Rauch-Tung-Striebel Smoothing Linear Multi-Target Tracking in Clutter

Most conventional data association based multi-target tracking (MTT) algorithms typically suffer from intractable computational complexities and could not perform in an environment where large number of closely spaced multiple targets move across each other in clutters. Unlike to the existing MTT systems, the linear multi-target (LM) algorithm modifies the measurement detection followed by neighbored tracks as a clutter, hence, it updates the track without the influence of other tracks. Thus, LM technique is a computationally efficient algorithm that allows the multi-target system to play like a single target tracking algorithm. Smoothing maximizes the state estimation accuracy and reduces estimation error based on future scan measurement. However, only few research paper focused on the LM algorithm without utilizing the benefits of the smoothing. This paper presents Rauch-Tung-Striebel Smoothing in the linear multi-target based on integrated probabilistic data association (RTS-LMIPDA). The RTS-LMIPDA algorithm fuses forward and backward LM track predictions to obtain the smoothing prediction which is required to calculate the smoothing multi-target state estimates in the forward track. Numerical analysis is presented to illustrate the estimation accuracy and false track discrimination (FTD) performances of RTS-LMIPDA in comparison to the existing MTT algorithms using the simulations.


I. INTRODUCTION
A radar detector returns uncertain measurements from multi-targets and various random sources to a multi-target tracking system. The measurements that originate from unknown object sources (e.g., reflections from terrain, target thermal noise, and various weather conditions such as clouds) are referred to as clutter measurements. In a radar surveillance region, the trajectory behavior, existence, and number of measurements that may have occurred from one of several targets are unknown. In addition, each feasible target measurement has a low detection probability PD, indicating that target tracking algorithms do not have prior information on the radar measurements. As a result, a target tracking algorithm initializes and updates the false (clutter) tracks as well as the true (target) tracks using the measurements received from a radar detector in each time scan. In this complex cluttered environment, the tracking algorithms employ false track discrimination (FTD) to identify and discriminate the false tracks [1]. FTD utilizes the target existence probability (TEP) as a track quality measure to confirm the true track and terminate the false track [2].
Conventional multi-target tracking (MTT) algorithms use the joint data association technique, which deals with target and clutter measurements by associating the measurements to the tracks in a probabilistic way. In a multi-target situation, the tracks may have joint (shared) measurements because the measurements may have originated from the potential target being followed by neighbor tracks. Therefore, the MTT algorithms such as joint probabilistic data association (JPDA) [3], joint integrated PDA (JIPDA) [4] [5], joint create all feasible joint measurement-to-track association assignment hypotheses and recursively calculate their joint a-posteriori probabilities in each scan. In this situation, a cluster is formed that groups the shared tracks with similar measurements of detection. The number of feasible joint measurement events grows combinatorically in terms of the number of tracks and measurements This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3134987, IEEE Access involved in a cluster. Thus, the numerical complexities of the JPDA and JIPDA algorithms often exceed the available computational resources, especially when there is a large number of targets in the cluttered environment. The joint data association is also used for multi-scan, multi-target tracking algorithms, such as joint integrated track splitting [6] [7]. These multi-scan MTT algorithms are even more complex filters, which form an exponentially growing number of track components. Similarly, a multi-scan multiple hypothesis tracker [8] [9] obtains the exponentially increasing number of multiple hypotheses by enumerating all feasible track-to-measurement association hypothesis. In addition to this complexity, each MHT hypothesis is followed by a global association which considering all tracks and measurements over a span of scans. Thus, MHT consumes more computation time than that of joint data association algorithms. The linear multi-target based on integrated probabilistic data association (LM-IPDA) method was developed to overcome these intractable computational complexities [10] [11]. LM-IPDA bypasses the enumeration of the feasible joint measurement events of the multi-targets using the modified clutter measurement density, which treats the neighbor tracks as a source of clutter. Thus, a track is updated by considering other target detection measurements as a modified clutter measurement being followed by neighbor tracks. Compared to joint data association algorithms [3][4] [5][6] [7], LM-IPDA is more computationally feasible and efficient for multi-target tracking in clutter. Subsequently, the linear multi-target algorithm was extended with an integrated track splitting filter [7]. The aforementioned algorithms use the probability of target existence as a track quality measurement for FTD, except for [3], [8], and [9]. Generally, a non-smoothing algorithm generates higher estimation errors due to the availability of the sensor measurement up to and including time scan k. This often limits the accuracy in the trajectory estimation and track quality measure of FTD. Therefore, a smoothing Rauch-Tung-Striebel (RTS) algorithm was developed, which evaluates the track state estimate in past scans based on the measurements available beyond time scan k [12]. In the recent development of single target tracking smoothing algorithms, researchers smoothed the track state estimate by fusing each individual forward track state prediction with multi-track backward predictions that were falling in the validation gate of a forward track [13][14] [15]. The multitrack backward predictions were recursively estimated in each scan using the sensor measurements, starting from the last scan N and ending at the first scan k of the smoothing interval. This made the smoothing algorithm efficient for target tracking but increased the smoothing time delay due to the time consumed in the initialization and estimation of the backward multi-tracks. These algorithms were extended for smoothing MTT by utilizing a joint data association algorithm, such as fixed interval smoothing based on JIPDA (FIsJIPDA) [16]. Recently, fixed interval smoothing was utilized in multi-scan, multi-target joint integrated track splitting to calculate the smoothing a-posteriori probabilities of the multiple track components associated in a cluster for smoothing state estimation [17]. RTS smoothing equations in a JIPDA algorithm (sJIPDA) use the forward track validated measurements to obtain a backward track prediction [18]. Thus, each updated backward track corresponds to each existing forward track for fusion. This makes the RTS smoothing algorithms, such as [18] and [19], optimal and computationally cost efficient. All smoothing algorithms discussed above use the smoothing probability of target existence for the FTD evaluation. However, the former MTT smoothing algorithms, such as the smoothing JPDA [20], smoothing multiple hypothesis tracker [21], and smoothing probability hypothesis density filter [22][23] did not produce FTD results.
This study utilizes RTS equations for smoothing a linear multi-target based on IPDA in a fixed-interval (RTS-LMIPDA). RTS formulas are used in the LM-IPDA to improve the estimation accuracy and FTD track quality measurement with a limited smoothing delay. Table 1 compares the contributions and motivations of the RTS-LMIPDA algorithm to the existing algorithms.  This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3134987, IEEE Access scan and k indicates the first scan of an interval. The forward tracks are initialized and updated using the Kalman Filter [24] recursively in each scan of the interval. When a forward track arrives at scan k = N, a backward track is initialized using a forward track validated measurement. Similar to sJIPDA, the backward prediction is obtained from the forward and RTS estimates. Finally, the RTS-LMIPDA algorithm fuses forward and backward predictions to obtain the smoothing linear multi-target state estimations. The a-posteriori smoothing probability of target existence is also calculated in each scan for FTD evaluation. Section II discusses the target and sensor measurement model. The RTS smoothing based on LM-IPDA is examined in Section III. The estimation statistic results of RTS-LMIPDA are verified using simulation in Section IV, followed by a conclusion in Section V.

II. TARGET PROPAGATION AND MEASUREMENT MODELS
It is assumed that a track has a hybrid state consisting of a target trajectory state k  x and target existence event k   , where τ indicates an index of targets and tracks. Without lost generality, the target existence is a random event which generates one measurement in each scan and is always detectable with the probability PD. The target model obeys the Markov chain one model [2] which uses state transition probability to maintain and update the existing track. The state transition probability is the probability that the target exists in scan k, provided that it existed in scan k-1, which is expressed as: where Ta denotes the average duration of the target existence and ΔTk,k-1 indicates the time between scan k-1 and k. It is assumed that each target state is a fourdimensional state vector, such as two-dimensional position and velocity vectors. Each target trajectory state k  x measured by a sensor in scan k-1 propagates to scan k using a state transition model F in the target dynamic equation, expressed as: where the target trajectory state is correlated by a zeromean white Gaussian process noise vk-1 that has a known covariance Q. The Q and F expressions are derived in [1] and [12], respectively, and are expressed as: where q represents the white Gaussian acceleration uncertainty received between two scan times T, I2x2 is the 22  identity matrix, and O2x2 is the 22  zeros matrix. The sensor observes each target in each scan k and if a τ th target exists and is detected with the probability PD, then it generates the position measurement k  y using: where Hk = [I2x2, O2x2], which calculates the target state position measurement. wk is a white Gaussian measurement noise with a zero-mean and known covariance matrix Rk. In addition, a random number of clutter measurements are observed and detected using a sensor that follows a nonhomogeneous Poisson distribution [25]. The density of clutter measurements is defined by the number of sensor measurements received from each scan at the x and y axes of the surveillance region, and it is assumed to be known in this study.
Yk represents the cardinality of a set of Mk sensor measurements received in scan k. The i th measurement in Yk is denoted by Yk,i and has a constant clutter measurement density, ρk,i ≡ ρ(Yk,i). {Φ} = Φ1, Φ2, … ΦN denotes the sequence of the consecutive measurement times in a smoothing interval, which has a length of N that denotes the last scan index in a smoothing interval. For example, Y {Φ} = {YΦ1, YΦ2, … YΦN} is the measurement interval that implies the sequence of consecutive sets of measurements.

III. RAUCH-TUNG-STRIEBEL SMOOTHING LINEAR MULTI-TARGET BASED ON IPDA (RTS-LMIPDA)
This paper utilizes RTS equations [12] in the LMIPDA algorithm to improve the estimation accuracy and FTD performance in comparison to other existing tracking algorithms for tracking closely moving multi-targets in clutter. Multiple forward tracks are initialized using a sensor measurement set Yk received in each scan k, assuming an overlapped fixed interval smoothing based on Y N , where N ≥ k. In addition, forward LMIPDA (f-LMIPDA) is used to estimate all existing tracks. One backward track is initialized and updated using the forward track validated measurement for each corresponding forward track if the existing (updated) tracks survived (as determined by the updated target existence probability) and they arrived at a predetermined interval length at time n. The backward LMIPDA (b-LMIPDA) track is propagated using RTS and forward estimates until the first scan k of the smoothing interval. Each corresponding forward and backward prediction are fused to obtain the smoothing prediction and innovation in each scan k moving forward in time. Subsequently, the RTS-LMIPDA track is estimated by updating the smoothing predictions based on the smoothing validation measurements , k i k  yY in each scan k. The probability of target existence is smoothed using Y N , which is utilized for track confirmation and termination in each scan k. This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3134987, IEEE Access

A. GENERALIZED LMIPDA (g-LMIPDA)
Tracks are initialized based on the sensor measurements Y {Φ} using a well-known approach referred to as two-point measurement initialization [1].
The LMIPDA track recursion starts with the hybrid state prediction, expressed as: The hybrid state prediction, with respect to the conditioning measurement sequence, is then expressed as: where the τ th state predicted PDF is expressed as: (9) and the τ th state predicted target existence probability is expressed as: Each track state prediction selects the validation measurement yk from Yk by applying the measurement selection condition provided in (11). If the τ th target is detected with the detection probability PD(τ), then the i th validated measurement yk,i is derived from a τ th target ( ,, Otherwise, it is a clutter measurement, expressed as: where superscript (·) T denotes the transpose and δ denotes the maximum threshold for validation measurement selection, which is calculated using the gating probability, PG(τ) [1], P G(τ) = 1e -0.5δ . Equation (11) builds an elliptical gate which surrounds the cardinality of measurements yk with a number of selected measurements mk within the gate volume V{Φ}\k [1]. This is expressed as: where |A| represents the determinant of the surveillance area A, d is the dimensionality of the surveillance space, and Γ(·) represents the Gamma function.
The LM-IPDA computes the likelihood , ki l  of the measurement yk,i (i > 0) with respect to a track τ in: The selected measurement yk,i consists of a τ th target measurement with the a-priori probability , ki P  [15], expressed as: , where i = 0 and  . These a-priori probabilities of the detected measurements are mutually exclusive and exhaustive, which ensures that only one measurement is a target detection. The LMIPDA modifies the clutter measurements density being observed by a τ th track in the coordinates of yk,I, expressed as: where μk,i represents the modified clutter measurements density, σ represents the neighbored tracks (other than τ), and τn indicates the total number of tracks. The LMIPDA algorithm uses (15) to update the τ th track in scan k by calculating the track likelihood ratio and data association probabilities of the measurements yk,i, respectively expressed as: ; Equation (16) estimates the predicted probability of the τ th target existence in: The LMIPDA estimates the state prediction based on yk,i by utilizing the Kalman filter estimation (KFE) [24], expressed as: Equation (17) approximates the τ th state estimates obtained in (19) using one Gaussian PDF mean and covariance, respectively expressed as: Similarly, the LMIPDA obtains the recursive multi-track estimation in each scan of a smoothing interval by using (8) to (20).

P
, and initial probability of target The updated track state PDF propagates from scan k -1 to scan k to obtain the state prediction for scan k using the Kalman Filter propagation equation, expressed as: An initial target existence probability is propagated from scan k -1 to scan k using (10) (20) and (18). The f-LMIPDA algorithm propagates the updated tracks in each scan k using (21), followed by (8) - (20) to recursively calculate the multitrack estimation in each scan k until arrival at the last scan index n of a smoothing interval. The values of state prediction and state estimation must be stored in each scan k required for the use of the b-LMIPDA and RTS-LMIPDA algorithms.

B. Backward LMIPDA (b-LMIPDA)
When a forward track arrives at scan k = N (last scan index of the smoothing interval), a backward track state PDF propagates from scan k + 1 to scan k using the validated measurements yk selected by a corresponding forward track. Unlike the FIsJIPDA algorithm, the backward state predictions are obtained by fusing the forward and smoothing estimates, followed by the RTS equations expressed in (22). First, the RTS equations (22) are utilized recursively to calculate the smoothing estimate | kN  x and its covariance |kN  P in each scan starting from scan N -1 and ending at the first scan k of the smoothing interval. Note that the smoothing estimate at scan N is equal to the forward estimate at scan k, for example, The backward track state covariance The b-LMIPDA state propagation begins with (23) and uses (13)

C. RTS Smoothing LMIPDA (RTS-LMIPDA)
The RTS-LMIPDA algorithm exploits the information fusion filter (IF) [11] and obtains the smoothing state prediction by utilizing the fusion of the forward and backward state predictions, expressed as: The predicted smoothing existence probability at scan k is the conditional probability on Y N\k , which indicates that the τ th target exists either in a forward interval 1: k -1 or backward interval k + 1: N. This is expressed as: Generally, a track is tentative and therefore, it needs to be verified with a track confirmation threshold. Each track is confirmed when its updated TEP, expressed in (18), exceeds a confirmation threshold and is terminated when its updated TEP becomes lower than the initial TEP. Note that a confirmed track stays confirmed until its termination. It is also possible that a confirmed track may follow the clutter. Therefore, a confirmed track is verified by the chi-squared statistical test [1], expressed as: where, || k denotes the estimation error, which is calculated by the difference of the true target state (2) and smoothed target state (20). P0|0 denotes the covariance matrix of the measurement noise of the target and γ denotes the maximum threshold of the chi-squared statistical test, which depends on the false alarm probability [1].
For the FTD verification, it is necessary to identify the confirmed true track (CTT) and confirmed false track (CFT) using (26). A track τ becomes a CTT if (26) is ≤ 20; otherwise, it becomes a CFT. Furthermore, a CTT could become a CFT due to closely located clutter and targets if (26) exceeds the false track test threshold (≥ 40). The 20 and 40 threshold values also depend on the target velocities and surveillance region. Therefore, the chi-squared test is calculated between each confirmed track and each target in each scan. In the multi-target situation, there could be many CTTs following the same target. However, there should be only one CTT for each corresponding target. Thus, the RTS-LMIPDA algorithm exploits the auction algorithm [27], which compares the minimum normalized distance squared value (left-hand-side term of (26)) corresponding to the CTTs. Subsequently, only one CTT is selected. However, if a track τ is identified as a CFT for all targets, then it is counted as a CFT; otherwise, it is counted as the CTT. The procedure and implantation of the RTS-LMIPDA algorithm is illustrated using a flow-chart, as shown in Fig. 1. The algorithm starts with a two-point track initialization at scan k = 2 such that each feasible measurement pair collected from consecutive scans (e.g., k = 1 and 2, k = 2 and 3, and so on) satisfies the maximum velocity threshold of the target [1]. Forward tracks are updated using (10) -(21), followed by track termination. The TEP is utilized to measure the track quality, which determines the FTD of the algorithms. Each track is terminated when its updated TEP (18) becomes lower than the initial TEP. Tracks are determined to be merged or not if they are following up the same target. If the tracks merge at scan k, then the termination is determined by comparing their updated TEPs [1]. If the number of scans k is equal to the predefined interval length N, then one backward is initialized for each forward track and the smoothing track data (estimate and its smoothing TEP) equals the forward track data (estimate and its updated TEP) at scan k = N. For scan k + ≥ first scan k, the RTS is exploited using (22) for the backward prediction calculation expressed in (23). This is followed by the fusion of the forward and backward predictions using (24), and the RTS-LMIPDA track update using (11) - (20). Otherwise, if k + < k, the algorithm proceeds to the next interval. The procedure is repeated until k + arrives at the end scan of the simulation run, as depicted in Fig. 1. The RTS-LMIPDA algorithm uses the fixed interval smoothing structure [1], that has a length of Nk + 1 scans. Figure 2 depicts first two overlapping smoothing interval structures. Assume the current smoothing interval consisted of scans k = 1, 2, 3, …, 8, and the next interval overlapped at scan N + 1 of the current interval, for example, k = 5, 6, 7, …, 12. That is, the first four scans were discarded after RMSE and FTD statistics accumulation and four new scans were appended in the measurement interval. In the last interval, the smoothing statistics were accumulated for all scans. The RTS-LMIPDA follows four step smoothing procedure: (1) Forward track recursion starts with state pdf prediction (i.e., 3|2  x ) and obtains the state estimate from scan k to scan N conditioned on scan k -1. (2) Smoothing and Backward tracks are simultaneously evaluated using (22) and (23), respectively, in each scan starting from scan N -1 and ending to scan k. Each backward prediction is conditioned on k|k+1, (e.g., 7|8  x ) and obtained using the fusion of the forward and smoothing estimate in each scan, except the scan N. (4) Finally, update the RTS track based on the sensor measurement up to the first half of interval, and then discard it after the statistic accumulation.
Repeat step 1-4 in the next interval and so on.

IV. EXPERIMENTAL ANALYSIS USING SIMULATIONS
The simulation analysis compares the estimation accuracy, root-mean square errors (RMSEs), FTD track quality measure, and track retention of the RTS-LMIPDA, FIsJIPDA [16], FIsJITS [17], sJIPDA [18], sJITS [19], and LMIPDA [10] algorithms. The RTS-LMIPDA algorithm is designed to track six targets that are crossing near the position (387, 300) m and are closely moving in a twodimensional surveillance region that is 700 m wide along the x-and y-axes, as shown in Fig. 3.   Table 2.  (1), which equals 0.97. The FIsJIPDA, FIsJITS, sJIPDA, and sJITS algorithms consumed a great deal of processing time compared to other algorithms due to the evaluation of joint data association and measurement probabilities of cluster tracks. More specifically, FIsJITS and sJITS that belongs to the computationally expensive ITS family, invested 23.38 secs and 16.4 secs, respectively, per run computation time for smoothing estimation of only six targets in clutters. In contrast, the proposed multi-target RTS-LMIPDA algorithm consumed only 1.45 secs per run computation time, thus the algorithm behaved as a single target tracker, similar to the LMIPDA at the cost of only 0.77 sec smoothing delay. It was also observed that simulation became halt when forcing the FIsJITS algorithm for tracking more than six targets in clutters due to excess of available resource's memory. The simulation time comparison of the algorithms is shown in Table 3. The same smoothing interval length was used for the RTS-LMIPDA, FIsJIPDA, FIsJITS, sJIPDA, and sJITS algorithms. Each algorithm is numerically analyzed using the same TEP, but its confirmation threshold was tuned to get almost the similar number of CFTs ( 25)  to ensure a fair comparison analysis. The track retention accounting following useful statistics are accumulated before and the after crossing of targets:  nCases: the number of CTTs following a target at scan 15  nOk: the number of CTTs still pursuing the original target at scan 30  nSwitched: the number of CTTs that end up original target but, now pursuing a different target at scan 30  nLost: the number of nCases tracks not pursuing any target at scan 30 (because, they became false, merged or were terminated due to low TEP).
The track retention statistics of each algorithm are accumulated from 200 simulation runs listed in Table 4. The RTS-LMIPDA provides highest nOk, and lowest nSwitched and nLost as compared with exiting algorithms. The number of CTTs corresponding to all targets is shown in Fig. 4. The proposed RTS-LMIPDA algorithm confirmed a corresponding target track in scan k = 3 and maintained tracking of the potential target. The LMIPDA algorithm took complete advantage of the RTS smoothing algorithm to obtain the highest number of CTTs compared to existing algorithms. Only a few of the CTTs were dropped by the RTS-LMIPDA due to the cross-section of the targets in scan k = 20. In the FIsJIPDA, FIsJITS, sJIPDA, and sJITS algorithms, the joint data association probabilities were evaluated only for the confirmed tracks to avoid using an excessive amount of computational resources. Both FIsJITS and sJITS performed well against FIsJIPDA and sJIPDA. However, their tracks were confirmed after scan 5 and have slower growth of confirmation rate as compared to the RTS-LMIPDA algorithm. Similarly, sJIPDA confirms its track in This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/ACCESS.2021.3134987, IEEE Access scan 5 but occasionally tracks the corresponding target and eventually missed the target due to nSwitched and nLost statistics. In both FIsJIPDA and sJIPDA algorithms, their CTTs were often distracted due to the association of the shared target measurements observed in scans k = 16-25. Because, in these scans, the targets are closely moving in clutters. Thus, the number of CTTs for the FIsJIPDA and sJIPDA algorithms drops nearly 25% of the total number of CTTs as depicted in Fig. 4. Thus, the joint data association based smoothing methods were not proved to be advantageous for tracking a large number of targets in a cluttered environment mainly due to excessive processing time compared to other algorithms (Table 3) and the regular distraction of the confirmed tracks due to the sharing of the joint measurements. Thus, the RTS-LMIPDA algorithm provides improved estimation accuracy and FTD as compared to existing smoothing algorithms as well as LMIPDA. The RMSEs of the target's position (τ = 1, 2, 4, 6) were calculated using the position estimation error statistics of the confirmed true tracks as shown in Fig. 5. The LMIPDA being a non-smoothing algorithm produced high estimation errors. It was observed that the joint data association based algorithms such as FIsJIPDA, FIsJITS, sJIPDA, and sJITS often fluctuating estimation errors due to the distraction of shared tracks within vicinity of the neighbored measurements. However, sJIPDA have a similar RMSE trend as compared to RTS-LIPDA, but it was only observed for targets 2 and 6. It can also be seen that the FIsJIPDA algorithm lost target 1 in scans k = 22-25 and k = 27, and confirmed the target 4 only in scans k = 21-23 and k = 27. Similarly, the FIsJITS algorithm lost target 1 in scan k = 26, and target 6 in scans k = 20-21 and k = 24-26. In addition, FIsJITS algorithm confirmed the targets 4 and 6 quite late in the simulation for example, it detected the target 4 in scan k = 13 and the target 6 in scan k = 17. In the situation of targets 2 and 6, sJITS algorithm consistently produced the high RMSEs as compared to other smoothing algorithms due to the weak growth of smoothing target existence probability in each scan k. In comparison, the RTS-LMIPDA algorithm maintained the smoothed target existence probability well in each scan k and obtain a significant reduction in the estimation errors, as depicted in Figs. 5(a)-(d). Thus, the proposed RTS-LMIPDA algorithm outperformed the existing algorithms and demonstrated superior performance in terms of multi-target state estimation accuracy, FTD track quality measurement, and RMSEs as shown in Figs. 4 and 5.

V. Conclusion
This study extended the LMIPDA algorithm using the RTS smoothing algorithm to improve its estimation accuracy and FTD for tracking multiple cross-over targets in a cluttered environment. Unlike the existing joint data association based multi-target algorithms (discussed in section IV), the LMIPDA did not form a cluster of tracks, thus ignoring entire evaluation steps of the joint multi-target data association probabilities required for multi-target estimations. This saved a great deal of processing time in the multi-target estimation compared to existing algorithms (Table 3). Similar to the sJIPDA algorithm, the RTS-LMIPDA algorithm calculated the backward prediction using forward and RTS estimates corresponding to each forward track. Consequently, smoothing estimates were calculated using the fusion of forward and backward predictions. The integrated RTS-LMIPDA algorithm was experimentally proven to provide a significant smoothing estimation and FTD performance. The RTS-LMIPDA algorithm is verified to be an efficient asset for practical implementation and demonstrated the track retention statistics with the highest nOk, and lowest nSwitched and nLost at the cost of smoothing time delay.