$ are
observed and thus known. It is therefore also known which data cells
$z_{k}^{(m)}$, $m\in{\cal{M}}$ have an intensity above threshold $\eta$ and
which do not. We here introduce the set
${\cal{M}}_{k}^{\text{D}}\subseteq{\cal{M}}$ of the indices of all detected
cells, i.e., all cell indices with $\mathsfbr{z}_{k}^{(m)}\geqslant\eta$ and
the set ${\cal{M}}_{k}^{\text{M}}={\cal{M}}\setminus{\cal{M}}_{k}^{\text{D}}$
of the indices of all missed detections, i.e., all cell indices with
$\mathsfbr{z}_{k}^{(m)}=\eta$.

After applying the update step, as we show in Appendix A, the posterior pdf
$f({\cal{X}}_{k}|\bm{z}_{1:k})$ is no longer a PMB pdf but a Poisson/MB
mixture (PMBM) pdf, which is given by

$f({\cal{X}}_{k}|\bm{z}_{1:k})\hskip
0.85358pt=\\!\sum_{{\cal{X}}_{k}^{(1)}\uplus\hskip
0.85358pt{\cal{X}}_{k}^{(2)}=\hskip
0.85358pt{\cal{X}}_{k}}f^{\text{P}}\big{(}{\cal{X}}_{k}^{(1)}\big{)}\hskip
0.85358ptf^{\text{MBM}}\big{(}{\cal{X}}_{k}^{(2)}\big{)}\hskip
0.85358pt.\vspace{-0.5mm}$ (9)

Here, $f^{\text{P}}\big{(}{\cal{X}}_{k}\big{)}$ is a Poisson pdf and
$f^{\text{MBM}}\big{(}{\cal{X}}_{k}\big{)}$ is an MBM pdf; expressions of both
are presented now.

The Poisson pdf $f^{\text{P}}\big{(}{\cal{X}}_{k}\big{)}$ is fully
characterized by the posterior PHD $\lambda(\bm{x}_{k})$, which is given by

$\displaystyle\lambda(\bm{x}_{k})=\bigg{(}\sum_{m\hskip 0.85358pt\in\hskip
0.85358pt{\cal{M}}^{\text{M}}_{k}}\delta^{(m)}(\bm{x}_{k})\hskip
0.85358pt\frac{1-p_{\text{D}}^{(m)}(\bm{x}_{k})}{1-p_{\text{FA}}^{(m)}}\hskip
0.85358pt\bigg{)}\lambda_{k|k-1}(\bm{x}_{k})\hskip 0.85358pt.$

Here, $p_{\text{D}}^{(m)}(\bm{x}_{k})$ and $p_{\text{FA}}^{(m)}$ are the
detection probability and false alarm probability, respectively, defined in
Section II-A and $\lambda_{k|k-1}(\bm{x}_{k})$ is the predicted posterior PHD
characterizing $f_{k|k-1}^{\text{P}}\big{(}{\cal{X}}_{k}\big{)}$ in (8).

In order to describe $f^{\text{MBM}}\big{(}{\cal{X}}_{k}\big{)}$ in (9), we
first introduce the random object-cell association vector
$\bm{\mathsfbr{a}}^{\prime}_{k}\triangleq[\mathsfbr{a}_{k}^{\prime(1)}\ldots\mathsfbr{a}_{k}^{\prime(J_{k})}]^{\text{T}}$
where $J_{k}=J_{k-1}+M^{\text{D}}_{k}$ and
$M_{k}^{\text{D}}\triangleq|{\cal{M}}_{k}^{\text{D}}|$. Differently to the
association vector $\bm{\mathsfbr{a}}_{k}$ defined in Section II-A, the random
vector $\bm{\mathsfbr{a}}^{\prime}_{k}$ accounts also for the possible non-
existence of objects, missed detections of existing objects, and the
appearance of new objects. More precisely, $\bm{\mathsfbr{a}}^{\prime}_{k}$
has entries $\mathsfbr{a}_{k}^{\prime(j)}\in\\{0\\}\cup{\cal{M}}$ for
$j\in{\cal{J}}_{k-1}$ and $\mathsfbr{a}_{k}^{\prime(j)}\in\\{0,1\\}$ for
$j\in{\cal{J}}_{k}\setminus{\cal{J}}_{k-1}$. Here, for $j\in{\cal{J}}_{k-1}$,
$\mathsfbr{a}_{k}^{\prime(j)}=\hskip 0.85358pt0$ indicates that the object
$\bm{\mathsfbr{x}}_{k}$ modeled by Bernoulli component $j$ does not exist,
$\mathsfbr{a}_{k}^{\prime(j)}\in{\cal{M}}^{\text{M}}_{k}$ that it exists but
did not generate a detection in cell $m\in{\cal{M}}^{\text{M}}_{k}$, and
$\mathsfbr{a}_{k}^{\prime(j)}\in{\cal{M}}^{\text{D}}_{k}$ that it exists and
generated detection $m\in{\cal{M}}_{k}^{\text{D}}$. Furthermore, for
$j\in{\cal{J}}_{k}\setminus{\cal{J}}_{k-1}$, $\mathsfbr{a}_{k}^{\prime(j)}=1$
indicates that detection $m\in{\cal{M}}_{k}^{\text{D}}$ was generated either
by an object $\bm{\mathsfbr{x}}_{k}$ modeled by the Poisson pdf or by clutter
and, conversely, $\mathsfbr{a}_{k}^{\prime(j)}=0$ that detection
$m={\cal{M}}_{k}^{\text{D}}$ was neither generated by an object modeled by the
Poisson RFS nor by clutter. Note that there is exactly one
$j\in{\cal{J}}_{k}\setminus{\cal{J}}_{k-1}$, i.e., a new Bernoulli component,
for each detected cell $m\in{\cal{M}}_{k}^{\text{D}}$. The specific assignment
of $m$ to $j$ is arbitrary. All valid association vectors
$\bm{a}^{\prime}_{k}$ form the set $\mathcal{A}^{\prime}_{k}$.

Using $\bm{\mathsfbr{a}}^{\prime}_{k}$, we can now express
$f^{\text{MBM}}\big{(}{\cal{X}}_{k}\big{)}$ as

$f^{\text{MBM}}\big{(}{\cal{X}}_{k}\big{)}=\sum_{\bm{a}^{\prime}_{k}\hskip
0.85358pt\in\hskip 0.85358pt\mathcal{A}^{\prime}_{k}}\hskip
0.85358ptp(\bm{a}^{\prime}_{k})\hskip 0.85358pt\hskip
0.85358ptf^{\text{MB}}_{\bm{a}^{\prime}_{k}}({\cal{X}}_{k})\hskip
0.85358pt.\vspace{-1mm}$ (11)

Note that here each mixture component corresponds to exactly one valid
association vector and its weight is given by the association probability
$p(\bm{a}^{\prime}_{k})$, which, in turn, is given by

$p(\bm{a}^{\prime}_{k})\propto\prod_{j\in{\cal{J}}_{k}}\hskip
0.85358pt\beta_{k}^{(j,a_{k}^{\prime(j)})}\vspace{0mm}$ (12)

for $\bm{a}^{\prime}_{k}\in\mathcal{A}^{\prime}_{k}$, and by
$p(\bm{a}^{\prime}_{k})=0$ for
$\bm{a}^{\prime}_{k}\notin\mathcal{A}^{\prime}_{k}$. Furthermore, the MB pdfs
$f^{\text{MB}}_{\bm{a}^{\prime}_{k}}({\cal{X}}_{k})$ consist of
$J_{k}=J_{k-1}+M^{\text{D}}_{k}$ Bernoulli components, where each of them is
parametrized by an existence probability $r_{k}^{(j,m)}$ and a spatial pdf
$f^{(j,m)}(\bm{x}_{k})$. In the following, we provide expressions for
$r_{k}^{(j,m)}$, $f^{(j,m)}(\bm{x}_{k})$, and $\beta_{k}^{(j,m)}$.

In fact, for Bernoulli component $j\in{\cal{J}}_{k-1}$ and $m=0$ , we have
$r_{k}^{(j,0)}=0$ , $f^{(j,0)}(\bm{x}_{k})$ undefined, and

$\displaystyle\beta_{k}^{(j,0)}$ $\displaystyle=1-r_{k|k-1}^{(j)}\hskip
0.85358pt.$ (13)

Here, $r_{k}^{(j,0)}=0$ indicates that the object $\bm{\mathsfbr{x}}_{k}$
modeled by Bernoulli component $j$ does not exist; $f^{(j,0)}(\bm{x}_{k})$
remains thus undefined. The likelihood of this event is given by (13).

Next, for $j\in{\cal{J}}_{k-1}$ and $m\in{\cal{M}}^{\text{M}}_{k}$, we have
$r_{k}^{(j,m)}=1$ and

$\displaystyle f^{(j,m)}(\bm{x}_{k})$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\frac{\delta^{(m)}(\bm{x}_{k})\hskip
0.85358pt\big{(}1-p_{\text{D}}^{(m)}(\bm{x}_{k})\big{)}\hskip
0.85358ptf_{k|k-1}^{(j)}(\bm{x}_{k})}{b_{k}^{(j,m)}}$ (14)
$\displaystyle\beta_{k}^{(j,m)}$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\frac{r_{k|k-1}^{(j)}\hskip
0.85358ptb_{k}^{(j,m)}}{1-p_{\text{FA}}^{(m)}}$ (15)

with
$b_{k}^{(j,m)}\triangleq\int\delta^{(m)}(\bm{x}_{k})\big{(}1-p_{\text{D}}^{(m)}(\bm{x}_{k})\big{)}\hskip
0.85358ptf_{k|k-1}^{(j)}(\bm{x}_{k})\hskip 0.85358pt\text{d}\bm{x}_{k}$. Here,
$r_{k}^{(j,m)}=1$ indicates that the object $\bm{\mathsfbr{x}}_{k}$ modeled by
Bernoulli component $j$ exists but was not detected in cell $m$. Its spatial
pdf is distributed according to (14), and the likelihood of this event is
given by (15).

Next, for $j\in{\cal{J}}_{k-1}$ and $m\in\mathcal{M}_{k}^{\text{D}}$, we have
$r_{k}^{(j,m)}=1$, and

$\displaystyle f^{(j,m)}(\bm{x}_{k})$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\frac{\delta^{(m)}(\bm{x}_{k})\hskip
0.85358ptp_{\text{D}}^{(m)}(\bm{x}_{k})\hskip
0.85358ptf_{1,\eta}(z_{k}^{(m)}|\bm{x}_{k})\hskip
0.85358ptf_{k|k-1}^{(j)}(\bm{x}_{k})}{c^{(j,m)}}$
$\displaystyle\beta_{k}^{(j,m)}$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358ptr_{k|k-1}^{(j)}\hskip 0.85358ptc_{k}^{(j,m)}$ (17)

where $c_{k}^{(j,m)}\triangleq\int\delta^{(m)}(\bm{x}_{k})\hskip
0.85358ptp_{\text{D}}^{(m)}(\bm{x}_{k})\hskip
0.85358ptf_{1,\eta}(z_{k}^{(m)}|\bm{x}_{k})\hskip
0.85358ptf_{k|k-1}^{(j)}(\bm{x}_{k})$ $\times\text{d}\bm{x}_{k}$. Here,
$r_{k}^{(j,m)}=1$ indicates that the object $\bm{\mathsfbr{x}}_{k}$ modeled by
Bernoulli component $j$ exists and was detected in cell $m$. Its spatial pdf
is distributed according to (LABEL:eq:upd_para3_2), and the likelihood of this
event is given by (17).

Furthermore, there are also new Bernoulli components
$j={\cal{J}}_{k}\setminus{\cal{J}}_{k-1}$ and $m\in{\cal{M}}_{k}^{\text{D}}$
with

$\displaystyle r_{k}^{(j,1)}$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\frac{d_{k}^{(j)}}{p_{\text{FA}}^{(m)}\hskip
0.85358ptf_{0,\eta}(z_{k}^{(m)})+d_{k}^{(j)}}$ (18) $\displaystyle
f^{(j,1)}(\bm{x}_{k})$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358pt\frac{\delta^{(m)}(\bm{x}_{k})\hskip
0.85358ptp_{\text{D}}^{(m)}(\bm{x}_{k})\hskip
0.85358ptf_{1,\eta}(z_{k}^{(m)}|\bm{x}_{k})\hskip
0.85358pt\lambda_{k|k-1}(\bm{x}_{k})}{d_{k}^{(j)}}$
$\displaystyle\beta_{k}^{(j,1)}$ $\displaystyle\hskip 0.85358pt=\hskip
0.85358ptp_{\text{FA}}^{(m)}\hskip
0.85358ptf_{0,\eta}(z_{k}^{(m)})+d_{k}^{(j)}$ (20)

where $d_{k}^{(j)}\triangleq\int\delta^{(m)}(\bm{x}_{k})\hskip
0.85358ptp_{\text{D}}^{(m)}(\bm{x}_{k})\hskip
0.85358ptf_{1,\eta}(z_{k}^{(m)}|\bm{x}_{k})\hskip
0.85358pt\lambda_{k|k-1}(\bm{x}_{k})$ $\times\text{d}\bm{x}_{k}$. Here,
$r_{k}^{(j,1)}$ in (18) is the probability that detection $m$ was generated by
object $\bm{\mathsfbr{x}}_{k}$ modeled by the Poisson RFS and not due to
clutter. The state of this object is distributed according to
(LABEL:eq:upd_para4_2), and the likelihood of the events that detection $m$
was generated either by an object modeled by the Poisson RFS or by clutter is
given by (20). Finally, we have $\beta_{k}^{(j,0)}=1$, $r_{k}^{(j,0)}=0$, and
$f^{(j,0)}(\bm{x}_{k})$ not defined.

###  III-C Approximate Update Step

We now approximate the MBM pdf in (11) by an MB pdf. We start by extending the
association alphabet $\mathcal{A}^{\prime}_{k}$ to
$\mathcal{A}^{\prime\prime}_{k}\triangleq(\\{0\\}\cup{\cal{M}})^{J_{k-1}}\times\\{0,1\\}^{M_{k}^{\text{D}}}$.
By setting $p(\bm{a}^{\prime}_{k})=0$ for
$\mathcal{A}^{\prime\prime}_{k}\setminus\mathcal{A}^{\prime}_{k}$, this
extension does not change the association pmf. Next, we approximate
$p(\bm{a}^{\prime}_{k})$ according to

$p(\bm{a}^{\prime}_{k})\approx\prod_{j\in{\cal{J}}_{k}}\hskip
0.85358ptp\big{(}a_{k}^{\prime(j)}\big{)}\hskip
0.85358pt,\quad\bm{a}^{\prime}_{k}\in\mathcal{A}^{\prime\prime}_{k}\vspace{-1mm}$
(21)

with

$p\big{(}a_{k}^{\prime(j)}\big{)}=\sum_{\sim\bm{a}_{k}^{\prime(j)}}\hskip
0.85358ptp(\bm{a}^{\prime}_{k})\hskip 0.85358pt.\vspace{-1.5mm}$

Here, $\sim\bm{a}_{k}^{\prime(j)}$ denotes the vector of all
$a_{k}^{\prime(j^{\prime})}$ with $j^{\prime}\in{\cal{J}}_{k}\setminus j$.
Note that a fast and scalable approximate solutions for
$p\big{(}a_{k}^{\prime(j)}\big{)}$ can be provided by the SPA [22, 10, 11].

As was shown in, e.g., [20], by inserting approximation (21) into (11), the
MBM pdf simplifies to an MB pdf whose existence probabilities and spatial pdfs
are given for $j\in{\cal{J}}_{k-1}$ by

$\displaystyle r_{k}^{(j)}$ $\displaystyle=\sum_{a_{k}^{\prime(j)}\in\hskip
0.85358pt\mathcal{M}}p(a_{k}^{\prime(j)})$ (22) $\displaystyle
f^{(j)}(\bm{x}_{k})$
$\displaystyle=\frac{1}{r_{k}^{(j)}}\sum_{a_{k}^{\prime(j)}\in\hskip
0.85358pt\mathcal{M}}p(a_{k}^{\prime(j)})\hskip
0.85358ptf^{(j,a_{k}^{\prime(j)})}(\bm{x}_{k})$ (23)

and for $j\in{\cal{J}}_{k}\setminus{\cal{J}}_{k-1}$ by

$\displaystyle r_{k}^{(j)}$ $\displaystyle=p(a_{k}^{\prime(j)}=1)\hskip
2.84526ptr_{k}^{(j,1)}$ (24) $\displaystyle f^{(j)}(\bm{x}_{k})$
$\displaystyle=\hskip 0.85358ptf^{(j,1)}(\bm{x}_{k})\hskip 0.85358pt.$ (25)

Note that according to (24) and (25), a new Bernoulli component is generated
for each detection in each update step. This results in a linear increase in
the number of Bernoulli components over time. In order to counteract this
increase and limit the number of Bernoulli components, we employ the concept
of recycling, where Bernoulli components with low existence probabilities are
transferred to the Poisson part of the multiobject state RFS. More precisely,
we transfer all Bernoulli components
$j\in\mathcal{J}_{k}^{\text{R}}\subseteq{\cal{J}}_{k}$ with an existence
probability $r_{k}^{(j)}$ below the threshold $\eta_{\text{R}}$. This yields
the approximated PHD

$\tilde{\lambda}(\bm{x}_{k})=\lambda(\bm{x}_{k})+\sum_{j\hskip
0.85358pt\in\hskip 0.85358pt\mathcal{J}_{k}^{\text{R}}}r_{k}^{(j)}\hskip
0.85358ptf^{(j)}(\bm{x}_{k})\vspace{-1.2mm}$

where $\lambda(\bm{x}_{k})$ is given by (LABEL:eq:upd_Pois), $r_{k}^{(j)}$ is
given by (22) or (24) and $f^{(j)}(\bm{x}_{k})$ by (23) or (25), respectively.
Note that, unlike of pruning Bernoulli components with low existence
probabilities recycling does not discard any information.

##  IV Multiobject Tracking with Point Measurements

In the following, we derive a likelihood function for point measurements that
also uses AM. Contrary to our cell measurement model where the measurements
are described by the random vector
$\bm{\mathsfbr{z}}_{k}=[\mathsfbr{z}_{k}^{(1)}\ldots\mathsfbr{z}_{k}^{(M)}]^{\text{T}}$
and where the measurement index represents the position information, point
measurements are typically modeled by an unordered set
$\mathsfbr{Z}_{k}=\\{\mathsfbr{z}_{k}^{(1)},\ldots,\mathsfbr{z}_{k}^{(\mathsfbr{M}_{k})}\\}$.
Here, position information is modeled explicitly by position-related
measurements and not by the measurement index.

###  IV-A Explicit Location Information and Clutter Cardinality

Let
$\big{[}\mathsfbr{z}^{(m)}_{k,1}\,\mathsfbr{z}^{(m)}_{k,2}\big{]}^{\text{T}}$
be the position-related measurement, e.g., in range-bearing or in Cartesian
coordinates, corresponding to amplitude measurement
$\mathsfbr{z}^{(m)}_{k}\in\mathsfbr{Z}_{k}$, and let
$\bm{\mathsfbr{y}}_{k}^{(m)}=\big{[}\mathsfbr{z}^{(m)}_{k}\hskip
0.85358pt\hskip 0.85358pt\hskip 0.85358pt\mathsfbr{z}^{(m)}_{k,1}\hskip
0.85358pt\hskip 0.85358pt\hskip
0.85358pt\mathsfbr{z}^{(m)}_{k,2}\big{]}^{\text{T}}$ be the corresponding
joint measurement vector. Position-related measurements can be obtained by
breaking up the cell structure and converting cell indices to position-related
measurements. A joint measurement vector $\bm{\mathsfbr{y}}_{k}^{(m)}$, can be
statistically described by the following joint pdfs (cf. (2) and (3))

$\displaystyle f_{1}(\bm{y}_{k}^{(m)}|\bm{x}_{k})$
$\displaystyle=f_{1,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)}|\bm{x}_{k})\hskip
0.85358ptf_{1,\eta}(z_{k}^{(m)}|\bm{x}_{k})\hskip 0.85358pt,$ (26)
$\displaystyle f_{0}(\bm{y}_{k}^{(m)})$
$\displaystyle=f_{0,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)})\hskip
0.85358ptf_{0,\eta}(z_{k}^{(m)}).$ (27)

where $f_{1,\eta}(z_{k}^{(m)}|\bm{x}_{k})$ and $f_{0,\eta}(z_{k}^{(m)})$ are
given by (2) and (3), respectively, and
$f_{1,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)}|\bm{x}_{k})$ and
$f_{0,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)})$ are the pdfs of the position-
related measurement for the cases that measurement
$\bm{\mathsfbr{y}}^{(m)}_{k}$ is generated by object $\bm{x}_{k}$ or by false
alarm/clutter, respectively. Given these basic definitions, we apply now a
series of manipulations and approximations in order to transform the
likelihood function for thresholded cell measurements in (4) and (5) into a
likelihood function for point measurements that also uses AM.

We start by noting that in the likelihood function for thresholded cell
measurements the information about the number of clutter measurements is
contained in the term $\prod_{{\cal{M}}_{\bm{a}_{k}}}f(z_{k}^{(m)})$ with
$f(z_{k}^{(m)})$ given by (7). Here $f(z_{k}^{(m)})$ can be rewritten as
follows: We first explicitly represent the events
$\mathsfbr{z}_{k}^{(m)}=\eta$ and $\mathsfbr{z}_{k}^{(m)}>\eta$ by the
Bernoulli random variable $\mathsfbr{b}_{k}^{(m)}\in\\{0,1\\}$, where
$\mathsfbr{b}_{k}^{(m)}=0$ models the event that $\mathsfbr{z}_{k}=\eta$ and
$\mathsfbr{b}_{k}^{(m)}=1$ the event that $\mathsfbr{z}_{k}^{(m)}>\eta$. Using
$\mathsfbr{b}_{k}^{(m)}$, we can now rewrite $f(z_{k}^{(m)})$ in (7) according
to

$f\big{(}z_{k}^{(m)},b_{k}^{(m)}\big{)}=f\big{(}z_{k}^{(m)}|b_{k}^{(m)}\big{)}\hskip
0.85358ptp\big{(}b_{k}^{(m)}\big{)}$

with

$p\big{(}b_{k}^{(m)}\big{)}=\begin{cases}1-p_{\text{FA}}^{(m)},&b_{k}^{(m)}=0\\\\[7.11317pt]
p_{\text{FA}}^{(m)},&b_{k}^{(m)}=1\hskip 0.85358pt.\end{cases}$ (28)

Furthermore,
$f\big{(}z_{k}^{(m)}|b_{k}^{(m)}=1\big{)}=f_{0,\eta}\big{(}z_{k}^{(m)}\big{)}$
and, since $f(z_{k}^{(m)}|b_{k}^{(m)}=0)$ is meaningless, it is not defined.

In point measurement models, the information about the number of clutter
measurements in (28) is typically represented by a single cardinality variable
$\mathsfbr{M}_{k}^{\text{FA}}$. In order to derive a likelihood function for
point measurements, we therefore define

$\mathsfbr{M}_{k}^{\text{FA}}\hskip 0.85358pt\hskip
0.85358pt\triangleq\sum_{m\hskip 0.85358pt\in\hskip
0.85358pt{\cal{M}}_{\bm{a}_{k}}}\mathsfbr{b}_{k}^{(m)}\hskip
0.85358pt.\vspace{-1.5mm}$ (29)

Since $\mathsfbr{b}_{k}^{(m)}$ is Bernoulli distributed according to (28), and
given the fact that the sum of Bernoulli distributed random variables is
Binomial distributed [24], $\mathsfbr{M}_{k}^{\text{FA}}$ in (29) is binomial
distributed too, i.e.,

$\displaystyle p(M_{k}^{\text{FA}})$ $\displaystyle=\hskip
0.85358pt{\cal{B}}\big{(}M_{k}^{\text{FA}};|{\cal{M}}_{\bm{a}_{k}}|,\hskip
0.85358ptp_{\text{FA}}\big{)}$
$\displaystyle\triangleq\binom{|{\cal{M}}_{\bm{a}_{k}}|}{M_{k}^{\text{FA}}}(p_{\text{FA}})^{M_{k}^{\text{FA}}}(1-p_{\text{FA}})^{|{\cal{M}}_{\bm{a}_{k}}|-M_{k}^{\text{FA}}}\vspace{-0.5mm}$
(30)

where $\binom{|{\cal{M}}_{\bm{a}_{k}}|}{M_{k}^{\text{FA}}}$ is the binomial
coefficient of $|{\cal{M}}_{\bm{a}_{k}}|$ and $M_{k}^{\text{FA}}$.

For later use, we now approximate the binomial pmf $p(M_{k}^{\text{FA}})$ in
(IV-A) by a Poisson pmf, i.e.,

$p(M_{k}^{\text{FA}})\approx\hskip
0.85358pt{\cal{P}}\big{(}M_{k}^{\text{FA}};\mu_{\text{FA}}\big{)}\triangleq\frac{(\mu_{\text{FA}})^{M_{k}^{\text{FA}}}e^{-\mu_{\text{FA}}}}{M_{k}^{\text{FA}}!}\vspace{0mm}$
(31)

with $\mu_{\text{FA}}=p_{\text{FA}}\hskip
0.85358pt|{\cal{M}}_{\bm{a}_{k}}|\approx p_{\text{FA}}\hskip 0.85358ptM$.
According to the Poisson limit theorem [24], the approximation error of (31)
is small if both $p_{\text{FA}}$ is small and $M$ is large.

###  IV-B Object-Measurement Associations and Likelihood Function

Next, we note that conventional point measurement models allow any possible
object-measurement association, theoretically, with non-zero probability. We
now define the random object-measurement association vector for point
measurements as
$\overline{\bm{\mathsfbr{a}}}_{k}=[\overline{\mathsfbr{a}}_{k}^{(1)},\ldots,\overline{\mathsfbr{a}}_{k}^{(n)}]$
with entries $\overline{\mathsfbr{a}}_{k}^{(i)}\in\\{0\\}\cup{\cal{M}}_{k}$,
${\cal{M}}_{k}\triangleq\\{1,\ldots,M_{k}\\}$, and where $M_{k}$ is the number
of point measurements $\bm{\mathsfbr{y}}_{k}^{(m)}$. Here,
$\overline{\mathsfbr{a}}_{k}^{(i)}=0$ indicates that no measurement is
associated with object $\bm{x}_{k}^{(i)}$ and
$\overline{\mathsfbr{a}}_{k}^{(i)}\in{\cal{M}}_{k}$ that measurement
$\bm{\mathsfbr{y}}_{k}^{(m)}$ is associated with object $\bm{x}_{k}^{(i)}$.
All valid association vectors $\overline{\bm{\mathsfbr{a}}}_{k}$ are collected
in the set $\overline{{\cal{A}}}_{k}$.

We now model all $\overline{\bm{\mathsfbr{a}}}_{k}$ as equally likely, i.e,

$p(\overline{\bm{a}}_{k}|{\cal{X}}_{k})=\frac{1}{N_{\overline{\bm{a}}_{k}}}$
(32)

where $N_{\overline{\bm{a}}_{k}}$ is the number of association vectors
$\overline{\bm{a}}_{k}$, i.e.,
$N_{\overline{\bm{a}}_{k}}=|\overline{{\cal{A}}}_{k}|$. Here,
$N_{\overline{\bm{a}}_{k}}$ can be determined by using basic results from
combinatorics. In fact, $N_{\overline{\bm{a}}_{k}}$ is equivalent to the
number of possible combinations of drawing $M_{k}^{\text{D}}$ object
detections out of $M_{k}$ measurements, where the draws are without
replacement and with the drawing order respected. This leads to

$N_{\overline{\bm{a}}_{k}}=\frac{M_{k}!}{(M_{k}-M_{k}^{\text{D}})!}=\frac{M_{k}!}{M_{k}^{\text{FA}}!}\hskip
0.85358pt.\vspace{1mm}$ (33)

Putting all together, we find the joint pdf of $\bm{\mathsfbr{y}}_{k}$,
$\overline{\bm{\mathsfbr{a}}}_{k}$, and $\mathsfbr{M}_{k}^{\text{FA}}$,
conditioned on ${\cal{X}}_{k}$, according to

$\displaystyle
f(\bm{y}_{k},M_{k}^{\text{FA}},\overline{\bm{a}}_{k}|{\cal{X}}_{k})$
$\displaystyle=\hskip
0.85358pt\frac{1}{N_{\overline{\bm{a}}_{k}}}\Big{(}\prod_{i\hskip
0.85358pt\in\hskip 0.85358pt{\cal{I}}_{k}}\hskip
0.85358ptf\big{(}\bm{y}_{k}^{(\overline{a}_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}\Big{)}$
$\displaystyle\hskip 2.84526pt\times\Big{(}\prod_{m\hskip 0.85358pt\in\hskip
0.85358pt{\cal{M}}_{\overline{\bm{a}}_{k}}}f_{0}(\bm{y}_{k}^{(m)})\Big{)}\hskip
2.84526ptp(M_{k}^{\text{FA}}).$ (34)

Analogously to (6),
$f\big{(}\bm{y}_{k}^{(\overline{a}_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}$ is
given by

$f\big{(}\bm{y}_{k}^{(\overline{a}_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}=\begin{cases}1-p_{\text{D}}\big{(}\bm{x}_{k}^{(i)}\big{)},&\overline{a}_{k}^{(i)}=0\\\\[7.11317pt]
p_{\text{D}}\big{(}\bm{x}_{k}^{(i)}\big{)}f_{1}(\bm{y}_{k}^{(m)}|\bm{x}_{k}),&\overline{a}_{k}^{(i)}\in{\cal{M}}_{k}.\end{cases}\vspace{1mm}$

Here, $f_{1}(\bm{y}_{k}^{(m)}|\bm{x}_{k})$ is given by (26), and
$f_{0}(\bm{y}_{k}^{(m)})$ and $p(M_{k}^{\text{FA}})$ by (27) and (IV-A),
respectively.

By finally inserting $N_{\overline{\bm{a}}_{k}}$ given by (33) and the Poisson
approximation of $p(M_{k}^{\text{FA}})$ given (31) into (34), we get the
approximate joint distribution
$\tilde{f}(\bm{y}_{k},M_{k}^{\text{FA}},\overline{\bm{a}}_{k}|{\cal{X}}_{k})$.
Due to the relation $M_{k}=M_{k}^{\text{D}}+M_{k}^{\text{FA}}$,
$\tilde{f}(\bm{y}_{k},M_{k}^{\text{FA}},\overline{\bm{a}}_{k}|{\cal{X}}_{k})$
is equivalent to
$\tilde{f}(\bm{y}_{k},M_{k},\overline{\bm{a}}_{k}|{\cal{X}}_{k})$, given by

$\displaystyle\tilde{f}(\bm{y}_{k},M_{k},\overline{\bm{a}}_{k}|{\cal{X}}_{k})$
$\displaystyle=\frac{e^{-\mu_{\text{FA}}}}{M_{k}!}\Big{(}\prod_{i\hskip
0.85358pt\in\hskip 0.85358pt{\cal{I}}_{k}}\hskip
0.85358ptf\big{(}\bm{y}_{k}^{(\overline{a}_{k}^{(i)})},\overline{a}_{k}^{(i)}|\bm{x}_{k}^{(i)}\big{)}\Big{)}\prod_{m\hskip
0.85358pt\in\hskip
0.85358pt\mathcal{M}_{\overline{\bm{a}}_{k}}}\lambda_{\text{FA}}\big{(}\bm{y}^{(m)}_{k}\big{)}$

with $\lambda_{\text{FA}}(\bm{y}_{k})\triangleq\mu_{\text{FA}}\hskip
0.85358ptf_{0}(\bm{y}_{k})$. By finally marginalizing out
$\overline{\bm{a}}_{k}$, we find the likelihood function for point
measurements as

$f(\bm{y}_{k},M_{k}|{\cal{X}}_{k})\hskip
0.85358pt=\sum_{\overline{\bm{a}}_{k}\in\hskip
0.85358pt\overline{\mathcal{A}}_{k}}\hskip
0.85358pt\tilde{f}(\bm{y}_{k},M_{k},\overline{\bm{a}}_{k}|{\cal{X}}_{k})\hskip
0.85358pt.\vspace{0mm}$ (36)

Note that here, $\bm{\mathsfbr{y}}_{k}$ is a random vector consisting of
$\mathsfbr{M}_{k}$ subvectors $\bm{\mathsfbr{y}}_{k}^{(m)}$,
$m\in{\cal{M}}_{k}$. However, in RFS-based MOT with point measurements, the
measurements are typically represented as a set
${\cal{Y}}_{k}=\\{\bm{\mathsfbr{y}}_{k}^{(1)},\ldots,\bm{\mathsfbr{y}}_{k}^{(M_{k})}\\}$.
The corresponding pdf is given by $f({\cal{Y}}_{k}|{\cal{X}}_{k})=M_{k}!\hskip
0.85358ptf(\bm{y}_{k},M_{k}|{\cal{X}}_{k})$ [7]. If the AM information
$\mathsfbr{z}^{(m)}_{k}$ is removed from $\bm{\mathsfbr{y}}_{k}^{(m)}$, the
corresponding point measurement model without AM is obtained. Based on
(LABEL:eq:meas2_5_3) and (36), a PMB filter using point measurements with AM
can be derived analogously as in [10].

##  V Simulation Study

In the following, we perform a simulation study in which we compare the
performance of our proposed PMB filter with thresholded pixel measurements
(PMB-CM) with that of a PMB filter that uses point measurements with (PMB-AM)
and without (PMB) amplitude information. But first, we start with a short
description of the underlying simulation setup.

###  V-A Simulation Setup

We consider a two-dimensional (2D) simulation scenario with a region of
interest (ROI) of
$[\text{0}\text{m},\text{32}\text{m}]\times[\text{0}\text{m},\text{32}\text{m}]$.
We simulated 10 objects during 200 time steps. Recap that the object state
vectors are defined according to
$\bm{\mathsfbr{x}}_{k}\\!=[\bm{\mathsfbr{p}}_{k}^{\text{T}}\,\,\sgamma_{k}]^{\text{T}}$,
where
$\bm{\mathsfbr{p}}_{k}=[\mathsfbr{p}_{k,1}\,\,\mathsfbr{p}_{k,2}\,\,\dot{\mathsfbr{p}}_{k,1}\,\,\dot{\mathsfbr{p}}_{k,2}]^{\text{T}}$
is the object’s 2D position and 2D velocity and $\sgamma_{k}$ the object’s
intensity (cf. Section II). The objects appear at various times before time
step 30 at randomly chosen positions in the ROI, and they disappear at various
times after time step 170 or when they leave the ROI. The object’s initial
velocities are drawn from
$f_{\text{v}}(\dot{p}_{k,1},\,\dot{p}_{k,2})={\cal{N}}(\dot{p}_{k,1},\,\dot{p}_{k,2};\bm{0}_{2},\sigma^{2}_{\text{v}}\bm{I}_{2})$
with $\sigma^{2}_{\text{v}}=10^{-2}$, and the object’s initial intensities are
set to $10$.

The object states’ evolution is modeled as follows: The kinematic part of the
object states, i.e., $\bm{\mathsfbr{p}}_{k}$, evolves according to a nearly
constant velocity motion model with independent and identically distributed
(iid) driving noise $\bm{\mathsfbr{w}}_{k}$, whose pdf is given by
${\cal{N}}(\bm{w}_{k};\bm{0}_{4},10^{-3}\bm{I}_{4})$, while the object
intensities remain constant over time.

The measurement is an image consisting of $32\times 32$ data cells of square
size, each of which has a side length of $1$m. The data is generated by
modeling the cell intensities as Rayleigh distributed, i.e.,
$f_{1}\big{(}c_{k}^{(m)}\big{|}\bm{x}_{k}\big{)}={\cal{R}}(c^{(m)}_{k};\sqrt{\gamma_{k}+\sigma^{2}_{\text{n}}}\hskip
0.85358pt)\vspace{-.5mm}$ and
$f_{0}(y_{k}^{(m)})={\cal{R}}\big{(}y^{(m)}_{k};\sigma_{\text{n}}\big{)}$ and
drawing samples from them. In RADAR applications, this model is referred to as
Swerling 1 [25]. We set $\sigma^{2}_{\text{n}}=1$ leading to a signal-to-noise
ratio (SNR) of $10$. If there are several objects in the same cell, lets say
$\bm{x}^{(m_{1})}_{k},\bm{x}^{(m_{2})}_{k},\dots,\bm{x}^{(m_{I})}_{k}$, then
the cell intensity $\mathsfbr{c}^{(m)}_{k}$ is sampled from
${\cal{R}}\Big{(}c^{(m)}_{k};\sqrt{\gamma^{(m_{1})}_{k}+\gamma^{(m_{2})}_{k}+\dots+\gamma^{(m_{I})}_{k}+\sigma^{2}_{\text{n}}}\Big{)}$.
In PMB-CM, we set
$f_{1}\big{(}c_{k}^{(m)}\big{|}\bm{x}_{k}\big{)}={\cal{R}}(c^{(m)}_{k};\sqrt{\gamma_{k}+\sigma^{2}_{\text{n}}}\hskip
0.85358pt)\vspace{-.5mm}$ and
$f_{0}(y_{k}^{(m)})={\cal{R}}\big{(}y^{(m)}_{k};\sigma_{\text{n}}\big{)}$. As
discussed in Section II-A, for a fixed threshold $\eta$, this directly induces
a statistical model for the thresholded data cells
$f_{1,\eta}\big{(}z_{k}^{(m)}\big{|}\bm{x}_{k}\big{)}$ and
$f_{0,\eta}\big{(}z_{k}^{(m)}\big{)}$.

For PMB and PMB-AM, position-related measurement information is represented
explicitly according to
$\big{[}\mathsfbr{z}^{(m)}_{k,1}\,\mathsfbr{z}^{(m)}_{k,2}\big{]}^{\text{T}}$,
which necessitates the further specification of
$f_{1,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)}|\bm{x}_{k})$ in (26) and
$f_{0,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)})$ in (27). We follow the
frequently used convention for point measurement filtering and model
$f_{1,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)}|\bm{x}_{k})$ as a Gaussian pdf
${\cal{N}}\big{(}z_{k,1}^{(m)},z_{k,2}^{(m)};[p_{k,1}\hskip
0.85358ptp_{k,2}]^{\mathrm{T}},\sigma^{2}_{\text{p}}\bm{I}_{2}\big{)}$. Here,
the variance $\sigma^{2}_{\text{p}}$ is obtained via moment matching by
setting $\sigma^{2}_{\text{p}}$ to the variance of a uniform distribution
defined by the side length of a data cell given by $1$m, i.e.,
$\sigma^{2}_{\text{p}}=1/12$. Note that in PMB and PMB-AM,
$f_{1,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)}|\bm{x}_{k})$ is evaluated by
setting $[z_{k,1}^{(m)},z_{k,2}^{(m)}]$ equal to the 2D cell center point.
Furthermore, we model $f_{0,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)})$ as
uniformly distributed over the ROI. It has been observed, that modeling the
likelihood function $f_{1,\text{p}}(z_{k,1}^{(m)},z_{k,2}^{(m)}|\bm{x}_{k})$
constant on the area of the corresponding data cell and zero otherwise leads
to almost identical performance compared to the aforementioned Gaussian model.

Figure 1: GOSPA error of PMB (blue curves) and PMB-AM (green curves) versus
time $k$ for different detection thresholds $\eta$.

###  V-B Filter Settings and Simulation Results

We employ particle implementations of PMB-CM, PMB-AM, and PMB. We represent
the spatial pdf of each Bernoulli component by $3,\\!000$ particles, the
posterior PHD by $50,\\!000$ particles, and the birth PHD by another
$50,\\!000$; the resulting $100,\\!000$ particles are reduced back to
$50,\\!000$ after the update step. More precisely, the birth PHD is given by
$\lambda_{\text{B}}(\bm{x}_{k})=\mu_{\text{B}}\hskip
0.85358ptf_{\text{B}}(\bm{x}_{k})$ with $\mu_{\text{B}}=5/32^{2}$ and
$f_{\text{B}}(\bm{x}_{k})=f(p_{k,1},p_{k,2})\hskip
0.85358ptf_{\text{v}}\big{(}\dot{p}_{k,1},\dot{p}_{k,2}\big{)}\hskip
0.85358ptf_{\text{I}}(\gamma_{k})$ where $f(p_{k,1},p_{k,2})$ is uniform over
the ROI, $f_{\text{v}}\big{(}\dot{p}_{k,1},\dot{p}_{k,2}\big{)}$ is defined in
the previous subsection, and $f_{\text{I}}(\gamma_{k})$ is uniform from $0$ to
$30$. We set the survival probability to $p_{\text{S}}=0.999$ and the
recycling threshold (cf. Section III-C) to $\eta_{\text{R}}=0.1$.

The tracking performance is evaluated by means of the generalized optimal
subpattern assignment (GOSPA) metric [26] with parameters $p=1$, $c=20$, and
$\beta=2$. We show the GOSPA errors for PMB and PMB-AM for different
thresholds $\eta\in\\{2,4,6\\}$ in Fig. 1. It can be seen that the use of AM
improves tracking performance. In particular, as Fig. 1(c) indicates,
surprisingly, the detection performance of newborn objects of PMB decreases
for small $\eta$, which in turn also increases the total GOSPA error. This can
be attributed to the fact that for low $\eta$ not only the possibility of
object detection but also the number of clutter measurements increases. Here,
AM helps the tracker to distinguish between object-originated and clutter
measurements.

Figure 2: GOSPA error of PMB-AM (green curves) and proposed PMB-CM (red
curves) versus time $k$ for different detection thresholds $\eta$.

Next, in Fig. 2 we compare the GOSPA errors of PMB-AM and PMB-CM. As can be
seen there, PMB-CM outperforms PMB-AM for all three values of $\eta$. This can
be attributed to the applied point measurement approximation described in
Section IV, which seems to lead to a higher likelihood that a measurement was
generated by a newborn object, which in turn to faster generation of new
Bernoulli components. As a result, PMB-AM detects newly appeared objects
slightly faster, as shown in Fig. 2(c), but on the other hand produces a
significant higher number of false tracks than PMB-CM, as shown in Fig. 2(d);
this effect is pronounced in scenarios with larger $\eta$.

Finally, we compare the three PMB filters also in terms of runtime/complexity;
the average number of detections per time step $k$, i.e.,
$\overline{N}_{\text{Pix}}=|{\cal{M}}_{k}^{\text{D}}|$, and the average filter
runtimes of PMB, PMB-AM, and PMB-CM for different values of $\eta$, obtained
with a Matlab implementation on an Intel Xeon Gold 5222 CPU and measured using
1000 Monte Carlo runs, are reported in Table I. The table shows that PMB-AM is
slightly slower than PMB due to the additional processing caused by AM. While
the runtime of PMB-CM increases only slightly for lower values of $\eta$, the
runtime increase of PMB and PMB-AM is significant. We can thus conclude that
the proposed PMB filter using thresholded cell measurements achieves a very
good accuracy/complexity tradeoff.

##  VI Conclusion

We proposed a Poisson Multi-Bernoulli (PMB) filter that uses thresholded cell
measurements. We furthermore established a link to the conventional point
measurement model by deriving the likelihood function for point measurements
using also amplitude information (AM). In a simulation experiment, we
demonstrated the advantages of our proposed PMB filter using thresholded cell
measurements compared to the conventional PMB with and without AM. Future work
includes a refined measurement model for cell measurements that explicitly
models multiple objects being in the same cell.

 Threshold $\eta$ |  6 |  4 |  2 |  0
---|---|---|---|---
$\overline{N}_{\text{Pix}}$ | 1.28 | 3.51 | 143.15 | 1024
PMB | 6.95 s | 10.05 s | 183.65 s |
PMB-AM | 7.02 s | 10.32 s | 198.98 s |
PMB-CM | 9.52 s | 9.6 s | 10.40 s | 11.13 s
TABLE I: Average number of detections $\overline{N}_{\text{Pix}}$ per time
step $k$ and average filter runtimes of PMB, PMB-AM, and PMB-CM for different
values of $\eta$.

## Appendix A

$\displaystyle f({\cal{X}}_{k}|\bm{z}_{1:k})\hskip 0.85358pt$
$\displaystyle\propto\sum_{{\cal{X}}_{k}^{(1)}\uplus\hskip
0.85358pt{\cal{X}}_{k}^{(2)}=\hskip
0.85358pt{\cal{X}}_{k}}\sum_{\bm{a}_{k}\in{\cal{A}}_{k}}\bigg{(}\prod_{j^{\prime}\in\hskip
0.85358pt{\cal{J}}_{k-1}^{(0)}}1-r_{k|k-1}^{(j^{\prime})}\bigg{)}\bigg{(}\prod_{j\hskip
0.85358pt\in\hskip 0.85358pt{\cal{J}}_{k-1}^{(1)}}r_{k|k-1}^{(j)}\hskip
0.85358pt\delta^{(a_{k}^{(i)})}(\bm{x}_{k}^{(i)})\hskip
0.85358ptf^{\prime}\big{(}z_{k}^{(a_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}\hskip
0.85358ptf_{k|k-1}^{(j)}(\bm{x}_{k}^{(i)})\bigg{)}$
$\displaystyle\times\bigg{(}\prod_{\bm{x}_{k}^{(i^{\prime})}\in\hskip
0.85358pt{\cal{X}}_{k}^{(1)}}\delta^{(a_{k}^{(i^{\prime})})}(\bm{x}_{k}^{(i^{\prime})})\hskip
0.85358ptf^{\prime}\big{(}z_{k}^{(a_{k}^{(i^{\prime})})}|\bm{x}_{k}^{(i^{\prime})}\big{)}\hskip
0.85358pt\lambda_{k|k-1}(\bm{x}_{k}^{(i^{\prime})})\bigg{)}\hskip
0.85358pt\prod_{m\hskip 0.85358pt\in\hskip
0.85358pt\mathcal{M}^{\prime}_{\bm{a}_{k}}}p_{\text{FA}}^{(m)}\hskip
0.85358ptf_{0,\eta}\big{(}z_{k}^{(m)}\big{)}$ (40)

In this appendix, we derive the exact update step applied in Section III-B.
The update step converts the predicted posterior pdf
$f({\cal{X}}_{k}|\bm{z}_{1:k-1})$ into the updated posterior pdf
$f({\cal{X}}_{k}|\bm{z}_{1:k})$ by applying Bayes theorem according to

$f({\cal{X}}_{k}|\bm{z}_{1:k})\propto f(\bm{z}_{k}|{\cal{X}}_{k})\hskip
0.85358ptf({\cal{X}}_{k}|\bm{z}_{1:k-1}).\vspace{0.5mm}$ (35)

Recap that the predicted posterior pdf $f({\cal{X}}_{k}|\bm{z}_{1:k-1})$ is of
PMB form and given by expression (8); it can be rewritten as

$\displaystyle f({\cal{X}}_{k}|\bm{z}_{1:k-1})\hskip 0.85358pt$
$\displaystyle=\sum_{{\cal{X}}_{k}^{(1)}\uplus\hskip
0.85358pt{\cal{X}}_{k}^{(2)}=\hskip
0.85358pt{\cal{X}}_{k}}f^{\text{P}}\big{(}{\cal{X}}_{k}^{(1)}\big{)}\prod_{j\in{\cal{J}}_{k-1}}f^{(j)}\big{(}{\cal{X}}_{k}^{(2,j)}\big{)}$

with ${\cal{X}}_{k}^{(2)}={\cal{X}}_{k}^{(2,1)}\uplus\hskip
0.85358pt\ldots\hskip 0.85358pt\uplus{\cal{X}}_{k}^{(2,J_{k-1})}$. Next, we
rewrite $f(\bm{z}_{k},\bm{a}_{k}|{\cal{X}}_{k})$ in (5) according to

$\displaystyle f(\bm{z}_{k},\bm{a}_{k}|{\cal{X}}_{k})$
$\displaystyle\propto\bigg{(}\prod_{i\hskip 0.85358pt\in\hskip
0.85358pt{\cal{I}}_{k}}\delta^{(a_{k}^{(i)})}\big{(}\bm{x}_{k}^{(i)}\big{)}\hskip
0.85358ptf^{\prime}\big{(}z_{k}^{(a_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}\bigg{)}$
$\displaystyle\hskip 8.53581pt\times\prod_{m\hskip 0.85358pt\in\hskip
0.85358pt\mathcal{M}^{\prime}_{\bm{a}_{k}}}p_{\text{FA}}^{(m)}f_{0,\eta}\big{(}z_{k}^{(m)}\big{)}$
(37)

with

$\displaystyle f^{\prime}\big{(}z_{k}^{(a_{k}^{(i)})}|\bm{x}_{k}^{(i)}\big{)}$
$\displaystyle\hskip
5.69054pt=\begin{cases}(1-p_{\text{D}}^{(a_{k}^{(i)})}\big{(}\bm{x}_{k}^{(i)}\big{)})/(1-p_{\text{FA}}^{(a_{k}^{(i)})}),&z_{k}^{(a_{k}^{(i)})}=\eta\\\\[5.69054pt]
p_{\text{D}}^{(a_{k}^{(i)})}\big{(}\bm{x}_{k}^{(i)}\big{)}\hskip
0.85358ptf_{1,\eta}\big{(}z_{k}^{(a_{k}^{(i)})}\big{|}\bm{x}_{k}^{(i)}\big{)},&z_{k}^{(a_{k}^{(i)})}>\eta\hskip
0.85358pt.\vspace{0.5mm}\end{cases}$ (38)

Here, we have first decomposed $\prod_{m\hskip 0.85358pt\in\hskip
0.85358pt\mathcal{M}_{\bm{a}_{k}}}f(z_{k}^{(m)})$ into $\Big{(}\prod_{m\hskip
0.85358pt\in\hskip
0.85358pt\mathcal{M}^{\prime}_{\bm{a}_{k}}}p_{\text{FA}}^{(m)}\hskip
0.85358ptf_{0,\eta}\big{(}z_{k}^{(m)}\big{)}\Big{)}\hskip
0.85358pt\prod_{m^{\prime}\hskip 0.85358pt\in\hskip
0.85358pt\overline{\mathcal{M}}_{\bm{a}_{k}}}1-p_{\text{FA}}^{(m^{\prime})}$
and then formally multiplied and divided by $\prod_{m\hskip 0.85358pt\in\hskip
0.85358pt\mathcal{M}^{\text{D}}_{k}}1-p_{\text{FA}}^{(m^{\prime})}$. Here,
$\mathcal{M}^{\prime}_{\bm{a}_{k}}$ and $\overline{\mathcal{M}}_{\bm{a}_{k}}$
are all cell indeces $m\in{\cal{M}}$ with $z_{k}^{(m)}=\eta$ and
$z_{k}^{(m)}>\eta$ , respectively.

Now, inserting (LABEL:eq:UpdEx_PredPost) and (37) together with (4) into (35)
yields the posterior pdf $f({\cal{X}}_{k}|\bm{z}_{1:k})$ in (40). Here,
${\cal{J}}_{k-1}^{(0)}$ comprises all Bernoulli components
$j\in{\cal{J}}_{k-1}$ with ${\cal{X}}_{k}^{(2,j)}=\emptyset$ and
${\cal{J}}_{k-1}^{(1)}$ all Bernoulli components $j\in{\cal{J}}_{k-1}$ with
${\cal{X}}_{k}^{(2,j)}=\\{\bm{x}_{k}^{(i)}\\}$. If we rewrite the terms in
(40) by weighted Bernoulli components [10, 20] and in terms of
$\bm{\mathsfbr{a}}^{\prime}_{k}$, as defined in Section III-B, we finally
obtain (9).

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