This section provides an overview of JIPDA with models of target dynamics and sensor measurements. It also introduces a reformulation of the posterior probabilities of the FJEs for JIPDA to aid understanding how the sequences of the Markov chain for each target can be used to represent the FJEs. In this paper, we use superscript to denote a track, and a true target that track follows.
2.2. JIPDA
Tracks may be initiated from target or clutter measurements. True tracks should be confirmed fast and kept confirmed, while false tracks should be terminated effectively through a proper track management method. JIPDA utilizes the PTE for track management.
The event that the target exists at time k is denoted by . Then, the PTE at time k conditioned on is denoted by , and the probability that the target does not exist satisfies
The Markov chain-one model [
8] for the propagation of the PTE for target
is given by
where the transition probabilities are defined as
The a posteriori probability density function (pdf) of the target state is given by
The estimated probability density function (pdf) of the target state is conditioned on
. This pdf can be divided into the sum of the data association probabilities for the set of measurements by using the total probability theorem, which is given by
where
is the hypothesis that the
th measurement of
is the measurement of target
(for
= 0, target
is not detected) and the data association probability
can be expressed as
A feasible joint event (FJE) is one possible mapping of the measurements to the tracks that follow targets. For each joint event, it is assumed that each track can be assigned to zero or one of the measurements which falls in the validation gate of the track, and each measurement can be allocated to zero or one of the tracks in order to be a FJE. Therefore, the FJE condition implies that no two tracks in a FJE share the same measurement.
Let and denote the jth FJE and the number of all the FJEs for data association at time k, respectively. Then, the sum of the a posteriori probabilities of all the FJEs satisfies
The data association probabilities of track
are obtained by summing over all the probabilities of FJEs that contain track
and the measurement of interest. Denote by
the set of FJEs in which track
is allocated to measurement
(0 means no measurement allocation), we have
where
is the gating probability of track
.
The a posteriori PTE for track in JIPDA is obtained from the sum of the joint probabilities by
Let denote the truncated measurement likelihood function of track for measurement in the validation gate of track ,
Now, the a posteriori probability of FJE
in JIPDA [
10] is defined. Denote by
and
the set of tracks allocated with no measurements and the set of tracks allocated with one measurement for the joint event
in Equation (11), respectively. The a posteriori probability of FJE
is defined by
where
is the index of the measurement allocated to track
in FJE
,
can be obtained by replacing the subscript
with
in Equation (16),
is the clutter measurement density, and the normalization constant
C is calculated from Equation (11).
In fact, tracks are partitioned into clusters [
10]. A cluster is a set of tracks and the measurements these tracks select. In other words, the tracks not belonging to the cluster do not share any of the cluster measurements. The purpose of clustering is to minimize the number of all the FJEs by limiting the numbers of tracks and measurements inside a cluster.
The following is an example to illustrate the set of all the FJEs of JIPDA and the a posteriori probability calculation for the set. Consider the two-dimensional multi-target tracking situation depicted in
Figure 1. There are two cluster tracks, labeled
and
, and three measurements
to
in the cluster. For this cluster, the total number of FJEs is 8.
Each track is assigned to zero or one measurement, and each measurement is allocated to zero or one track. Two FJEs are different if at least one track-to-measurement assignment is different. All the FJEs for the cluster shown in
Figure 1 are listed in
Table 1. Note that we use
j to denote the
j-th FJE in
Table 1.
Let denote the total number of tracks in a cluster. Let and denote the total number of measurements in the cluster and the number of measurements in the validation gate of track in the cluster, respectively. The set of tracks in the cluster, and the sets of and measurements are defined by
The number of unique assignments of
measurements to
tracks, assuming that all tracks select all measurements satisfies [
10]
The number of all the FJEs depends only on the number of measurements, the number of tracks, and the measurements.
Since
and
in Equation (17) are mutually exclusive and exhaustive in the set
of cluster tracks,
The tracks in
are assigned to non-detection and the tracks in
are assigned to one of the cluster measurements that is not shared by other tracks in
. The a posteriori probability
is assigned to the tracks in
and the a posteriori probability
is assigned to the tracks in
.
Therefore, the a posteriori probability of FJE
can be expressed by
where
denotes the measurement state of measurement allocated to track
in the FJE
, and the measurement state can be no detection or a member of
, as described in Equation (25).
If we denote that the number of all FJEs for
and
is
, then the following
matrix of which the element represents the a posteriori probability of track
,
of FJE
. The matrix is called the a posteriori probability matrix of FJEs (PMFJE) and is denoted by
such as
where the rows represent tracks and the columns represent FJEs. The
th element of
represents the a posteriori probability of track to measurement association for track and the measurement with state
for FJE
. The a posteriori probability of FJE
is calculated by multiplying all the elements in the column
j of
such as
where
is an element in the
th row and the
th column of
. Each column
j of
represents the collection of elements in
of FJE
. Any two measurement states
and
that are assigned to track
and
, respectively, in FJE
should be different according to the FJE condition described in
Section 2.1, i.e.,
if
. The PMFJE for the cluster tracks shown in
Figure 1 can be obtained as follows: