Fixed lag smoothing target tracking in clutter for a high pulse repetition frequency radar
- Uzair Khan^{1},
- Yi Fang Shi^{1} and
- Taek Lyul Song^{1}Email author
https://doi.org/10.1186/s13634-015-0234-x
© Khan et al. 2015
Received: 16 February 2015
Accepted: 27 May 2015
Published: 7 June 2015
Abstract
A new method to smooth the target hybrid state with Gaussian mixture measurement likelihood-integrated track splitting (GMM-ITS) in the presence of clutter for a high pulse repetition frequency (HPRF) radar is proposed. This method smooths the target state at fixed lag N and considers all feasible multi-scan target existence sequences in the temporal window of scans in order to smooth the target hybrid state. The smoothing window can be of any length N. The proposed method to smooth the target hybrid state at fixed lag is also applied to the enhanced multiple model (EMM) tracking algorithm. Simulation results indicate that the performance of fixed lag smoothing GMM-ITS significantly improves false track discrimination and root mean square errors (RMSEs).
Keywords
1 Introduction
It is well known that, when a Pulse-Doppler radar operates in high pulse repetition frequency (HPRF) mode, the target range information is ambiguous due to the aliasing range [1]. In the absence of measurement noise, multiple possible target ranges project onto the same range measurement.
Nagel and Hommel [2] propose the range-gated HPRF method to resolve range ambiguity; however, because of the limited number of bursts, the number of range gates is often not sufficient, especially in a cluttered environment with uncertain target detection. The authors in [3] present a multiple model algorithm to eliminate the range ambiguity problem in an HPRF radar. These references estimate the trajectory state without providing a track quality measure for false track discrimination (FTD). The Gaussian mixture measurement likelihood-integrated track splitting algorithm (GMM-ITS) [4] and the enhanced multiple model algorithm (EMM) (it incorporates the track quality measure in a multiple model algorithm (MM) [3]) are investigated in [5] for single-target tracking in clutter using an HPRF radar; both algorithms are capable of trajectory estimation and false track discrimination.
The application of smoothing is quite effective in the situation awareness and threat assessment applications. The past state of the target is updated using all measurement information received till the current scan. Smoothing produces a reduction in estimation error and improves the FTD. The fundamental techniques of smoothing in estimation are provided in [6].
The idea of fixed lag smoothing is proposed in [7]. Later, its application in target tracking is considered in different ways. [8] consider the multi-scan data association for tracking a target and applies fixed lag smoothing; however, it does not consider the track quality measure. [9] applies the RTS to the MHT algorithm, but it also ignores the track quality measure. Chakravorty and Challa [10] propose augmented state integrated probabilistic data association (ASIPDA). In ASIPDA, however, the smoothing probability of target existence uses the smoothing innovation obtained only in the current scan. In sIPDA [11], the authors use the Fraser-Potter [12] approach for smoothing with IPDA. Another extension of smoothing considers track splitting for single-target smoothing [13].
This paper presents a new method to smooth the target hybrid state at fixed lag N. It applies the proposed smoothing algorithm on both GMM-ITS and EMM algorithms to obtain the smoothing benefits for an HPRF radar. In this technique, a relatively small window of scans is defined in each smoothing interval. The target trajectory state and target existence state are smoothed at fixed lag N in the smoothing interval. This technique considers all feasible multi-scan target existence events and smoothed state estimates (using the augmented state GMM-ITS update) calculated at all intermediate scans in the smoothing interval in order to smooth the target hybrid state at fixed lag N.
The fixed lag smoothing-based tracking algorithms proposed in this work also provide a significant improvement over existing online algorithms in terms of false track discrimination and root mean square errors (RMSEs).
This paper is organized as follows. The basic models used are discussed in Section 2, while in Section 3 an overview of GMM-ITS and EMM algorithms is provided. The GMM-ITS algorithm for the augmented target state is extended in Section 4. The fixed lag smoothing GMM-ITS (FLs GMM-ITS) algorithm is proposed next in Section 5. The simulation results are presented in Section 7, followed by concluding remarks in Section 8.
2 Mathematical model
Fundamental nomenclature used in the proposed work
Z ^{ k } | Cumulative set of measurements from the sensor from the initial scan to scan k |
Z _{ k } | Measurement set at scan k from the sensor |
Z _{ k,i } | i-th measurement of Z _{ k } |
z _{ k,i } | i-th validated measurement from Z _{ k } |
\(\textbf {y}_{k,i}^{g}\) | g-th measurement component of Z _{ k,i } |
\(\textbf {y}_{k,i}^{g,p}\) | Position component of \(\textbf {y}_{k,i}^{g}\) |
\(\textbf {y}_{k,i}^{g,w}\) | Decorrelated Doppler component of \(\textbf {y}_{k,i}^{g}\) |
\(\textbf {y}_{k,i}^{g,{c_{{k_{\tau }} - 1}}}\) | g-th measurement component of Z _{ k,i } selected by the track component \({c_{{k_{\tau }} - 1}}\) |
\({\bar {c}}_{k}\) | Newly formed track component at scan k |
c _{ k−1} | Track component at previous scan k−1 |
\(\xi _{{~}_{{k }}}^{{{\bar c}_{{k }}}}\) | Probability of track component \({\bar {c}}_{k}\) |
ξ _{ k−1} c _{ k−1} | Probability of track component c _{ k−1} |
2.1 Target model
where the scalar q denotes the root mean square of the acceleration plant noise and T is the sampling time.
2.2 Measurement model
2.2.1 Target measurement
Due to the ambiguous range, GMM-ITS regenerates a set of Gaussian measurement components to update the track. Measurement components G _{ k,i } with respect to each measurement Z _{ k,i } are created, and each Gaussian component is defined by its mean \(\textbf {y}_{k,i}^{g}\), covariance \(\textbf {R}_{k,i}^{g}\), and relative weight \(\gamma _{k,i}^{g}\) [5].
2.2.2 Clutter measurement
where ρ ^{ p } denotes the clutter measurement position density and ρ ^{ d } is the Doppler component density of clutter measurement [5].
where \({\mathbf {y}_{k,i}^{g,p}}\) is the position component of measurement \({\mathbf {y}_{k,i}^{g}}\) and \({w_{k,i}^{g}}\) is the decorrelated Doppler component of measurement \({\mathbf {y}_{k,i}^{g}}\).
3 An overview on GMM-ITS algorithm and EMM algorithm
This section presents a brief description about the GMM-ITS algorithm and the EMM tracking algorithm.
3.1 Gaussian mixture measurement ITS
The GMM-ITS algorithm is first introduced in [4], later it is applied to the problem of single-target tracking in clutter using an HPRF radar [5, 18]. In GMM-ITS algorithm, the non-linear (non-Gaussian) measurement likelihood is approximated by a Gaussian mixture of measurement components, which corresponds to the ambiguous measurement components due to the aliasing target range in the application of an HPRF radar.
3.1.1 GMM model for an HPRF radar
3.1.2 GMM model integration with ITS algorithm
Once the GMM model for HPRF radar is formed, the GMM-ITS algorithm absorbs the integrated track splitting (ITS) to proceed subsequent tracking procedure. The ITS [15] algorithm is a multi-scan tracking algorithm, which updates the target trajectory state and the target existence state at each scan k using multi-scan data association. The GMM-ITS algorithm uses ITS for data association to obtain the improvement in the performance of tracker in clutter using an HPRF radar [5].
3.2 Extended multiple model (EMM)
The multiple model (MM) tracking algorithm for an HPRF radar is proposed in [3], where each model proceeds in the probabilistic data association (PDA) [19] sense independently. The MM tracking algorithm is modified by incorporating the probability of target existence update to develop enhanced multiple model (EMM) tracking algorithm [5].
4 Augmented state GMM-ITS
This section presents one complete recursion cycle for augmented state GMM-ITS (AS GMM-ITS). At each scan, the augmented state is smoothed, and the results are later used in Section 5.1 to determine the smoothed target state at a fixed lag of N using the proposed method.
The GMM-ITS algorithm [4] details are omitted, and only its application to the augmented state is discussed in order to minimize the complexity at this stage. Let k _{ τ }=k−N+r, (0≤r≤N) be the variable to address each scan in the smoothing interval. Each measurement in each scan not selected by the already established track initializes a new track [5].
where \(\left [ {{\hat {\mathbf {x}}_{k - N|k - N}},{{\mathbf {P}}_{k - N|k - N}}} \right ]\) is the filtered target trajectory state and its error covariance matrix updated at scan k−N. The track trajectory state is approximated by a single Gaussian component c _{ k−N } at initialization. Here, the probability of the component c _{ k−N } is \({\xi _{k-N}^{c_{k-N}}}=1\).
4.1 State augmentation
where I _{ n } is an n-dimensional identity matrix, and 0 _{ n,n } and 0 _{ m,n } are matrices of zeros, where n is the order of the target state vector, and m is the order of the position measurement vector. Terms w _{ k−1}, v _{ k }, F _{ k−1}, and Q _{ k−1} are defined in Section 2.1. The order of matrices F ^{AS} and Q ^{AS} is ((N+1).n×(N+1).n), while H ^{AS,p } has dimensions of (m×(N+1).n), and H ^{AS,w } has dimensions of (1×(N+1).n).
4.2 State prediction
4.3 Measurement selection and likelihood calculation
To reduce the computational requirements, a subset of validated measurements from all the measurements received by the sensor is selected at each scan k _{ τ } in the smoothing interval. A gating procedure[20] is used to select the validated measurements for each measurement component g corresponding to each track component \({c_{k_{\tau } - 1}}\). A gating test (28) is applied to the position component (\({\textbf {y}_{k_{\tau },i}^{g,{p}}} \)) of each measurement received at each scan in the smoothing interval.
where \({{\textit {\textbf {S}}}}_{{k_{\tau } },i}^{s^{p}}\) is as defined in (30).
4.4 State update
The track augmented trajectory state \({\hat {\textit {\textbf {X}}}}_{k_{\tau }}^{\text {AS}}\) and its augmented error covariance matrix \({{\textit {\textbf {P}}}}_{k_{\tau }}^{\text {AS}}\) provide filtered and smoothed trajectory states \(\left \{{\textbf {\textit \^{x}}}_{{k_{\tau } }|{k_{\tau } }},{{{\textbf {\textit \^{x}}}}_{{k_{\tau }} - 1|{k_{\tau } }}}, \ldots,{{\textbf {\textit \^{x}}}}_{k_{\tau }} - N|{k_{\tau } } \right \}\) and their corresponding error covariance matrices \(\left \{ {{{{\textit {\textbf {P}}}}_{{k_{\tau } }|{k_{\tau } }}},{{{\textit {\textbf {P}}}}_{{k_{\tau }} - 1|{k_{\tau } }}}, \ldots,{{{\textit {\textbf {P}}}}_{{k_{\tau }} - N|{k_{\tau } }}}} \right \}\) at scan k _{ τ } in the smoothing interval.
5 Fixed lag smoothing GMM-ITS
The fixed lag smoothing GMM-ITS (FLs GMM-ITS) algorithm for any size of smoothing lag N is proposed. The fixed lag smoothing IPDA (FLs-IPDA) [21] uses a single-scan data association in the smoothing interval. It considers only two scans in the smoothing interval and does not provide any generalization to smooth the target state at any size of fixed lag.
In the FLs GMM-ITS algorithm, the concept of fixed lag smoothing is extended for a more general case of smoothing for any size of lag N, and it utilizes the benefits of multi-scan data association. The results of the augmented state GMM-ITS algorithm derived in Section 4 are used in the smoothing interval. The smoothed state with respect to scan k−N, obtained in (44) and (45), at each scan is weighted by the probabilistic weights calculated using multi-scan target existence events. In Sections 5.1 and 5.2, the FLs GMM-ITS algorithm updates the target trajectory state p(x _{ k−N }|χ _{ k−N },Z ^{ k }) and target existence state P{χ _{ k−N }|Z ^{ k }} in the smoothing interval.
In the next smoothing interval, the target state with respect to scan k−N is ignored (as it was updated in the last smoothing interval), and a future scan k+1 is added in the smoothing interval to smooth the track state at scan k−N+1.
In the smoothing interval, there are N+1 feasible multi-scan target existence events.
- ᅟ
(1) Target exists at all scans k _{ τ } in the smoothing interval, where 1≤r≤N.
- ᅟ
(2) Non-existence of target at any scan k _{ τ } implies target non-existence at all following scans.
- ᅟ
(3) Target does not exist at any scan k _{ τ } in the smoothing interval.
These conditions considers N+1 feasible multi-scan target existence events in the smoothing interval. The number of feasible multi-scan target existence events increases in a linear manner with the increase in smoothing lag. The next sections present the formulas for smoothed hybrid target state for any fixed lag N.
5.1 Smoothed target trajectory state at fixed lag N
The target trajectory state is composed of position and velocity components. The information of N future scans are used to smooth the target trajectory state at scan k−N. The feasible multi-scan target existence events are used to weight the smoothed target state estimates received at each scan in the interval. The smoothed state estimates are obtained by augmented trajectory state update (44) and augmented state error covariance matrix update (45) at each scan k _{ τ } in the interval.
The first term in the summation on the right-hand side is the smoothed trajectory state estimate \({\textbf {\textit \^{x}}}_{k-N|{k_{\tau }}}\) at each scan in the smoothing interval k _{ τ }=[k − N, k − N + 1, …,k] in Section 4 using (44) and (45). Υ(s) are the probabilistic weights calculated using the multi-scan target existence events in the smoothing interval.
To maintain the clarity, the index variable s is used to address each scan in the interval, 1≤s≤N+1. The factor Υ(s) is
The weighting factor Υ(s) satisfies
The term in (51) for s=1 is
where p _{ ρ,k−s+1} is the likelihood that a measurement belongs to clutter, and μ _{ F }(m _{ k−s+1}) is the Poisson distribution function for the clutter measurements. Δ _{ k−s+1} is calculated using (47) by replacing k _{ τ } with k−s+1.
and p _{11} is the state transition probability.
5.2 Smoothed target existence state at fixed lag N
Compared to (64), in (51), the multi-scan target existence probabilities are conditioned on the target existence event χ _{ k−N }, as target trajectory state for non-existence event does not mean anything.
At the start of recursion at scan k−N, the known target existence probability is P{χ _{ k−N }|Z _{ k−N }}. Equation (64) is expanded following the similar procedure that is used to obtain the Υ(s) in (51), to propose a procedure to obtain the smoothed target existence state at any scan k _{ τ }=k−N+r in the smoothing interval for 0≤r≤N:
where Λ _{ k−n } is calculated using (47) by replacing k _{ τ } with k−n. In the next smoothing interval, the recursion starts at scan k−N+1 with target existence probability P{χ _{ k−N+1|k−N+1}} calculated by (48) by replacing k _{ τ } with k−N+1. The target existence state is then smoothed using (65) and (66).
The smoothed trajectory state and error covariance matrix [̂x _{ k−N|k−s+1 }, P _{ k−N|k−s+1}] for 1≤s≤N+1 in the smoothing interval are obtained in Section 4 using augmented state smoothing algorithm. At the final scan k in the smoothing interval, all these smoothed trajectory state estimates are weighted by the smoothing weights Υ(s) to obtain the smoothed target trajectory state at fixed lag N using (50). The smoothed target existence state at fixed lag N is also calculated (64), which is an outcome of total probability theorem.
6 Fixed lag smoothing enhanced MM HPRF tracker
The fixed lag smoothing approach presented above is also applied to extended multiple models (EMM) HPRF tracker. The EMM tracker incorporated the track quality measure in multiple model HPRF tracker [3]. The augmented state is also calculated for the EMM algorithm, and then fixed lag smoothing is used to obtain the smoothed target hybrid state at any scan. The fixed lag smoothed enhanced multiple model HPRF tracker (FLs EMM) is also simulated in the proposed work to compare its performance.
7 Simulation study
This simulation sections compares the tracking performance of smoothing algorithms (FLs GMM-ITS and FLs EMM) with online tracking algorithms (GMM-ITS and EMM). The results provide the information on the relative benefits of the proposed fixed lag smoothing algorithm.
where p _{ ij } denotes the transition probability from event i to event j. Event 1 and event 2 represents χ _{ k } and \(\bar {\chi _{k}}\), respectively.
The confirmation thresholds for the algorithms are tuned to deliver the same number of confirmed false tracks (CFT) (≈7) for all 1000 Monte Carlo simulation runs. In both simulation scenarios, the CFT are set to be same for a fair comparison. Both smoothing algorithms consider the lag size of 2.
FLs GMM-ITS and FLs EMM algorithms looks to have similar RMSEs, although one expects better performance for the FLs GMM-ITS algorithm. The reason is that the true tracks which are confirmed earlier in the FLs GMM-ITS compared to FLs EMM generate less favorable performance in terms of estimation errors.
The computational time for the EMM algorithm is set as a reference to determine the percentage of extra computational time needed by other algorithms. The total sampling time is 80,000 s, which is greater than the computation time of the aforementioned tracking algorithms; therefore, all algorithms are capable of working in real time.
Computation time [sec]
Algorithm | Execution time | Percentage |
---|---|---|
EMM | 49,874 | Reference |
GMM-ITS | 50,994 | 2.3 % |
FLs EMM | 55,156 | 10.5 % |
FLs GMM-ITS | 59,256 | 12 % |
8 Conclusions
This paper provides a new procedure to calculate the smoothed target hybrid state at fixed lag N. The benefits of the smoothing algorithm are compared for target tracking algorithms in the clutter using the HPRF radar. The smoothed augmented states (obtained at each scan in the smoothing interval) and their respective weights (calculated using multi-scan target existence events) are used to obtain the fixed lag smoothed trajectory state estimate for the target. The target existence state is also smoothed at fixed lag N using all feasible multi-scan target existence events in the smoothing interval.
The FLs GMM-ITS performs much better than FLs EMM, GMM-ITS, and EMM algorithms to track the target using an HPRF radar in terms of RMSEs and false track discrimination. The smoothed target hybrid state provides small estimation errors and produces excellent false track discrimination in the simulation conditions used in the proposed work.
Declarations
Acknowledgments
This work was conducted at High-Speed Vehicle Research Center of KAIST with support of Defense Acquisition Program Administration (DAPA) and Agency for Defense Development (ADD).
Authors’ Affiliations
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