 Research
 Open Access
Joint integrated track splitting for multipath multitarget tracking using OTHR detections
 Yuan Huang^{1},
 Taek Lyul Song†^{1}Email authorView ORCID ID profile and
 Joo Hyun Lee^{2}
https://doi.org/10.1186/s1363401805824
© The Author(s) 2018
 Received: 2 January 2018
 Accepted: 4 September 2018
 Published: 20 September 2018
Abstract
In target tracking applications where an overthehorizon radar (OTHR) is used to gather measurements, one target may generate more than one measurement at each scan due to the multipath propagation effect. However, traditional tracking methods obtain the data association probabilities based on tracktomeasurement data association events under the assumption that each target can generate at most one detection at each scan, leading to poor performances if these methods are applied to multiple detection applications such as OTHR applications. In this paper, we develop a multipath multitarget tracking algorithm entitled multiple detection joint integrated track splitting (MDJITS). This novel algorithm jointly solves the measurement origin uncertainty and measurement path model uncertainty. The probability of target existence (PTE) is utilized in the OTHR application as a track quality measure for true track confirmation and false track discrimination. The data association algorithm of MDJITS the proposed algorithm is realized based on measurement cells that each measurement cell consists of one or more of the validated measurements, while considering the measurement path model. The proposed algorithm is compared with the multiple detection joint integrated probabilistic data association (MDJIPDA) algorithm in a multitarget crossing scenario, implementing the OTHR system in the presence of clutter and failed target detections, to demonstrate the desired effectiveness.
Keywords
 Multitarget tracking
 OTHR
 Multiple detection
 Track management
1 Introduction
The core objective for the remote sensing target localization and tracking is to generate target state estimates using measurements processed by the sensors, such as the radar [1–3]. The measurements can originate from targets or from clutter and lead to both true tracks (following the targets) and false tracks (following the clutter) existing in the surveillance area. Therefore, false track discrimination [4], involving true track confirmation and false track termination, is crucial for tracking algorithms. The probability of target existence (PTE) was first introduced in [4] as a soft decision of the existence of a target, which can be used to alleviate the false track discrimination issue.
Most target tracking algorithms assume that the sensor can generate at most one detection per target with a detection probability (usually less than unity) at each scan [5–10]. After the measurement selection procedure, multitarget tracking (MTT) algorithms will generate the joint tracktomeasurement assignments across the tracks. The main difference among the MTT algorithms lies in the way assignments are realized and how the corresponding probabilities are evaluated [11].
Among various MTT algorithms, the multiple hypothesis tracking (MHT) is a theoretically optimal tracking algorithm that enumerates and evaluates all possible tracktomeasurement association hypotheses [5, 12]. Each hypothesis represents a global association considering all tracks and measurements over a number of scans. Based on the different hypothesis generation criteria, different versions of MHT can be divided into two main groups. Measurementoriented MHT [5] generates hypotheses based on measurements, while trackoriented MHT [12] generates hypotheses based on tracks. The maximum likelihood probabilistic multiple hypothesis tracking (MLPMHT) is a powerful nonBayesian algorithm that uses a generalized likelihood ratio test to confirm target existence [13]. Ciuonzo and Horn [14] use a Merkle tree to store and organize hypotheses for a distributed trackoriented multiple hypothesis tracking (TOMHT) resulting in an efficient structure for implementation.
The joint integrated probabilistic data association (JIPDA) tracker is a singlescan MTT algorithm that incorporates the probability of target existence into the JPDA algorithm as a track quality measure [15]. JIPDA is widely used mainly because it is easy to implement and has low computational cost, due to how all the data association events for a track are compressed into one track state at each scan. In the JIPDA framework, the probability of target existence is used to solve the false track discrimination problem, and the track state is generated under the assumption of target existence. Compared to the multiscan global hypothesis generated by the MHT tracker, the JIPDA forms tracktomeasurement data association events based only on current scan circumstances, which significantly lowers the computational expense. As suggested in [12], under adverse circumstances, especially when the clutter density is high and the detection probability is low, JIPDA may have poor tracking performances. In [16, 17], the maximum likelihood probabilistic data association (MLPDA) algorithm is utilized to identify the sensor states as well as target states.
The joint integrated track splitting (JITS) algorithm is a multiscan MTT tracker [11]. Each track generates a hypotheses tree consisting of a set of hypotheses. Each hypothesis is treated as a track component, representing one assertion of the possible target detections history for multiple scans. Every component selects measurements and spawns new components, and these new components propagate to the next scan. The track state is updated by a Gaussian mixture of all the components belonging to that track. The probability of target existence is an inherent part of the tracking state and is recursively updated based on the measurement likelihood ratio, considering all the components. Updating the estimated state and the probability of target existence are two core aspects of JITS. As the number of hypotheses exponentially grows, component reduction methods are used. Pruning and merging are the efficient and most widely used methods [18, 19] for controlling the number of components. Both JIPDA and JITS use the probability of target existence to solve the false track discrimination issue, and the comparison results showed that JITS outperforms JIPDA at the expense of computational complexity [11]. Instead of the global hypotheses being generated by the MHT, JITS forms a hypotheses tree for each track, which significantly reduces the structure complexity. Some of the differences and tracking performance comparisons of IPDA, ITS, and MHT are shown in [20]. Kim and Song [21] proposed the smoothing JITS for multitarget tracking in clutter.
These algorithms assume that for each scan, one target can give rise to at most one detection with a given detection probability of P_{D}<1. However, many practical applications are plagued by the multiple detection problem. This problem is especially prominent when a special kind of radar, called the overthehorizon radar (OTHR) [22–24], is used. In [22–24], the multipath probabilistic data association (MPDA) method is introduced for singletarget tracking. However, the multipath tracks are generated as it lacks a means of discrimination. In the OTHR tracking system [25–28] which is widely used in remote sensing applications, transmission and receiving signals can be scattered by different ionospheric layers which results in different measurement paths (models). The multipath approach leads to multiple detections generated by the same target, and the relationship between measurements and paths is not prior known which results in the measurement path model uncertainty. If the targets in the surveillance region are closely spaced and cross each other, localization and tracking become more difficult. Moreover, due to the incorrect model and measurement combination, multipath tracks are generated and need to be identified. In [25, 27], the multiple detection multiple hypothesis tracking (MDMHT) algorithm and the multidetection probability hypothesis density (MDPHD) algorithm are derived by combining the multiple hypothesis tracking and the random finite set framework. In [29], the Gaussian mixture probability hypothesis density (GMPHD) filter is extended for multipath multitarget tracking with the overthehorizon radar system. But the MHT and the PHD algorithms lack a specialized track quality measure for true and false track management. In [26, 28], the Bernoulli filter and the cardinality balanced multitarget multiBernoulli filter are applied to the OTHR applications. However, the Bernoulli filterbased methods are recommended to use in the environment with high detection probability and low clutter density. In [30], the authors developed the multiradar multitarget tracking algorithm for maritime surveillance at overthehorizon (OTH) distances.
In multiple detection scenarios, the classic tracking algorithms, which assume at most one target originated measurement, lose a large amount of the target motion information contained in the measurements. This weakness is caused by the fact that each of the data association events only considers at most a single measurement per track. Each of the data association events assumes that one of the validated measurements is generated by a target or clutter, and then, the track state is updated based on those hypotheses. If multiple measurements can be considered in one date association event, the tracking performance can be improved significantly due to the target information being more efficiently extracted. The measurement partition method is introduced to generate the possible targetoriented measurement cells and was embedded in the JPDA and PHD frameworks [27, 31]. The measurement partition method assumes that more than one measurement can be generated by one target, forming all the possible targetoriented measurement combinations based both on the validated measurements and on the restriction of the maximum number of target detections. Those measurement cells are then used in the data association events instead of a single measurement. The algorithms in [27, 31] are multitarget tracking methods but do not have the track quality measure which is essential for distinguishing true and false tracks.
Both JIPDA and JITS assume that at most one measurement per target at each scan, which leads to information loss in multiple detection scenarios. By jointly considering the measurement origin uncertainty and the measurement path model uncertainty, two novel structures named multiple detection JIPDA (MDJIPDA) and multiple detection JITS (MDJITS) are designed especially for use in the OTHR tracking system. In these two multiple detection structures, after the measurement selection step, the measurement cells are first generated such that one measurement cell may contain one or more of these validated measurements. Then, the data association events of MDJIPDA and MDJITS are generated using the measurement cells. Combined with prior information from the last scan, the measurement cells are used to update the track state and the probability of target existence.
The track state of MDJIPDA is generated by a Gaussian mixture based on all the data association events, and then, this track propagates to the next scan. The output of MDJIPDA for each track at each scan contains one Gaussian distribution for the track state and the probability of target existence. As for MDJITS, at each scan, each component selects measurements and generates the measurement cells separately. Each component generates new components using any one of its own measurement cells, where each component is represented by a Gaussian distribution for its state and a component probability. The final output for the track state is obtained by a Gaussian mixture based on all the components that belong to the track. Then, the components of the track propagate to the next scan where each component can be seen as a measurement cell assignment history. Compared to MDJIPDA, in which each track propagates as one Gaussian pdf, each MDJITS track propagates as a set of many Gaussian components. The number of components of a track exponentially increases with the number of scans, but actually not all the components are significant for the track. Thus, the components with the component probabilities lower than a given threshold are pruned, and if any two components consecutively use the same measurement cells to update the component states for several scans, these two components should be merged into one [18, 19]. Since the MDJITS track maintains many track components which can be seen as the targetgenerated measurement hypotheses, it outperforms MDJIPDA in the multitarget crossing scenarios.
Initial results of MDJITS are reported in [32] in which the multiple detection problem occurs from a highresolution sensor. The algorithm proposed in [32] only needs to solve the manytoone measurementstotrack associations. However, in order to associate multiple measurements to one track, the work proposed here should jointly consider the measurement origin uncertainty and the measurement path model uncertainty caused by the unknown signal propagation paths of the OTHR. The same measurement with different measurement path models provides completely different target state information. We provide here the rigorous algorithm derivations of MDJITS for the OTHR system by utilizing the combinations of the measurement cells and the path models. We also introduce a multipath track discrimination method for the data association structures.
The problem statement is given in Section 2. The mathematical derivation of MDJITS for the OTHR system and the complexity analyses are given in Section 3. Section 4 contains the simulation studies, followed by conclusions in Section 5.
In this paper, the following abbreviations, notations, and the assumption are employed:
A. Abbreviations:
CFT: Confirmed false tracks; CTTs: Confirmed true tracks; EKF: Extended Kalman filter; JIPDA: Joint integrated probabilistic data association; JITS: Joint integrated track splitting; MDJIPDA: Multiple detection joint integrated probabilistic data association; MDJITS: Multiple detection joint integrated track splitting; MDMHT: Multiple detection multiple hypothesis tracking; MDPHD: Multidetection probability hypothesis density; MLPDA: Maximum likelihood probabilistic data association; MLPMHT: Maximum likelihood probabilistic multiple hypothesis tracking; MPDA: Multipath probabilistic data association; MHT: Multiple hypothesis tracking; MTT: Multitarget tracking; OTHR: Overthehorizon radar; PTE: Probability of target existence; RMSEs: Root mean square errors
B. Notations: \(\chi _{k}^{t}\) The event that track t is tracking a target i.e., target t exists. m_{k} The number of validated measurements at scan k. Φ_{t,max} The maximum number of target originated measurements in the m_{k} validated measurements, Φ_{t,max}= min(L,m_{k}). \({z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)\) One measurement cell of track t at time k. Φ_{t} The number of measurements originated from target t such that 1≤Φ_{t}≤Φ_{t,max}. \({n_{{\Phi _{t}}}} \) An index that indicates is the measurement cell \({z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)\) is one of the possible \({c_{{\Phi _{t}}}}\) combinations where there are Φ_{t} measurements originated from target t. \({n_{{\Phi _{t}}}} \in \left \{ {1,2, \ldots,{c_{{\Phi _{t}}}}} \right \}\) and \({c_{{\Phi _{t}}}} = C_{{\Phi _{t}}}^{{m_{k}}} = \frac {{{m_{k}}!}}{{{\Phi _{t}}!\left ({{m_{k}}  {\Phi _{t}}} \right)!}}\) (the notation \(C_{m}^{n}\) in this paper represents the number of mcombinations from a given set of n elements). z(k) The measurement set that of all the validated measurements at scan k, \(z\left (k \right) = \left \{ {{\textbf {z}_{1}}\left (k \right),{\textbf {z}_{2}}\left (k \right),...,{\textbf {z}_{{m_{k}}}}\left (k \right)} \right \}\). Z^{k} A collection of validated measurement sets that includes all the validated measurements up to and including scan k, \({Z^{k}} = z \left (k \right)\bigcup {{Z^{k  1}}}\).
The superscript or subscript t indicates that the parameter is specified by track t. k always represents the time index.
C. Assumption:
There are L measurement paths, and each target can be detected at most one time through each of those paths. This suggests that there are at most L target measurements from each target at each scan.
2 Problem statement
Here, x(k) represents the target state at scan k, f is the state propagation function, and v(k−1) is the process noise.
In the OTHR system, the high frequency wave reflects through ionospheric layers to detect a target located beyond the horizon. The signal is emitted by the transmitter and reflected by one of the ionosphere layers before it reaches the target, and then, it is reflected by the target and bounces off one of the ionosphere layers before it is received by the receiver. Therefore, the m different ionosphere layers form m^{2} different measurement paths (models). In order to demonstrate the measurement geometry and measurement generation process more concisely, only two ionosphere layers are considered in this paper. Here, two assumptions are made: one is the flat earth model and the other is that the heights of the ionospheric layers are known and fixed. Those two assumptions are employed in most OTHR studies [22, 27, 31] for simplicity. In order to consider the shape of the earth, the earth is introduced as an ellipsoid based on the WGS84 coordinate system. Then, the spherical OTHR measurement geometry can be modeled according to this coordinate system [33]. In realistic situations, the ionosphere state evolves intermittently. The noisy height information can be introduced by tacking into account that the ionosphere state changes slowly and stays invariable within the sampling period. Then, OTHR target tracking with the noisy ionosphere state is performed by joint estimation of fastupdating multitarget states and intermittentupdating the ionosphere state [34–36].
Propagation models
Path_{i}  Model  Transmit layer h_{t}  Receive layer h_{r} 

i=1  EE  h _{ E}  h _{ E} 
i=2  EF  h _{ E}  h _{ F} 
i=3  FE  h _{ F}  h _{ E} 
i=4  FF  h _{ F}  h _{ F} 
where h_{tr} (t,r∈{E,F}) is the measurement function associated with the transmit layer t and receive layer r. w_{tr} (t,r∈{E,F}) is the zero mean Gaussian measurement noise in the corresponding paths. Both h_{tr} and w_{tr} are related to the corresponding model.
where h_{t} and h_{r} represent the height of transmit layer and receive layer associated with different measurement models.
3 Multiple detection JITS
In the MDJITS data association step, one has to consider the possibility that multiple measurements might originate from the same target. This special issue of the MDJITS data association process involves jointly assigning the measurement cells to the tracks. At each scan, each component of each track employs the gating method [1, 2] to validate measurements. After obtaining validated measurements, the measurement partition method is used to generate the measurement cells from these validated measurements. Each measurement cell contains one or some of the validated measurements. Then, a measurement cell assigned to a tracker should not share any elements with other tracks in a feasible joint event [3].
The joint events are used to jointly assign the measurement cells to the tracks, and each joint event is composed of tracks with disjoint measurement cells. In a joint event, each track is assigned to one measurement cell or unassigned. Then, the joint event probabilities are calculated for the corresponding joint events.
3.1 Measurement partition method
In order to assign multiple measurements to one track, the measurement cells are generated. Each measurement cell can be treated as a measurement set which contains at most Φ_{t,max} elements of the validated measurements. In a data association event, if a measurement cell is assigned to a track, the measurements contained in this measurement cell are used for the state estimation and the data association probability calculation. The measurement cells are generated using the validated measurements based on the assumption that there are Φ_{t} measurements originated from target t.

Suppose that only one of the validated measurements is the target detection (Φ_{t}=1) and n_{1}∈{1,2,3}. The measurement cells are:$$ \begin{aligned} {z_{1,1}}\left(k \right) &= \left\{ {{\textbf{z}_{1}}\left(k \right)} \right\};\\ {z_{1,2}}\left(k \right) &= \left\{ {{\textbf{z}_{2}}\left(k \right)} \right\};\\ {z_{1,3}}\left(k \right) &= \left\{ {{\textbf{z}_{4}}\left(k \right)} \right\}. \end{aligned} $$

Suppose that two of the validated measurements are the target detections (Φ_{t}=2) and n_{2}∈{1,2,3}. The measurement cells are:$$ \begin{aligned} {z_{2,1}}\left(k \right) = \left\{ {{\textbf{z}_{1}}\left(k \right),{\textbf{z}_{2}}\left(k \right)} \right\};\\ {z_{2,2}}\left(k \right) = \left\{ {{\textbf{z}_{1}}\left(k \right),{\textbf{z}_{4}}\left(k \right)} \right\};\\ {z_{2,3}}\left(k \right) = \left\{ {{\textbf{z}_{2}}\left(k \right),{\textbf{z}_{4}}\left(k \right)} \right\}. \end{aligned} $$

Suppose that all these three measurements are the target detections (Φ_{t}=3) and n_{3}∈{1}. The measurement cell is:$$ \begin{aligned} {z_{3,1}}\left(k \right) = \left\{ {{\textbf{z}_{1}}\left(k \right),{\textbf{z}_{2}}\left(k \right),{\textbf{z}_{4}}\left(k \right)} \right\}. \end{aligned} $$
After the measurement partition process, seven measurement cells are generated. These measurement cells are utilized in the data associations for track t. When measurement cell z_{2,2}(k) is considered in a data association event, the measurements contained in z_{2,2}(k), which are z_{1}(k) and z_{4}(k), are treated as target detections.
The number of measurement cells increases with the number of validated measurements m_{k} and the maximum number of target originated measurements Φ_{t,max}. After the measurement partition process, the number of measurement cells becomes \({\sum \nolimits }_{{\Phi _{t}} = 1}^{{\Phi _{t,\max }}} {C_{{\Phi _{t}}}^{{m_{k}}}} \).
Here, the measurement cells are generated without considering the measurement cell path model. The measurement cell path model, which contains path models for each measurement in the measurement cell, will be specified in Section 3.2.
3.2 MDJITS
When there are tracks that share measurements, the feasible joint events are used to solve the tracktomeasurement assignment issue. The feasible joint events cover all possible tracktomeasurement assignments for all the tracks and the measurements in a cluster. A cluster at scan k is a set of the tracks and the measurements these tracks select. A track inside the cluster should share measurements with one or more different tracks in the cluster, and the tracks not belonging to the cluster do not select any of the cluster measurements. In the multiple detection case, the measurement cells composed of different validated measurements are assigned to the tracks in one feasible joint event.
In this paper, for reasons of clarity and without loss of generality, it is assumed that all the measurements in the cluster are validated by all the tracks belonging to that cluster and that the detection probability of each path is considered identical, i.e., P_{DEE} = P_{DEF} =P_{DFE}=P_{DFF}=P_{D}.
the event that none of the validated measurements are the detections of target t.
where each κ^{t}(i), i∈{1,2,…,k} can be of the form from (5) or (6). Each κ^{t,k} represents one component for track t at time k.
Here, one recursion of MDJITS at scan k is illustrated. At scan k, the following posterior information at scan k−1 is assumed to be available: (1) The track state probability density function (pdf) for track t, \(p\left ({{\textbf {x}^{t}}\left ({k  1} \right)\chi _{k  1}^{t},{Z^{k  1}}} \right)\); (2) the probability of target existence for track t, \(P\left ({\chi _{k  1}^{t}{Z^{k  1}}} \right)\), which represents the probability of target existence of a target tracked by track t; (3) the component state for each component of track t, \(p\left ({{\textbf {x}^{t}}\left ({k  1} \right)\chi _{k  1}^{t},{\kappa ^{t,k  1}},{Z^{k  1}}} \right)\); and (4) the component probability for each component belongs to track t, which is P(κ^{t,k−1}χ_{k−1},Z^{k−1}).
where the Kalman filter prediction can be used for a linear state propagation model. If κ^{t,k−1} is omitted, this equation can be used for MDJIPDA track state prediction.
the details can be found in [12, 20].
The following describes the measurement selection of each component of track t. Each component of track t selects the measurements in its validation gate, and all the components of track t form a common validation gate and common validated measurements of track t [20, 37]. Based on (8) and the selection process of the validated measurements, each component of track t generates the predicted measurement for each measurement Path_{i} and selects the measurements inside the validation gate corresponding to Path_{i} using an ellipse gating method [3].
In (10), P_{G} is the gating probability of a single detection. The Gaussian pdf for measurement cell \({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)}\) is employed, and each measurement cell is allocated with one measurement cell path model \({M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}\). Note that \({{\mathbf {\zeta }_{M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}}}\) represents the measurement prediction based on the state prediction (8) and measurement cell path model \({M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}\). Note also that \({{\textbf {S}_{M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}}}\) represents the measurement cell innovation covariance corresponding to measurement cell \({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)}\) given measurement cell path model \({M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}\). One can apply the extended Kalman filter (EKF) to obtain state estimates for the equations. The details for generating these estimates can be found in [22, 23].
where this combined measurement likelihood function utilizes all the components that use measurement cell \({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)}\) with the allocated path model \({M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}\). This combined measurement cell likelihood is used to calculate the joint event probability.
Then, the data association step is processed. In the following derivations, “track” and “measurement” mean cluster tracks and cluster measurements in a cluster, respectively. The validated measurement set z(k) is the union of the validated measurements validated by each of the tracks belonging to the cluster. A joint event ε is an event of assigning measurement cells including nondetection to all the tracks. One joint event should satisfy the following: (1) Each track is assigned to at most one measurement cell and (2) each measurement cell is assigned to at most one track.
where c is a normalization constant and m_{k} is the number of validated measurements in the cluster. p(z(k)ε,m_{k},Z^{k−1}) is the joint measurement likelihood function for measurement set z(k); P(m_{k}ε,Z^{k−1}) represents the a priori probability for the number of the measurements; and P(εZ^{k−1}) is the a priori probability of the joint event.
where V_{k} is the volume of the surveillance region.
For the joint event ε, the relation between measurement cell \({z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)\) and measurement cell path model \(M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)\) for every track involved in ε is already known.
where \({\hat {m}}_{k}\) is the mean number of clutter measurements such that \({\hat {m}_{k}}=\lambda {V_{k}}\) with the clutter measurement density λ.
where \(P\left ({\chi _{k}^{t}{Z^{k  1}}} \right)\) is the predicted probability of the existence of target t. The probability that there is at least one target detection in the validation gate is given by P_{Dec}=1−(1−P_{D}P_{G})^{L}.
Equation (17) represents the prediction process for the probability of target existence where \({P_{ij}} = P\left [ {e_{k}^{t} = je_{k  1}^{t} = i} \right ]\), j∈{1,2} represents the transitional probability between the existence states. P_{ij} satisfies \(\sum \limits _{j = 1}^{2} {{P_{ij}} = 1} \) for i∈{1,2}.
Then, those joint events and the corresponding event probabilities are used to generate the data association probabilities of measurement celltotrack assignments for MDJIPDA and MDJITS.
The details for deriving (22) can be found in [11, 20].
where \(\sum \limits _{{\kappa ^{t}}\left (k \right)}\) denotes the summation through all feasible joint events that contain the measurement cells with the allocated measurement cell path models of track t including κ^{t}(k)=〈00〉. The probability of nonexistence \(P\left ({\bar \chi _{k}^{t}{Z^{k}}} \right)\) is equal to \(1P\left ({\chi _{k}^{t}{Z^{k}}} \right)\).
where the likelihood functions \(p\left ({z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right){\chi ^{t}_{k}}, {\kappa ^{t,k  1}},\right. \left. {M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)}, {Z^{k1}} \right)\) and \({{p\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right){\chi ^{t}_{k}}, M_{{j_{k}}}^{t}}\right.}} \left.{\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)}\right)}, {Z^{k1}} \right)\) are presented in (10) and (11), respectively.
where \(p\left ({{\textbf {x}^{t}}\left (k \right)\chi _{k}^{t},{\kappa ^{t,k}},{Z^{k}}} \right)\) is the a posteriori state pdf for each component, which can be updated by the existing nonlinear filters. If the EKF is used, the details to calculate the Jacobian matrix can be found in [22, 23]. \(P\left ({{\kappa ^{t,k}}{\chi ^{t}_{k}},{Z^{k}}} \right)\) is the a posteriori component probability, presented in either (28) or (29) depending on context. Equation (30) is different from Eq. (27) since the association history, which is κ^{t,k}, is considered in (30) and the association at current scan, which is κ^{t}(k), is considered in (27).
Pseudocodes for MDJITS and MDJIPDA
MDJITS track update process 
for each track 
1. Probability of target existence prediction, \({P\left ({\chi _{k}^{t}{Z^{k  1}}} \right)}\), (17) 
2. Track component state prediction, \(p\left ({{\textbf {x}^{t}}\left (k \right)\chi _{k}^{t},{\kappa ^{t,k  1}},{Z^{k  1}}} \right)\), (8) 
3. Measurement selection based on track components 
4. Measurement cells generation using measurement partition method 
5. Data association event probabilities, 
\(\beta _{k}^{t}\left ({\left \langle {00} \right \rangle } \right)\) and \(\beta _{k}^{t}\left ({\left \langle {{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)} \right \rangle } \right)\), (25) and (26) 
6. Track component probability, P(κ^{t,k}χ_{k},Z^{k}), (28) and (29) 
7. Component state generation using nonlinear filter, 
\(p\left ({{\textbf {x}^{t}}\left (k \right)\chi _{k}^{t},{\kappa ^{t,k}},{Z^{k}}} \right)\) 
8. Track state at scan k, \(p\left ({{\textbf {x}^{t}}\left (k \right)\chi _{k}^{t},{Z^{k}}} \right)\), (30) 
9. The probability of target existence at scan k, \(P\left ({\chi _{k}^{t}{Z^{k}}} \right)\), (24) 
end for 
MDJIPDA track update process 
for each track 
1. Probability of target existence prediction, \({P\left ({\chi _{k}^{t}{Z^{k  1}}} \right)}\), (17) 
2. Track state prediction, \(p\left ({{\textbf {x}^{t}}\left (k \right)\chi _{k}^{t},{Z^{k  1}}} \right)\), (8) 
3. Measurement selection based on track 
4. Measurement cells generation using measurement partition method 
5. Data association event probabilities, 
\(\beta _{k}^{t}\left ({\left \langle {00} \right \rangle } \right)\) and \(\beta _{k}^{t}\left ({\left \langle {{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)} \right \rangle } \right)\), (25) and (26) 
6. Track state generated by each data association event using 
existing nonlinear filter 
7. Track state at scan k, \(p\left ({{\textbf {x}^{t}}\left (k \right)\chi _{k}^{t},{Z^{k}}} \right)\), (27) 
8. The probability of target existence at scan k, \(P\left ({\chi _{k}^{t}{Z^{k}}} \right)\), (24) 
end for 
3.3 Complexity analyses
Multiple detection multitarget tracking algorithms allow for manytoone measurementstotrack assignments in the joint events. The number of feasible joint assignments (i.e., the number of the joint events) is combinatorial in the number of the measurements and the number of the tracks. In multiple detection multitarget tracking, most of the computational load is generated by these joint assignments. The computational load generated by this part is identical in MDJITS and MDJIPDA, because the same joint assignment process is used in these two algorithms.
The main difference between MDJITS and MDJIPDA lies in the track structure. In MDJITS, each track is modeled as a set of track components that each component has a unique measurement assignment history that consists of zero or some of the validated measurements at each scan. A track is the union of its mutually exclusive components. However, each track in MDJIPDA is expressed by one Gaussian pdf. The track structure difference makes MDJITS more robust but more timeconsuming compared to MDJIPDA.
In order to analyze the complexity of MDJITS, the joint measurementstotrack assignments and the track structure are discussed.
A. Complexity of joint measurementstotrack assignments
where N is the number of tracks considered.
where l_{t} (t=1,⋯,N) represents the number of the pathassigned measurements allocated to track t.
Number of joint events
M _{ k}  Number of tracks  

1  2  3  
9  130  6799  136,348 
10  176  8980  398,476 
11  232  11,603  1,079,464 
12  299  14,710  2,715,541 
B. Complexity of the track structure
In this part, the complexities of the track propagation of a track in MDJITS and MDJIPDA are analyzed.
Let B_{k} denote the number of the pathassigned measurement cells represented by \(\left \langle {z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)M_{{j_{k}}}^{t} \right. \left.\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right) {\vphantom {M_{{j_{k}}}^{t}}}\right \rangle \) including the special case 〈00〉. In MDJITS, each measurement cell and path model combination (\(\left \langle {{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)M_{{j_{k}}}^{t}\left ({{z_{{\Phi _{t}},{n_{{\Phi _{t}}}}}}\left (k \right)} \right)} \right \rangle \) or 〈00〉) is combined to the parent component κ^{t,k−1} to make a new component. Thus, component κ^{t,k−1} becomes B_{k} new components after processing the combinations. Finally, the total number of new components becomes K·B_{k} after processing all the combinations with all the K parent components. For each component, as well as for each track, the state estimate and the a posteriori component probability are computed recursively. The MDJITS track propagation has the complexity of \( \mathcal {O}\left ({K \cdot {B_{k}}} \right)\) at scan k. The complexity of this algorithm increases as the increasing of the number of measurement.
The number of components grows exponentially in time. Practical implementation of MDJITS must include procedures to control the number of components. Component merging and pruning techniques have been described in [37].
Each track in MDJIPDA keeps only one Gaussian pdf, representing the track state estimate. At scan k, B_{k} combinations are associated to a track. This association problem can be solved with the complexity \(\mathcal {O}\left ({B_{k}} \right)\).
4 Results and discussion
Surveillance area parameters
Parameter  Value 

Slant range size  1000–1400 km 
Rate of slant range size  0.013889–0.22222 km/s 
Apparent azimuth size  0.069813–0.17453 rad 
T to R distance d  100 km 
Hight of layer E  100 km 
Hight of layer F  260 km 
Target parameters
Parameter  Value 

Target1  [1050 km;0.15 km/s;0.10472 rad;8.72665×10^{−5} rad/s] 
Target2  [1225 km;−0.14 km/s;0.11472 rad;7.72665×10^{−5} rad/s] 
Target3  [1300 km;−0.17 km/s;0.14701 rad;−4.72665×10^{−5} rad/s] 
Target4  [1050 km;0.1 km/s;0.18701 rad;0 rad/s] 
P _{ D}  0.4 
P _{ G}  0.9966 
The rate of bearing is set to be 0 in the initialization step.
Some of those parameters are the same as given in [39].
4.1 False track discrimination and track accuracy

A track is initialized by the onepoint initialization method. Each track is assigned an initial probability of target existence in the track initialization step. The values of the initial probability of target existence for MDJIPDA tracks and MDJITS tracks are different, ensuring that those two trackers have the same number of confirmed false tracks.

The probability of target existence of a track is updated scan by scan, and once it exceeds the confirmation probability of target existence (0.98), the track becomes a confirmed track.

The confirmed track is tested as to whether it is a confirmed true track or a confirmed false track by the following criteria:$$ \begin{aligned} {\left(\tilde{\textbf{x}}\left({kk} \right)\right)}^{T} \textbf{P}_{00}^{ 1}\tilde{\textbf{x}}\left({kk} \right) < 20~ &{{\text{for~comfirmed~true~track}}},\\ {\left({\tilde{\textbf{x}}}\left({kk} \right)\right)}^{T} \textbf{P}_{00}^{ 1}\tilde{\textbf{x}}\left({kk} \right) > 40~&{{\text{for~comfirmed~false~track}}}. \end{aligned} $$(40)
Here, \(\tilde {\textbf {x}}\left ({kk} \right) = \textbf {x}\left (k \right)  \hat {\textbf {x}}\left ({kk} \right)\) is the state estimation error of a confirmed track at time k and P_{00} is the initial state covariance given in (32).

If the probability of target existence of a track is lower than the termination probability of target existence (initial probability of target existence/5), the track is terminated.
Tracker parameters
MDJIPDA  MDJITS  

Initial PTE  0.000001  0.0017 
Confirmation PTE  0.98  0.98 
Termination PTE  0.000001/5  0.0017/5 
Number of CFT  9  9 
The RMSEs in the bearing of target 2 and target 3 are shown in Fig. 5. The RMSEs in the bearing first increase due to the bearing rate being set as 0 in the track initialization step, and so both of these two trackers need some time to recover this parameter. Then, the estimation error in the bearing estimation is reduced as the number of scans increases. All these simulation results demonstrate the benefits of using MDJITS. MDJITS outperforms MDJIPDA, especially in terms of false track discrimination, while MDJITS maintains the same estimation accuracy in terms of RMSE in both range and bearing estimates.
This simulation is implemented on a 4.00 GHz, Intel Core i7 PC and run with MATLAB. The CPU time per each Monte Carlo run for MDJIPDA and MDJITS are 345.7360 s and 349.3160 s, respectively. The simulation time is similar since most of the computational load is assigned to the joint measurementstotrack assignment part, which is identical for both MDJIPDA and MDJITS. Since MDJITS needs to maintain track components, the computational cost of MDJITS is a little bit higher than that of MDJIPDA.
All the simulation results suggest that MDJITS has better true track confirmation performances compared to MDJIPDA. However, in order to achieve this benefit, MDJITS is more timeconsuming.
4.2 A measure for the multipath track discrimination
In the OTHR system, there is a special kind of tracks called the multipath track [39]. The multipath track is generated due to an incorrect combination between the measurements and the propagation models at each scan. The multipath track is different from the false track since the multipath track uses target measurement to update the track state and propagate parallel to the true target trajectory, while the false track is updated by using the clutter measurement and wanders away from the true target trajectory. Thus, one of the tasks when using the OTHR tracking system is to distinguish between the multipath track and the true track. The mechanism analysis details are given in [26]. Following this, we propose a method for both MDJIPDA and MDJITS to distinguish the multipath tracks.
The maximum number of measurement paths (models) for each target is defined as L. Here, we define a variable vector \(\left [ {a_{k,1}^{t},a_{k,2}^{t}, \ldots,a_{k,L}^{t}} \right ]\) (\(a_{k,i}^{t} \in \left \{ {0,1} \right \},i = 1,2, \ldots L\)) that is used to count the active paths at scan k for track t. \({a_{k,i}^{t}} = 1\) means that path i is active at scan k, and \({a_{k,i}^{t}} = 0\) means that path i is not active at scan k. In MDJIPDA, the state estimation is generated by a Gaussian mixture considering all of the data association events, where each event represents the track state pdf for the track involved in the event. Among all the data association events of a track, the one with the highest data association probability is selected and the measurement path models considered in this data association event are counted. For example, in the event that has the highest data association probability, it turns out that z_{1}(k) and z_{3}(k) are considered as target detections. z_{1}(k) comes from path 1 and z_{3}(k) comes from path 2; thus, the corresponding active model vector is \(\left [ {a_{k,1}^{t},a_{k,2}^{t},a_{k,3}^{t},a_{k,4}^{t}} \right ] = \left [ {1,1,0,0} \right ]\) (path 1 and path 2 are active). As for MDJITS, the measurement path models considered in the component with the highest component probability are counted. Then, after each simulation run, we obtain a statistic of the active paths over all the scans \(\left [ {a_{1}^{t},a_{2}^{t}, \ldots,a_{L}^{t}} \right ]\), where \(a_{i}^{t} = {\sum \nolimits }_{k} {a_{k,i}^{t}} \). The active path statistic \(\left [ {a_{1}^{t},a_{2}^{t}, \ldots,a_{L}^{t}} \right ]\) is used to distinguish between the true track and the multipath track. The proposed method is used in Section 4 to discriminate the multipath tracks.
Active model statistic
Track  Active path statistic [EE, EF, FE, FF] 

T1  [16, 16, 17, 15] 
T2  [19, 13, 2, 0] 
5 Conclusions
In this paper, in order to jointly resolve the measurement origin uncertainty and measurement path model uncertainty in the OTHR system, we developed the MDJITS filter. The main benefit of this approach is that it generates all possible targetoriented measurement combinations with validated measurements considering possible measurement propagation models. The target state information contained in measurements are more efficiently extracted due to onetomany, tracktomeasurement cell association for this multiple detection problem instead of the existing onetoone association for single detection. Compared to the widely used PDA framework, MDJITS is a multiscan tracker that maintains information on the measurement cell history and propagates based on the components, making it more robust in harsh tracking environments.
To demonstrate the superiority of the proposed MDJITS filter, we compared it with the MDJIPDA filter in a MTT environment using the OTHR system. The simulation results indicate that the new filter outperforms the MDJIPDA filter for true track confirmation and maintenance and shows the same state estimation accuracy as the MDJIPDA filter.
There are several aspects worthy for the further work: the method which considers the time varying heights of the ionospheric layers can be inserted into the MDJITS structure; the more computationally efficient structure of the MDJITS algorithm is expected to be developed.
Notes
Declarations
Acknowledgements
The authors would like to thank the anonymous reviewers for the improvement of this paper.
Funding
This work was supported by Hanwha Systems, Radar R&D Center through Grant U17017.
Availability of data and materials
The datasets supporting the conclusions of this article are included within the article (and its additional file(s)).
Authors’ contributions
YH made the main contributions to conception and tracking algorithms’ design, as well as drafting the article. TLS offered critical suggestions on the algorithms’ design, provided significant revising for important intellectual content, and gave final approval of the current version to be submitted. JHL helped to answer the reviewers’ questions and revise the manuscript. All the authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Publisher’s Note
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Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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