The Modified Probability Hypothesis Density Filter With Adaptive Birth Intensity Estimation for Multi-Target Tracking in Low Detection Probability

The existing Probability Hypothesis Density (PHD) filters with birth intensity estimation only operate on single or two consecutive scan data for multi-target tracking. However, for those targets with low detection probability, it is hard to achieve a satisfactory level of track initiation and maintenance. To overcome the weakness above, we propose a modified PHD filter with adaptive birth intensity estimation. The core of the proposed filter is to define two state sets as the formal set and the temporary set. In the framework of measurement driven estimation, we classify the measurements into three categories depending on whether it is in the neighborhood of the state in above two sets. And the birth states of the formal set and the temporary set are generated by the classified measurements respectively. In addition, if there is no matching measurement for the state in the formal set, duplicate the corresponding state as the birth state of the temporary set. For each state in temporary set, we introduce a forgetting factor and a dynamic detection probability in filter to cope with the rapid decrease of its intensity due to the absence of measurement. If its forgetting factor is over the dead threshold, the state will be deleted from the set. Based on the principles above, we derive the Gaussian-mixture (GM) implementation of the PHD filter proposed in this paper. Experiment results show that, in low detection probability scenario, the modified PHD filter outperforms other PHD filters with birth intensity estimation.


I. INTRODUCTION
With the rapid development of sensors, multi-target track has become a research hotspot in recent years [1]- [12]. Generally, there are two branches in multi-target track method. One branch is the track methods with the help of the traditional data association techniques to find the exact relationship between the sensor measurement and the target state. Typical methods in this branch are nearest neighbor (NN) algorithm [3], joint probabilistic data association (JPDA) algorithm [4], [5], multiple hypothesis tracking (MHT) [6], [7] and etc. The other branch is the track method in the framework of the random finite set (RFS). As the RFS theory is a systematic and direct procedure to estimate the numbers and states of the multiple targets without dealing with the complex association between target The associate editor coordinating the review of this manuscript and approving it for publication was Zhen Ren . states and measurements, it draws considerable attentions in multi-target track area [8], [9]. The Probability Hypothesis Density (PHD) filter is the most popular and typical filter among those track methods based on RFS [10]- [12]. It is the first order statistical moment of the RFS, which is an approximation solution to the optimal multi-target Bayes filter.
The traditional PHD filter assumes the birth intensity as prior information. However, in many real cases, multiple targets occur at random in the surveillance field. And the distribution of new born target is unknown. The tracking performance will degrade significantly without target birth prior information. In existing literature, the PHD filter with birth intensity estimation is based on the principle of measurement driven. Ristic et al. designed an adaptive target birth intensity estimation strategy for PHD and CPHD filter [13], [14]. The target birth intensity is approximated by the newborn particles drawn from each of the measurement with same weight. And this idea is also applied into the Cardinality Balanced Multi-Target Multi-Bernoulli (CBMeMBer) filter [14]- [16] only presents the SMC (Sequential Monte Carlo) implementations of the filter proposed. In [17], Houssineau and Laneuville deduced its Gaussian mixture GM-PHD version under the assumption that the received measurements can entirely embody new born targets. Inspired by [13] and [17], Zhu et al. [18] proposed an improved version of the algorithm. To make it closer to the truth, the proposed algorithm directly estimates the unmeasured component, velocity, with the help of the two-point measurement difference in consecutive scan [19]. To cope with strong clutter situation, [20] presents a dual threshold particle PHD filter with unknown target birth intensity. Different from above solutions based on direct measurement-driven estimation, [21] and [22] give another way to deal with the target birth intensity. Before the PHD update operator, both introduce extra steps to distinguish the newborn targets originated measurements and other measurements. In [21], the birth intensity is estimated by two steps. The first step is based on the criterion of maximum a posterior (MAP) relating to the entropy distribution of the intensity weight. The second step is based on the coverage rate. In [22], the birth intensity is estimated based on iterative random sample consensus (RANSAC) algorithm within a sliding window.
All the above PHD filters with birth intensity estimation only operate on the current or the current and previous consecutive scan data for multi-target tracking, except for [22]. But in the iterative RANSAC based filter, in essence, the line model is constructed by using the measurements of the current scan and the previous scan. Measurements of other scans only play a role of verification. Thus, for those targets with low detection probability, it is hard to maintain the accuracy and time cost of track initiation to a satisfactory level when continuously missing measurements. In recent years, passive radar has attracted great interests due to its low-cost and feasibility of various illuminators [23]. One of its main challenges is the low detection probability due to the difficulties in accurate synchronization [24]. It is necessary to explore an effective track filter under the low detection probability situation. On the basis of the existing algorithms, we propose a PHD filter with modified adaptive birth intensity estimation in this paper. The core of the proposed filter is to define two state sets as the formal set and the temporary set. In the framework of measurement driven estimation, we classify the measurements into three categories depending on whether it is in the neighborhood of the state in the above two sets. And the birth states of the formal set and the temporary set are generated by the classified measurements respectively. In addition, if there is no matching measurement for the state in the formal set, duplicate the state as the birth state of the temporary set. For each state in temporary set, we introduce a forgetting factor and a dynamic detection probability in filter to cope with the rapid decrease of its intensity due to the absence of measurement. If its forgetting factor is over the dead threshold, the state will be deleted from the set. Based on the principles above, we derive the Gaussian-mixture (GM) implementation of the PHD filter proposed in this paper. Experiment results show that, in low detection probability scenario, the modified PHD filter outperforms other PHD filters with birth intensity estimation.
The remaining part is organized as follows. Section II introduces the background on the multi-target tracking algorithm under RFS. Section III is the modified PHD filter with adaptive birth intensity estimation. Section IV introduces the Gaussian-mixture implementation of the proposed PHD filter. Section V is the simulation and result. Section VI is conclusion.

II. MULTI-TARGET TRACKING ALGORITHM UNDER RFS
Multi-target Bayes filter under RFS framework has been a popular systematic approach to multi-target tracking. For multi-target track, the main goal is to track and estimate the target states correctly based on sensor measurements.

A. THE RANDOM FINITE SET MODEL FOR MULTI-TARGET TRACKING
Both target states and measurements can be modeled in RFS. Suppose X k and Z k represent the RFS of target states and measurements at time step k respectively.
where card ( * ) denotes the cardinality of the set.
(2) and (3) describe the composition of X k and Z k respectively.
The target states set X k is the union of the surviving target states X P k|k−1 from X k−1 , the extended target states X E k|k−1 from X k−1 and the new born target states X B k .
The measurements set Z k is the union of the measurements set ξ k generated by targets and the false alarm measurements set C k .

B. BAYESIAN TRACKING THEORY
In the Bayesian track theory, there are two main procedures, that is prediction and updating.
For prediction, For updating, where P k|k−1 (X k |Z 1:k−1 ) is the prediction probability of the multi-targets. P k|k (X k |Z 1:k ) is the updated probability of the VOLUME 8, 2020 multi-targets. f k|k−1 (X k | X) denotes the multiple target transition density and g k (Z k | X k ) is the likelihood function of the measurement Z k .

C. THE STANDARD PHD FILTER
Due to the set integral issue in the Bayes formula, it becomes intractable for the RFS to involve in the Bayesian track theory.
To reduce the complexity, Mahler adopted the first order statistical moment D(x) of the RFS to estimate the multi-target states in each iteration and deduced the standard PHD filter [11]. In the framework of the RFS and the Bayesian theory, the recurrence form of the PHD filter is given by (6) and (7). For prediction, For updating, where D k|k−1 (x) denotes the predicted state intensity from time step k-1 to time step k. D k|k (x) denotes the updated state intensity at time step k. γ k (x) is the state intensity function of new born targets. p s,k (x) is the survival probability. β k|k−1 (x|x ) is the intensity function of extended targets. p D,k (x) is the detection probability. κ k (z) is the intensity function of clutter.

III. THE MODIFIED PHD FILTER WITH ADAPTIVE BIRTH INTENSITY ESTIMATION
Low detection probability leads to the absence of lots of measurements generated by targets, which can degrade the tracking performance significantly, especially in track initiation. To ensure the performance of the PHD filter, we propose a modified PHD filter with adaptive birth intensity estimation in this section. In the standard PHD filter, the state intensity will drop rapidly when there is no measurement in the neighborhood. To cope with the absence of measurements, the strategy of the proposed algorithm is to keep the states generated by the unconfirmed measurements as long as possible for future verification, while considering the algorithm efficiency at the same time. Fig. 1 shows the flowchart of the proposed algorithm.
There are three processing modules in the modified PHD filter. First of all, a preprocessor in the measurement set is designed before the PHD predictor. In the framework of measurement driven estimation, we classify the measurements Z k into three categories (Z P0 k , Z B2 k and Z B k ) depending on whether it is in the neighborhood of the state in the formal set and the temporary set (X FM k|k−1 and X TP k|k−1 ). Then, we define the state set generated by Z B2 k and Z B k above as the formal set and the temporary set respectively. The left module describes the procedure in the formal set. The right module describes the procedure in the temporary set. There is a transfer state set X FT k|k from the right to the left. The processing progress in the left module is similar to the traditional PHD filter. In the right module, we introduce a forgetting factor λ, relating to the length of survival time. And change the original detection probability into the dynamic form based on the forgetting factor. After the PHD updater, the states in temporary set need to update its forgetting factor. Screen out those states whose forgetting factor is over the dead threshold. The survival states are ready for the target extraction and the next recurrence.

A. PREPROCESSING IN THE MEASUREMENT SET
In the strategy of measurement-driven birth intensity estimation, the birth intensity is directly determined by the measurements. Considering the efficiency of the algorithm, the first step is to select the measurements generated by the persistent targets from the whole measurement set according to the spatial information. The rest is primarily supposed to be the measurements generated by the new born targets. To further distinguish the residual measurements, consider the previous residual measurements and classify the residual into two groups. One is the measurement set generated by the confirmed new born targets. The other is the measurement set generated by the suspected new born targets. The confirmation principle is the continuity of the measurement with the previous new born states. Thus, based on the above analysis, (3) can be derived into (8).
In (8), X P0 k denotes the state set at time step k, in which every state is obtained from the transition of the persisting target state at time step k-1. X B2 k denotes the state set at time step k, in which every state is obtained from the transition of the previous new born state. X B1 k denotes the new born target state set at time step k.
So Z k can be rewritten as (9).
The target measuring model is assumed as where h ( * ) is the measuring function. w is the measuring Gaussian noise with covariance matrix R.
In the proposed PHD filter, we define two state sets as the formal state set X FM and the temporary state set X TP . The former is the state set, in which each target state has been confirmed by the association with previous states. So, the source of X FM is the states generated by Z B2 . The latter is the state set, in which each target state has not been confirmed. So, the main source of X TP is from the states generated by Z B . Moreover, in low detection probability scenario, not all states in X FM have matching measurements in Z P0 k . To maintain the track, we duplicate those states as the birth states of the X TP , which is another source of X TP .
Next is going to introduce the primary classification method for Z k . Based on the definitions above, Z P0 k , Z B2 k and Z B k can be calculated by formula (11), (12) and (13) in Table. 1 respectively.
VOLUME 8, 2020 Besides, there is an attachment parameter set β k,B2 with the Z B2 k , as (15) shown. β k,B2 denotes the association index of element in X TP k|k−1 . For example, if β (j) k,B2 = p, that means the p-th element in X TP k|k−1 is most closely related to the j-th element in Z B2 k .

B. MEASUREMENT-DRIVEN BIRTH INTENSITY ESTIMATION
In this sub-section, we design the method of birth intensity estimation based on the measurement categories above. Assume the target state x consists of the position vector p and the kinematic vector u. The former can be calculated through the measurement set directly. The latter requires further calculations relating to the whole kinematic procedure. Inspired by E-PHD [18] and two-point track initiation algorithm [19], we consider the kinematic vector during the birth intensity estimation.
In the measurement driven framework, Z B2 k generates the confirmed new born intensity γ k,2 (x) in the formal set. Z B k generates the suspected new born intensity γ k,1 (x) in the temporary set. Table 2 gives the estimation methods of both γ k,1 (x) and γ k,2 (x).

C. DEDUCTION OF THE MODIFIED PHD FILTER
As the proposed birth intensity estimation method has introduced in the above sub-section, we deduce the modified PHD filter in this sub-section. There are two state sets, that is the formal set and the temporary set, in the modified PHD filter. Suppose the target state intensity at time step k is D k (x). So D k (x) can be represented by formula (20).
where D TP k (x,λ) and D FM k (x) are the intensity of the target state in the temporary set and the formal set respectively.
The standard PHD filter has two steps, that is prediction and updating. We deduce the proposed modified PHD filter into (23) and (30), as shown at the bottom of page 7. In general, researches on the PHD filter with birth intensity estimation omit the extended target states for facilitation of deduction [15]- [22].
Assume the intensity of the replica is D * 0 (x, λ). The corresponding forgetting factor λ is set as 0. The prediction equation of the modified PHD filter is given by (21).
For facilitation of following derivation, combine D * 0 (x, λ) and D TP k−1|k−1 (x, λ) together before the prediction. Therefore, we introduce an updating step in temporary set before the prediction, given by (22).
So, the prediction equation can be rewritten as: Assumption 1: To simplify the model, we assume the detection probability is supposed to be a constant value p D . Since the birth intensity is totally generated by the measurements at time step k, the detection probability for the new born targets is equal to 100%.
Besides, in low detection probability situation, the absence of measurements leads to great difficulties in track initiation. The state intensity for new born targets will drop significantly, if there is no associated measurement at next time step. So, the aim of the proposed algorithm is trying to make full use of the historical unassociated measurements and keeping the survival life of the states generated by them as long as possible. From (7), we can find that the intensity updating equation of PHD filter has two parts. One is the previous state intensity under the assumption that target is not detected. The other is to update the state intensity with measurements, supposing the target is detected. To keep the historical states in temporary set for a long period, we suppose the detection probability of the state in the temporary set is much lower than the constant detection probability p D . And assume that the longer survival life of the state in the temporary set, the higher detection probability of the state. Thus, the proposed solution is to change the detection probability into the dynamic form considering the forgetting factor.
Based on the above analysis, p D,k (x) is given by formula (24).
where X b2,k and X b1,k are the birth state sets estimated through Z B2 k and Z B k respectively. λ is the forgetting factor corresponding to the state x in X TP k|k−1 . µ is the constant coefficient to adjust the change rate of the detection probability in temporary set. To maintain good performance, the value of µ is related to the original detection probability p D . If p D is getting lower, µ should be upper. And µ should not be lower than 1.
Assumption 2: It can be found that the i-th Gaussian component in γ k,1 (x) or γ k,2 (x) is closely related to the i-th measurement in Z B k or Z B2 k and is unrelated to other measurements. So, for convenience of calculation, the likelihood function for the new born target state can be approximated as (25) shown. where . g 0 is a fix value relating to the likelihood function g( * ). Eq. (25) is also applicable to Z B2 k and γ k,2 (x).
Assumption 3: Due to the involvement of the temporary set, the computation amount of the proposed filter increases greatly. Considering the efficiency, we decide to make some modifications on the calculating strategy of the likelihood function. In [16], the solution is to set a noise threshold for the likelihood function. If the likelihood function value is below the threshold, it is assumed as zero. However, it still requires calculating the likelihood function value before judgement. In the proposed filter, as the measurement set Z k has been classified according to the association with different state sets in sub-section III.A, it is unnecessary to make full use of the whole state sets to calculate the likelihood function of Z k . The measurement set Z k is the union of Z P0 k , Z B k and Z B2 k in (9).The likelihood function of z k (z k ∈Z k ) about the state x in the mismatched state set can be approximately regarded as zero. In addition, the intention of the temporary set is to estimate birth intensity with the help of historical unassociated states. So, it only requires for the contribution of the likelihood function about Z B k ∪Z B2 k to update D TP k|k−1 (x). Based on the analysis above, the calculating strategy of the likelihood function is given by (26) ∼ (29).
Take (23) ∼ (29) into (7), and obtain the updating equation of the modified PHD filter as (30). For the specific derivation and analysis process, see appendix B.
At the meantime, update the forgetting factor of the state in temporary set, as (34) shown.
And delete those states in the temporary set whose forgetting factor exceeds a preset threshold.
After each loop of prediction and updating, extract the target state whose intensity is over a preset threshold. The target extraction has three steps. Firstly, the extraction of target state performs on the formal set and the temporary set respectively. VOLUME 8, 2020 Then, merge the extracted states and half down the intensity. Finally, extract the target from the merging result, if its intensity is over the preset threshold.

IV. GAUSSIAN MIXTURE IMPLEMENTATION OF THE PROPOSED FILTER
In this section, we derive the Gaussian mixture implementation method of the proposed filter.
Assume the survival probabilities is the constant value p s,k . And the state transition function and the measurement likelihood function are both linear Gaussian model, as formula (35), (36) shown [12].
Suppose the posterior intensity at time k-1 is a Gaussian mixture form: Update the temporary set by (22) and obtain the new For prediction, The weights and the Gaussian components are given by (40).
For updating, where L 1 z (i) , L 2 z (i) and L 3 z (i) are given by (43).
The target state extraction has three steps: a. Respectively extract the states in the X FM k|k and X TP k|k by (44).
where Th is a preset weight threshold for extraction. X FM k|k and X TP k|k are the updated formal state set and the updated temporary set at time step k. b. Merge theX kF andX kT . Obtain the merged state set X kM and its corresponding merged weights. Reduce the weight by half. c. Extract the target states in the merged setX kM by (45).
X k and N k are the extracted state set and the cardinality of the set at time step k.
In the end, we provide the detailed pseudocode of the GM-implementation of the proposed filter in Appendix C for readers.

V. SIMULATION AND RESULTS
In this section, we conduct experiments to test the performance of the proposed filter. To make it an intuitive performance comparison, we choose three representative PHD filters based on adaptive birth intensity estimation. The information about those comparative filters will be given in detail in sub-section V.A. The experiments focus on the issue of multi-targets tracking in the low detection probability.

A. PARAMETERS SETTING
In this section, we designs a scenario in which an unknown number of targets appear and die at different time steps. The dimension of the state is set as four. The targets move in the Cartesian coordinates. The state of a target at time t can be denoted by x t = x t , v x,t , y t , v y,t T .The state transition model is a constant velocity model with (46) and (47).
where T = 1s is the interval time of each scan.
where σ v = 5 is the standard deviation of the process noise. The number of clutter points obey poison distribution with an average rate ρ c per scan. The survival probability p s of the state is set as 0.99. The maximum number of the gaussian components in both the formal set and the temporary set are set as 100. The surveillance time length is 100s. We set five targets in simulation scenario. Table 3 shows the parameter setting of these five tracks. For the proposed filter, we set w b1 and w b2 as 10 −2 and 10 −2 respectively. The setting of the maximum forgetting factor λ max is assumed to be related to the detection probability p D . Because if λ max is larger, the false alarm rate is getting greater in low p D . To make a balance, (50) gives the setting principle of λ max .
where * denotes round down. The constant coefficient µ in (24) is also related to p D , given by (51).
We choose three representative PHD filters based on adaptive birth intensity estimation for comparison. They are the extended PHD filter (E-PHD) [18], the PHD filter based on entropy distribution (ED-PHD) [21] and the PHD filter based on iterative RANSAC (IR-PHD) [22]. The E-PHD combines the single-point and two-point difference track initialization with the adaptive birth intensity estimation. In [21], the ED-PHD is the PHD filter with entropy distribution and coverage rate-based birth intensity estimation. And it mainly applies in the area of multi-target visual tracking. However, our research focus on the point-target track. The coverage rate makes no contribution in this simulation. Thus, the ED-PHD in simulation only considers the entropy distribution-based birth intensity updating by the criterion of the maximum a posterior (MAP). The IR-PHD adopts iterative random sample consensus (I-RANSAC) with a sliding window to estimate the birth intensity. The unique parameter settings of three comparative filters in simulation are given in Table 4. The common parameter settings in simulation are given in Table 5.

1) MULTI-TARGET TRACKING RESULTS IN LOW DETECTON PROBABILITY SCENARIO
In this subsection, we conduct the simulation with the detection probability p D = 0.3. And the average rate ρ c of    posion distribution for clutters is set as 2. Fig. 3 illustrates all measurements of five tracks in this scenario. Blue 'x' denotes the location of measurements.    Fig. 4(a)(c)(e)(g) illustrate the tracking results in Cartesian coordinates. At meantime, to make a reference in visual, we also plot the true track and the measurements generated by true targets. Red line denotes the true track. Red circle 'o' denotes the start position of the track. Green' * ' denotes measurements generated by true targets. Blue 'x' denotes the tracking results. Fig. 4 Table 6, we can find that the proposed PHD filter requires less scan data to make correct track initiation. Thus, the proposed PHD filter can start the track with the fastest speed among all above filters in low detection probability scenario.
From Fig. 4, it is obvious that the proposed filter has the best performance on track formation and maintenance in low detection probability. The tracking performance of E-PHD and ED-PHD are at same level, while IR-PHD performs worst. Besides, the number of false track points in the result of proposed PHD and IR-PHD are less than the one in the results of E-PHD and ED-PHD.
To evaluate the tracking error, we choose the optimal sub-pattern assignment (OSPA) distance [25], which captures the cardinality error, as well as the distance of individual elements, of two finite sets. The OSPA distance of two finite sets X = {x 1 , . . . ,x m } and Y = {y 1 , . . . , y n } is given by (52) where k is the set of permutations on {1, 2, . . . , k} , k > 0, and the distance d (c) (x, y) p = min(c, x − y ). p(p ≥ 1) is the order of the distance, and c is the cut-off parameter. Moreover, to illustrate the track error in more detail, we also utilize (53) and (54) to respectively demonstrate the OSPA cardinality error and the OSPA location error.
Set the cut-off parameter c as 100 and the order p as 1.
The OSPA distance of tracking results, as well as the OSPA location and the OSPA cardinality, are illustrated in Fig.5. Use different colors to distinguish the filter. Red line denotes the tracking error of the proposed PHD filter. Black line denotes the tracking error of the E-PHD filter. Blue line denotes the tracking error of the ED-PHD filter. Green line denotes the tracing error of the IR-PHD filter. In Fig. 5, it is obvious that the OSPA distance of the tracking results of the proposed filter is generally smallest among all filters. Moreover, the OSPA cardinality error of the tracking results of the proposed filter is significantly less than the one of the other filters' results, even though the OSPA location error of the tracking results of the proposed filter is similar in size with the one of other filters' results. Fig. 6 shows how the cardinality of the tracking results of different filters changes over time. The black line denotes the true cardinality for reference. Red dots, black dots, blue dots and green dots are the cardinality of the results of the proposed PHD, the E-PHD, the ED-PHD and the IR-PHD respectively. It is clear that the proposed PHD filter is superior in the performance of cardinality estimation. Change the detection probability p D into 0.5 and repeat the above experiment again. Fig. 7 shows the OSPA index of the tracking results. Increase the number of clutters and change the average rate ρ c of posion distribution into 4. Keep the detection probability p D = 0.3 as the original value. Repeat the above experiment again. Fig. 8 shows the OSPA index of the tracking results. From Fig. 7 and Fig. 8, it is obvious that the OSPA distance of the proposed filter's result is the smallest in general, no matter how the clutter density and the detection probability change. Thus, the tracking performance of the proposed filter is superior over other comparative filters in low detection probability scenario. VOLUME 8, 2020
In this sub-section, the simulation scenario is same as the above subsection. The only variable parameter is the detection probability. Change the detection probability p D from 0. 25    Monte-Carlo results, it is obvious that the proposed filter requires the least number of scans for correct track initiation, especially in low detection probability situation.

C. TEST BY THE FIELD DATA
In this part, we use the field data to test the proposed PHD filter, as well as other three comparative filters. The field data is the radar data from a passive radar exploring uncooperative radar illuminator. As such radar is hard to achieve the synchronization accuracy to a satisfactory level, the detection probability is relatively low. Fig. 12 illustrates a piece of passive radar data with the time length of 4 minutes around. The time step is 2.5 s. And the data consists of 100 scans. According to the data from the active radar, we confirm two tracks of true targets in the surveillance scope, which is pointed out in Fig. 12. The detection probability in this scenario is about 0.5.
Besides, there are a few scans with no measurements. And the distribution of the clutters is not uniform. In addition, dense clutters and interferences are distributed in the right upper region of the scope. In the field experiment, the baseline of the illuminator and the receiver is relative short to the detecting range. And both face in the similar direction for surveillance. So, the observation model is approximately linear with (48)  The E-PHD filter can partially track these targets, but there are too many false-alarm clutters in the result. Also, the ED-PHD filter can partially track these targets. Its track maintenance ability is relatively weak under the real low detection probability scenario, especially no measurements and clutters in several consecutive scans. The IR-PHD filter is fail to track in this field experiment. The reason we analysis is that the birth state estimation in IR-PHD filter requires two true measurements in two consecutive scans for the construction of line model, while the absence of measurement in several scans leads to the difficulty of track initiation of IR-PHD.
In summary, the calculational complexity of the proposed PHD filter is O (N f ,k−1 + N t,k−1 )(L k C1) .

VI. CONCLUSION
In this paper, we proposed a modified PHD filter with adaptive birth intensity estimation to improve the tracking performance in the low detection probability scenario. The core of the proposed filter is to define two state sets as the formal set and the temporary set. And introduce a forgetting factor into the temporary set. The measurement-driven birth intensity estimation is modified based on the proposed mechanism. We derive the Gaussian mixture implementation of the proposed PHD filter in this paper. The experiment result shows the proposed PHD filter outperforms other filter in the low detection probability scenario, especially in OSPA distance and the track initiation. However, due to the introduction of the temporary set, the time-consuming of the proposed filter is increased. For better application in engineering, it is meaningful to explore the improvements on the efficiency of the propose filter in the future work.

APPENDIXES APPENDIX A THE ANALYSIS ON S AND THE DETAILED DERIVATION OF (14)
The primary source of the state in temporary set is distinguished by its velocity vector. If the velocity is over a threshold, we assume the source of the state is the formal set. Otherwise, the source is the measurements in Z B . According to different situation, we derive the correlation matrix S respectively.
For those states originated from the formal set, its velocity information is meaningful and has acted on the previous Kalman filtering loop. So, the classification principle of Z B2 k follows (10). As (z − h (x)) T R −1 (z − h (x)) obeys the Chi-squared distribution, S equals to R. And given the maximum probability value, U 1 can be obtained through inverse of the chi-square cumulative distribution function.
For those states originated from the measurements set Z B , its velocity is usually very low, since there are no measurements in the neighborhood in previous filtering steps. So the classification principle of Z B2 k should consider the forgetting factor λ of the state. And the measurements model is constructed by (55).
T is the interval time step. α is the screment vector of the state in each time step. α consists of two parts, that is position vector α [p] and velocity vector α [v] . ( * [p] denotes the position vector of * . * [v] denotes the velocity vector of * .) To simplify the analysis, we suppose the states in temporary set follow CV (constant velocity) model and the measuring function h( * ) is partial linearization. So (55) can be rewritten as (56).
As the velocity is unknown, we assume α [p] obeys the gaussian distribution N(0,C), where C is is the maximum velocity value. N is the dimension of the position of the state. Suppose obeys the Chi-squared distribution. And S equals to C T . Based on the analysis above, S(x, λ) can be calculated by (57).

S(x,λ)
where U V is the threshold of velocity. | * | denotes the modulus of vectors.

APPENDIX B THE DETAILED DERIVATION OF (30)
The updating equation D k|k (x) can be derived into (58), as shown at the bottom of this page. Assume So, D k|k (x) can be further derived into (60), as shown at the bottom of this page. The further derivation of D FM * k|k (x) and D TP * k|k (x,λ) are given by (61) and (62), respectively, as shown at the bottom of the next page. Now analysis the L z (j) : Assume So, D FM * k|k (x) and D TP * k|k (x,λ) can be derived into (67) Regard the last part of the D TP * k|k (x,λ) as the transferred state intensity from the temporary set to the formal set, that is D FT k|k (x).
So, the final updating formula of both sets are given by (70) and (71), respectively, as shown at the top of page 20.