Multi‐sensor fusion for multi‐target tracking using measurement division

IET Radar, Sonar & NavigationVolume 14, Issue 9 p. 1451-1461 Research ArticleFree Access Multi-sensor fusion for multi-target tracking using measurement division Long Liu, Long Liu orcid.org/0000-0003-4662-6829 School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this authorHongbing Ji, Hongbing Ji School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this authorWenbo Zhang, Corresponding Author Wenbo Zhang wbzhang@xidian.edu.cn School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this authorGuisheng Liao, Guisheng Liao School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this author Long Liu, Long Liu orcid.org/0000-0003-4662-6829 School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this authorHongbing Ji, Hongbing Ji School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this authorWenbo Zhang, Corresponding Author Wenbo Zhang wbzhang@xidian.edu.cn School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this authorGuisheng Liao, Guisheng Liao School of Electronic Engineering, Xidian University, P.O. BOX 229, Xi'an, 710071 People's Republic of ChinaSearch for more papers by this author First published: 07 August 2020 https://doi.org/10.1049/iet-rsn.2018.5567AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract The iterated-corrector probability hypothesis density (IC-PHD) filter propagates the posterior intensity of each sensor at one time step to improve tracking accuracy. However, targets cannot be estimated by the IC-PHD filter, if the detection probability of the last update sensor is low. To deal with this problem, this study presents a new multi-sensor multi-target tracking method. Analysing the iterative process of this filter, it can be observed that the measurements obtained by sensors can be divided into several measurement subsets. Then, the similarity among the measurements is described by two parts, the similarity by combining the posterior intensity updated by the measurement, and the credibility of the posterior intensity. The similarity can be used to determine whether the measurements are from one target and whether the measurement selection meets the real situation. Based on the similarity among measurements, a two-way selection approach is presented to find out measurement subsets corresponding to the true targets. In this approach, measurement and measurement subset are mutually selected. Two measurements are also mutually selected. Thus only the measurement subsets corresponding to true targets have large weights. The simulation results show that the proposed method can reduce the miss-detection and false alarm effectively. 1 Introduction The task of multi-target tracking (MTT) [1] aims to use the information received by sensors to obtain some information of targets, such as the number of targets, the state of targets (including position, speed, acceleration etc.), tracks and other features. Two challenging issues in MTT are the target uncertainty and the measurement uncertainty. The former means that the number and state of targets change due to the target appearance and disappearance, and it is difficult to obtain the number of targets. The latter means that sensors may get inaccurate measurements or even miss-detect targets due to sensor performance and the environment. In addition, for the MTT with clutter, it is difficult to determine whether a measurement is derived from a real target or a clutter. The classic MTT methods are based on data association. They associate measurements and targets through data association and then use single-target Bayesian filtering algorithms to estimate target states. MTT problems are converted into multiple single-target tracking problems. The representative MTT methods are the multiple hypothesis tracking [2] filtering algorithm and the joint probabilistic data association [2] filtering algorithm. Since the number of associations [3] increases exponentially with the increase of the number of targets and observations, the computational complexities of these methods are high. Although many research works have been proposed to improve the efficiency of data association, most of them are at the cost of performance. In recent years, Mahler [4] proposed an effective approach for the MTT problem, namely the random finite set (RFS) theory. The most commonly used RFS methods are the probability hypothesis density (PHD) filter [5], the cardinalised PHD (CPHD) filter [6] and the multi-target multi-Bernoulli (MeMBer) filter [4]. Above algorithms need no data association. To refine these algorithms, many improved versions have been proposed [7-12]. An acceleration method for the PHD filter was described in [7]. In [9], the miss-detection problems of the CPHD filter were studied. The cardinality balanced MeMBer (CBMeMBer) filter [10] was proposed to deal with the cardinality bias of the original MeMBer filter. Although the RFS-based algorithms can achieve well performance, they need to obey the assumptions that the birth intensity, the clutter intensity and the probability of detection are known as a priori. Hence, many researchers studied these issues and achieved significant results. The problem of clutter estimation was described in [13]. The unknown birth intensity for the PHD and CPHD filters was discussed in [14, 15], respectively. For the MeMBer filter, Vo et al. took both the clutter intensity and probability of detection into consideration and presented the robust multi-Bernoulli filter [16]. In [17], the variational Bayesian approximation method was used to estimate the unknown observation noise. With the development of sensor technologies, considerable attention has been focused on the multi-sensor solution for both civil and military applications. Compared with single-sensor target tracking, multi-sensor target tracking [18, 19] can improve the detection, the stability and the field of view of the tracking system effectively. The iterated-corrector PHD (IC-PHD) filter, an approximate multi-sensor PHD filter, was described in [4]. In addition, the idea of iterated-corrector was also applied in the CPHD filter [6] and the MeMber filter [20]. Since the iterated-corrector structure is easy to understand and implement, it has been used in many other areas, such as tracking sound source targets [21], multi-sensor registration errors [22], and radiation source location [23]. Although the iterated-corrector structure performs well in most tracking environments, it is sensitive to the sensor order and the probability of detection of the last updated sensor [24]. For example, a target cannot be estimated correctly, if it is only undetected by the last updated sensor. To deal with the above problems, the improved IC-PHD filter (IIC-GM-PHD), a calculation method for joint probabilities of miss-detection and detection, was described in [25]. The joint probabilities of miss-detection and detection are used to replace the original probabilities of miss-detection and detection of each sensor and reduce their influence on the tracking results. Moreover, the product multi-sensor PHD (PM-PHD) filter was proposed in [26], and a cardinality-modified coefficient was introduced to maintain the cardinality estimation. Since the modified coefficient is a scalar quantity and only maintains the magnitude of cardinality, the PM-PHD filter may not estimate the state of targets correctly. An improved PM-PHD filter, the cardinality-modified PM-PHD (CM-PM-PHD) filter [27], was proposed to handle the problem of state estimation. In the CM-PM-PHD filter, a weight calculation method was presented to improve the accuracy of state estimation by reassigning the weight of Gaussian components corresponding to targets. Both the IIC-GM-PHD filter and the CM-PM-PHD filter are based on the relationship of Gaussian components in the linear Gaussian mixture (GM) model, and they are not applicable to non-linear situations. Furthermore, another multi-sensor filter was described in [28], named the General PHD filter. The original general PHD filter was applied to the two-sensor case, and the multi-sensor case was presented in [29]. Unlike the IC-PHD filter and the PM-PHD filter, the general PHD filter needs to partition the measurements received by all sensors and traverse all the possible measurement partitions. The applications of the General PHD filter are limited by the huge computation burden caused by the measurement partition. A multi-sensor multi-Bernoulli filter with measurement partition was proposed in [30]. Moreover, a greedy measurement partitioning mechanism was introduced in [30] to solve the computationally intractable problem. Analysing the iterative process in the IC-PHD filter, it can be found that measurements obtained by different sensors are divided into several measurement subsets. All the measurement subsets can be grouped into three categories: the right measurement subset of targets, the unnecessary measurement subset of targets and the measurement subset of non-targets. If the last element of the right measurement subset of targets is the measurement of miss-detection, the weight of this measurement subset will be small and incorrect. In this situation, the corresponding target will be undetected. Addressing this problem, an improved weight calculation method is proposed in this paper. The new weight consists of two parts: the weight of 'subset to measurement' and weight of 'measurement to subset'. Thus, the right measurement subsets of targets have large weights even they contain the measurements of miss-detection. Furthermore, the weights of the measurement subsets of non-targets decrease. Simulation results show that the heuristic method is an effective solution to the miss-detection problem. The rest of this paper is organised as follows. The RFS and the PHD filter are described in Section 2. The IC-PHD filter is introduced in Section 3. Section 4 discusses the heuristic weight calculation method in details. Simulation results are performed in Section 5. Finally, Section 6 gives conclusions and future works. 2 Background In the multi-target motion model, the state set and the measurement set can be written by (1) (2) Here, and denote the numbers of targets and measurements at time k, respectively. Given a state RFS at the time , the state RFS at the time k can be expressed by (3) Here, is the RFS of targets which still survive at time k from a multi-target state . is the RFS of targets spawned by the previous multi-target state . is the RFS of new targets which appear instantly at the time k. The observation RFS at the time k can be expressed by (4) Here is the RFS of the clutter and is the measurement set generated by the state . The Bayesian filter in the RFS framework consists of two parts, the multi-target prediction and the multi-target update. Given a multi-target posterior density at time , the multi-target prediction density at the time k can be expressed by (5) Here, is the state transition function. Then, the multi-target posterior density at the time k can be expressed by (6) Here, is the observation likelihood function. Due to the high complexity and heavy computational burden in processing the joint probability density function of and directly, Mahler proposed a PHD, which approximates the probability density of multi-target RFS with its first-order moment. For an RFS on with probability distribution , its first-order moment is a non-negative function on , called the intensity, such that for each region (7) In other words, the integral of D over any region gives the expected number of the elements of in s. Hence, the total mass gives the expected number of the elements X. The local peaks of the intensity D are the points of the highest local concentration of the expected number of elements in , thus the estimation of the element state of . The simplest approach is to round and choose the resulting number of the highest peaks from the intensity. From the above analysis, PHD theory can be used to model MTT. Let denote the intensities associated with the multi-target posterior density and the multi-target predicted density , respectively, and then the posterior intensity can be derived by the PHD recursion (8) (9) Here, and denote the PHD of and at the time k, respectively. denotes the intensity of . and are the probabilities of survival and detection, respectively. 3 IC-PHD filter The IC-PHD filter propagates the updated PHD of each sensor at the time k. It seems that measurement sets of all sensors are updated separately. Actually, analogous to the general PHD filter, the IC-PHD filter fuses the measurements received by different sensors and divides them into several measurement subsets. This section gives another explanation for the IC-PHD filter. 3.1 Basic theory Assume that there is a homogeneous sensor network with s sensors. The sensor network has been calibrated in time and space. Measurements received by the sensors in their respective coordinate systems have been converted to the same time and spatial reference system. The sensors are supposed to collect their measurement sets simultaneously. The update formulas for the IC-PHD filter are defined as (10) (11) (12) (13) Here, , and denote the probability of detection, likelihood function and clutter intensity of the ith sensor, respectively. is the predicted PHD, is the updated PHD of the ith sensor and is considered as the tracking results of the IC-PHD filter. From (10), (14) and (15) are obtained as mentioned below (14) (15) In (15), the combined pseudo-likelihood of and can be defined as (16) Here, is the rth measurement of , and is given below (17) Supposed that measurement is generated by the miss-detection of the ith sensor. Note that is different from the measurements in , and it may correspond to multiple targets and clutter. Then, the combined pseudo-likelihood of and is (18) (19) and the combined pseudo-likelihood of and is (20) (21) and the combined pseudo-likelihood of and is (22) (23) From (10), (11) and (15), it can be obtained as (24) (25) 3.2 Problem of miss-detection Generally, targets can be tracked by the above-mentioned weight calculation method effectively. However, this method may lead to miss-detection in some situations. Supposed that the probability of detection of the sth sensor is large, and a target is only undetected by the sth sensor. The measurement of the target obtained by the sth sensor is denoted by , and the measurement obtained by the ith sensor is denoted by . The weight of measurement subset is (26) The state of the target corresponding to the measurement subset is estimated by the highest peaks of [19]. Here, (27) By (26) and (27), the number and state of the target can be estimated accurately by . For measurement subset , and are calculated by (28) and (29). (28) (29) Assumed that the probabilities of detection are independent of the state. Then we have (30) (31) Equation (31) indicates that the state estimation is not affected by the miss-detection. However, the target cannot be estimated by , because is much smaller than . 4 Improved weight calculation method A possible solution to the miss-detection problem is to change the sensor update order [24]. This method improves the tracking results of the IC-PHD filter effectively. However, the cardinality estimation still depends on the performance of the last update sensor. Addressing this problem, an improved weight calculation method is described in this section. 4.1 Motivation Assume that is the extended measurement set of the ith sensor and the number of measurements in is . Here, is the number of measurements in . At the time k, the updated PHD of the ith sensor is (32) and the updated PHD of the sensor is (33) The measurement subsets of the ith sensor and the sensor are denoted by and , respectively. The numbers of measurement subsets in and are and , respectively. Both and can be grouped into three categories: the right measurement subset of targets , the unnecessary measurement subset of targets and the measurement subset of non-targets . All these three categories are defined as follows: : a measurement subset which satisfies (i) if one target is detected by the ith sensor, the ith element in the measurement subset will be the measurement of the target; (ii) if one target is undetected by the ith sensor, the ith element in the measurement subset will be the measurement of miss-detection . : a measurement subset that consists of measurement of miss-detection and measurements generated by the same target. It is noted that does not belong to . : a subset that consists of measurements of miss-detection, different targets and clutter. Equations (32) and (33) indicate that the updated PHD of each sensor is the weighted sum of of all measurement subsets. The state estimation of targets is mainly determined by of and affected less by of . Moreover, of has a negative effect on the state estimation of targets. Therefore, the weights of should be large, the weights of should be small, and the weights of should be zero. However, it is difficult to determine the category that a measurement subset belongs to, as measurements in the measurement subset can be generated by miss-detection, targets and clutter. Fortunately, the probability that the measurement subset belongs to a certain category is available. The main idea of the new weight calculation method is to replace the original weight with the probability that a measurement subset belongs to . 4.2 Basic theory For a measurement subset , if the similarity among the measurements in is large, these measurements are most likely generated by the same target and vice versa. In other words, if the similarity among the measurements in is large, the probability that corresponds to a target will be large; and if the similarity among measurements in is small, the probability that corresponds to a target will be small. The similarity of can be obtained by computing the probability that corresponds to a target. The posterior intensity updated by is (34) In the state space, corresponds to at most one target and . Then, can be considered as the probability that corresponds to a target in the state space. For a given state , is the probability that corresponds to a target with the state . In some situations, may be incorrect. Supposed that the probability of detection . A target with the state is likely to be undetected by the ith sensor. If the value is large and , is most likely generated by clutter. Then the state estimation of the target is incorrect. Therefore, , the credibility of , is introduced to modify . For a target with state , is the probability that the target is detected by the ith sensor. In other words, is the probability that there is a measurement corresponding to the target. The credibility that corresponds to a target with the state is . Similarly, is the credibility that corresponds to a target with the state . Then, is calculated by (35) (36) In (35), is the credibility that corresponds to a target with the state . The modified probability that corresponds to a target with the state is (37) Thus, the similarity of is (38) In general, the values of similarities of and are larger than that of . For the measurement subsets of the same target, the value of similarity of is larger than that of . Based on , the probability that belongs to can be computed as follows: Supposed that is the measurement subset in . Here, . In , the measurement subset which consists of and is denoted by . If is the right measurement subset of a target, it can satisfy two conditions (i) is the right measurement subset of the target; (ii) is the measurement of the target or miss-detection. Since these two conditions are independent of each other, their effects on are discussed separately. (i) Probability of 'subset to measurement': can generate measurement subsets with . If is the right measurement subset of a target, a right measurement subset of this target exists in these measurement subsets. The probability that is the right measurement subset can be computed by (39) (ii) Probability of ' measurement' to subset: can generate measurement subsets with . If is the measurement of a target and , a right measurement subset of this target exists in these measurement subsets. The probability that is the right measurement subset can be computed by (40) Here, . If , one or more right measurement subsets of targets exist in these measurement subsets. The probability that is the right measurement subset can be computed by (41) Then, (40) and (41) are rewritten as (42) (43) Then, the new weight of is obtained by fusing and (44) Equations (39), (42) and (44) are illustrated in Figs. 1a–c, respectively. Fig. 1Open in figure viewerPowerPoint Weights of measurement subsets (a) Subset to measurement, (b) Measurement to subset, (c) Fusion weights Fig. 1 shows two targets. Target 1 is undetected by the sensor. and denote the right measurement subsets of target 1 and target 2, respectively. and denote the unnecessary measurement subsets of target 1 and target 2, respectively. denotes the measurement subset of non-targets. Here, . , and denote measurements of miss-detection, target 2 and clutter, respectively. Here, . The thickness of the line denotes the size of the weights. The dashed line means that the weights of measurement subsets are extremely small. The two weights of each measurement subset are illustrated in Figs. 1a and b. It can be seen that most measurement subsets have two appropriate weights, such as the right measurement subset of targets , the unnecessary measurement subset of targets , and the measurement subset of non-targets . Moreover, some measurement subsets have one large weight and one small or extremely small weight, such as . Actually, weights of these measurement subsets should not be large, because they are not the right measurement subset of targets. In (44), the negative effects of the large weights on the fusion weights are reduced by the small weights. Thus, each measurement subset has an appropriate fusion weight, which can be observed in Fig. 1c. 4.3 GM implementation There are two major approaches to implement the RFS filters, known as the sequential Monte Carlo [31] and the GM [32], respectively. In this paper, the MD-IC-PHD filter is compared with the IC-PHD filter [4], the IIC-GM-PHD filter [25], the PM-PHD filter [26], the CM-PM-PHD filter [27] and the MS-MeMber filter [30]. Since the IIC-GM-PHD filter and the CM-PM-PHD filter are only based on the linear GM model, the GM implementation of the MD-IC-PHD filter is given below. Based on the assumptions given in [26], both the predicted PHD and the updated PHD are assumed to obey the GM distribution. Then, the predicted PHD at the time k is given as (45) and the updated PHD of the ith sensor is given by (46) Here, , and denote the weight, the state mean vector and the covariance matrix of jth Gaussian component, respectively. J denotes the number of Gaussian components, and . Then, the updated PHD of the sensor is given by (47) when (48) (49) when (50) (51) (52) (53) (54) Here, . In (47), corresponds to several Gaussian components. The original weight of is computed by summing the weights of these Gaussian components (55) Here, . Suppose that is independent of the state. Equations (35), (42), (39) and (43) can be rewritten as (56)–(61). (56) (57) (58) (59) Then, the new weights of Gaussian components which correspond to are given by (60) Equation (47) can be written as (61) 5 Simulation results In this section, the simulation experiments are studied to analyse the proposed method. The state transition model is described as (62) Here, is the target state vector, and represent the planar position coordinates of the target, and represent their velocities and (63) (64) (65) Here, T=1 s is the sampling period and is the standard deviations of state noise . The measurement obtained by each sensor is given in Cartesian coordinates as follows (66) The covariance matrix of observation noise is . To reduce the amount of calculation, some similar Gaussian components close to each other will be merged into one Gaussian component, and the Gaussian components with negligible weights will be pruned. Moreover, the number of Gaussian components is limited. Based on experience gained from simulations, the merging and pruning thresholds are chosen as and , respectively, and the maximum number of Gaussian components is [26]. Note that the thresholds mentioned above are not necessary for the filter, and suitable thresholds have no effect on the performance of the filter. Some suggestions for threshold selection are given here. (i) When the state transition model and the measurement model are accurate, U, and have a small value; (ii) when the models are not accurate and the state and measurement noises are large, U, and have a large value or not function in the filter; (iii) should be much bigger than the number of targets. The optimal sub-pattern assignment (OSPA) [33] distance is employed to evaluate the MTT performance (67) In this experiment, two parameters are set to be and (see [27] for the meanings of these parameters), respectively. The experiment results are obtained by averaging the results of 200 Monte Carlo runs. Experiment. Considering that six targets with different initial positions move in the surveillance region . The setting of targets is shown in Table 1. The target trajectories are shown in Fig. 2. Table 1. Initial position and moving duration Target Initial position, m Start time, s End time, s 1 (−4000, −2000) 1 30 2 (−4000, 2000) 11 30 3 (4000, 2000) 21 50 4 (4000, −4000) 31 60 5 (4000, −4000) 21 60 6 (4000, 1500) 41 70 Fig. 2Open in figure viewerPowerPoint True target trajectory. 'Δ' — locations at which targets born; '□' — locations at which targets die The intensity of the birth Poisson RFS is described as (68) Here, and . The intensity of the clutter RFS is assumed to be (69) Here, is the clutter rate of the ith sensor and is the uniform density over the surveillance region. In this experiment, different parameters are used to verify the effectiveness of the proposed method, such as the number of sensors, the probability of detection of the last sensor, observation noise and clutter rate. The setting of these parameters is given in Table 2. Moreover, two different sensor orders are considered here. The sensor with is the last update sensor in the first case, while it is the second update sensor in the second case. The probabilities of detection of the other three sensors are . Table 2. Setting of different parameters s number of sensors 3, 4, 5, 6 0.99 10 10 probability of detection 4 0.99, 0.95, 0.9, 0.85, 0.8 10 10 observation noise 4 0.99 10, 30 10 clutter rate 4 0.99 10 10, 30 The simulation results of different numbers of sensors, probabilities of detection, observation noises and clutter rates are shown in Figs. 3-6, respectively. The running time of different filters is given in Table 3. The estimated number of targets, OSPA distance and mean squared error (MSE) of target numbers for different sensor orders are shown in Figs. 7a–c, respectively. Table 3. Running time of different filters (s) s IC-PHD IIC-GM-PHD PM-PHD CM-PM-PHD MS-MeMber MD-IC-PHD 3 0.05 0.05 0.07 0.21 0.06 0.07 4 0.07 0.07 0.10 0.27 0.07 0.09 5 0.15 0.15 0.15 0.38 0.07 0.15 6 0.16 0.16 0.21 0.52 0.08 0.18 Fig. 3Open in figure viewerPowerPoint Tracking results with different numbers of sensors (a) OSPA distance, (b) MSE Fig. 4Open in figure viewerPowerPoint Tracking results with different probabilities of detection (a) OSPA distance, (b) MSE Fig. 5Open in figure viewerPowerPoint Tracking results with different observation noises (a) OSPA distance, (b) MSE Fig. 6Open in figure viewerPowerPoint Tracking results with different clutter rates (a) OSPA distance, (b) MSE Fig. 7Open in figure viewerPowerPoint Tracking results with different sensor orders (a) Estimated number of targets, (b) OSPA distance, (c) MSE In Fig. 3, the performances of all filters are getting better as the number of sensors increases, except the IIC-GM-PHD filter. This is because the IIC-GM-PHD filter is mainly used to deal with the problem of miss detection in the IC-PHD filter. False alarms will be retained in the filtering process of the IIC-GM-PHD filter, and it results in a cardinality overestimation. This problem will become worse as the number of sensors increases. Moreover, changes in the OSPA distance are small when , but the running time is increasing. In Table 3, the running time of the MS-MeMBer filter is the smallest of these algorithms, because the MS-MeMBer filter uses thresholds to remove unnecessary multi-sensor subsets and partitions. The IC-PHD filter, the IIC-GM-PHD filter and MD-IC-PHD filter have similar running time. The running time of the CM-PM-PHD filter is more than that of the above filters. Because the CM-PM-PHD filter needs to reassign the weight of Gaussian components under different situations. Figs. 4 and 5 show that the IIC-GM-PHD filter, the CM-PM-PHD filter and the MD-IC-PHD filter are more stable than the IC-PHD filter and the PM-PHD filter. Both the cardinality estimation and state estimation of the IC-PHD filter are affected by the probability of detection. Although the PM-PHD filter performs well in estimating the cardinality, it may estimate some false targets easily when the true targets are undetected. The MS-MeMber has the best cardinality estimation, but its state estimation is slightly less accurate than that of the CM-PM-PHD filter and the MD-IC-PHD filter. Fig. 6 shows that the proposed method is insensitive to the clutter rate, for the reason that the weight of false alarms is reduced. , , and which are calculated by (39), (42), (44) and (26), respectively. is a measurement subset which consists of and measurement of the target, and the weight of is . Then the weight of the target is computed by the summation of and . In the original weight calculation method, both and are close to 1, and the number of targets is overestimated. In the proposed method, is determined by and . The weights corresponding to measurements are normalised in (39). Then, and ( and ). Here, and are two weights of . Thus, the number of targets is estimated correctly. A specific example is shown in Figs. 8 and 9. Fig. 8Open in figure viewerPowerPoint Gaussian components at 45 s (a) Original method, (b) Proposed method Fig. 9Open in figure viewerPowerPoint Weights of Gaussian components of 'target 5' (a) Subset to measurement, (b) Measurement to subset, (c) Fusion weights In Fig. 8, 'target 5' is overestimated by the original method at , but it is estimated correctly by the MD-IC-PHD filter. The weights of Gaussian components computed by (39), (42) and (44) are shown in Figs. 9a–c, respectively. In Figs. 9a–c, the number, the mean vector and the covariance matrix of Gaussian components are the same. Some Gaussian components cannot be displayed in Figs. 9b and c, because their weights are tiny. In Fig. 9a, the summation of weights of and is approximately equal to 1. In Fig. 9b, one Gaussian component corresponds to an unnecessary measurement subset of 'target 5'. The other two Gaussian components correspond to and , respectively, and their weights are both approximately equal to 1. In Fig. 9c, only the Gaussian components corresponding to and are retained, and the summation of their weights is approximately equal to 1. As shown in Fig. 8b, the two Gaussian components are merged into one Gaussian component. Fig. 7 shows that the proposed method performs well in both two sensor update orders, and the method described in [19] ('IC-PHD 1432') can improve the performance of the IC-PHD filter effectively. Essentially, the method described in [19] is still the IC-PHD filter, and thus its tracking results depend on the last update sensor. Although the probability of detection of the last update sensor is large, targets will not be estimated if they are undetected by last update sensor. A specific example is shown in Figs. 10 and 11. The Gaussian components with sensor order '1234' () are shown in Fig. 10. It can be seen that the weight of 'target 2' is underestimated by the original method and estimated by the MD-IC-PHD filter. The three weights computed by (36), (39) and (41) are shown in Figs. 11a–c, respectively. In Fig. 11, each Gaussian component corresponds to a measurement subset. is a measurement subset which consists of the measurements of 'target 2' in and the measurement of miss-detection in . is the right measurement subset of 'target 2'. In Fig. 11a, the weights of four measurement subsets are approximately equal to 1. It is difficult to determine the right measurement set of 'target 2'. However, in Fig. 11b, only is retained, and its weight is approximated to be equal to 1. Then 'target 2' can be estimated by fusing the results of Figs. 11a and b, and the fusion results are shown in Fig. 11c. Fig. 10Open in figure viewerPowerPoint Gaussian component of targets at 27 s (a) Original method, (b) Proposed method Fig. 11Open in figure viewerPowerPoint Weights of Gaussian components of 'target 2' (a) Subset to measurement, (b) Measurement to subset, (c) Fusion weights 6 Conclusions In original IC-PHD filter, it is difficult to calculate the weights of measurement subsets which include the measurement of miss-detection. The inaccurate weights of these measurement subsets may lead to the miss-detection problem. In the improved weight calculation method, these measurement subsets can be given proper weights as long as they are the right measurement subsets of targets. In addition, the proposed method can reduce the negative influence of false alarm. The new weight of false alarm is mainly determined by the ratio of two weights of 'measurement to subset', where these two weights correspond to the false alarm and target, respectively. It makes the values of the weight of false alarm small. The effectiveness of the proposed method is verified by the simulation results. In this paper, some assumptions are made to avoid the influence of multi-sensor system bias and facilitate the filter design, such as the measurements have been calibrated in time and space. However, the practical application of the proposed filter may be limited by these assumptions. To deal with the problem of system bias, we will relax these assumptions and take the blind calibration [34] into consideration in the future works. 7 Acknowledgment This work was supported by the National Natural Science Foundation of China under grant No. 61803288, the Postdoctoral Science Foundation of China under grant No. 2018M633467 and the Fundamental Research Funds for the Central Universities under Grant No. XJS190201. 8 References 1Wax, N.: 'Signal-to-noise improvement and the statistics of track populations', J. Appl. Phys., 1955, 26, (5), pp. 586– 595CrossrefWeb of Science®Google Scholar 2Blackman, S.S.: 'Multiple hypothesis tracking for multiple target tracking', IEEE Aerosp. Electron. Syst. Mag., 2004, 19, (1), pp. 5– 18CrossrefWeb of Science®Google Scholar 3Fortmann, T., Bar-Shalom, Y., Scheffe, M.: 'Sonar tracking of multiple targets using joint probabilistic data association', IEEE J. Ocean. Eng., 1983, 8, (3), pp. 173– 184CrossrefWeb of Science®Google Scholar 4Mahler, R.: ' Statistical multisource-multitarget information fusion' ( Artech House Inc., USA, 2007) Google Scholar 5Mahler, R.: 'Multitarget Bayes filtering via first-order multitarget moments', IEEE Trans. Aerosp. Electron. Syst., 2003, 39, (4), pp. 1152– 1178CrossrefWeb of Science®Google Scholar 6Mahler, R.: 'PHD filters of higher order in target number', IEEE Trans. Aerosp. Electron. Syst., 2007, 43, (4), pp. 1523– 1543CrossrefWeb of Science®Google Scholar 7Li, T., Sun, S., Sattar, T.P.: 'High-speed Sigma-gating SMC-PHD filter', Signal Process., 2013, 93, (9), pp. 2586– 2593CrossrefWeb of Science®Google Scholar 8Tang, X., Wei, P.: 'Multi-target state extraction for the particle probability hypothesis density filter', IET Radar Sonar Navig., 2011, 5, (8), pp. 877– 883CrossrefWeb of Science®Google Scholar 9Ulmke, M., Franken, D., Schmidt, M.: ' Missed detection problems in the cardinalized probability hypothesis density filter'. Proc. 11th Conf. Information Fusion, Cologne, Germany, June 2008, pp. 1– 7Google Scholar 10Vo, B.T., Vo, B.N., Cantoni, A.: 'The cardinality balanced multi-target multi-Bernoulli filter and its implementations', IEEE Trans. Signal Process., 2009, 57, (2), pp. 409– 423CrossrefWeb of Science®Google Scholar 11Ouyang, C., Ji, H.B., Li, C.: 'Improved multi-target multi-Bernoulli filter', IET Radar Sonar Navig., 2012, 6, (6), pp. 458– 464CrossrefWeb of Science®Google Scholar 12Yang, J.L., Ji, H.B., Ge, H.W.: 'Multi-model particle cardinality-balanced multi-target multi-Bernoulli algorithm for multiple manoeuvring target tracking', IET Radar Sonar Navig., 2013, 7, (2), pp. 101– 112Wiley Online LibraryCASWeb of Science®Google Scholar 13Chen, X., Tharmarasa, R., Pelletier, M. et al: 'Integrated clutter estimation and target tracking using Poisson point processes', IEEE Trans. Aerosp. Electron. Syst., 2012, 48, (2), pp. 1210– 1235CrossrefWeb of Science®Google Scholar 14Ristic, B., Clark, D., Vo, B.N. et al: 'Adaptive target birth intensity for PHD and CPHD filters', IEEE Trans. Aerosp. Electron. Syst., 2012, 48, (2), pp. 1656– 1668CrossrefWeb of Science®Google Scholar 15Beard, M., Vo, B.T., Vo, B.N. et al: 'A partially uniform target birth model for Gaussian mixture PHD/CPHD filtering', IEEE Trans. Aerosp. Electron. Syst., 2013, 49, (4), pp. 2835– 2844CrossrefWeb of Science®Google Scholar 16Vo, B.T., Vo, B.N.: 'Robust multi-Bernoulli filtering', IEEE J. Sel. Top. Signal Process., 2013, 7, (3), pp. 399– 409CrossrefWeb of Science®Google Scholar 17Yang, J., Ge, H.: 'An improved multi-target tracking algorithm based on CBMeMBer filter and variational Bayesian approximation', Signal Process., 2013, 93, (9), pp. 2510– 2515CrossrefWeb of Science®Google Scholar 18Khaleghi, B., Khamis, A., Karray, F.O. et al: 'Multisensor data fusion: a review of the state-of-the-art', Inf. Fusion, 2013, 14, (1), pp. 28– 44CrossrefWeb of Science®Google Scholar 19Fantacci, C., Vo, B.N., Vo, B.T.: 'Robust fusion for multisensor multiobject tracking', IEEE Signal Process. Lett., 2018, 25, (5), pp. 640– 644CrossrefWeb of Science®Google Scholar 20Vo, B.T., See, C.M., Ma, N. et al: 'Multi-sensor joint detection and tracking with the Bernoulli filter', IEEE Trans. Aerosp. Electron. Syst., 2012, 48, (2), pp. 1385– 1402CrossrefWeb of Science®Google Scholar 21Pham, N.T., Huang, W., Ong, S.H.: ' Multiple sensor multiple object tracking with GMPHD filter'. Proc. 10th Conf. Information Fusion, Quebec, Canada, July 2007, pp. 1– 7Google Scholar 22Li, W., Jia, Y., Du, J. et al: 'Gaussian mixture PHD filter for multi-sensor multi-target tracking with registration errors', Signal Process., 2013, 93, (1), pp. 86– 99CrossrefWeb of Science®Google Scholar 23Zhang, H., Jing, Z., Hu, S.: 'Localization of multiple emitters based on the sequential PHD filter', Signal Process., 2010, 90, (1), pp. 34– 43CrossrefWeb of Science®Google Scholar 24Nagappa, S., Clark, D.: ' On the ordering of the sensors in the iterated-corrector probability hypothesis density (PHD) filter'. Proc. Conf. Signal Processing, Sensor Fusion Target Recognition, Orlando, Florida, May 2011, pp. 1– 6Google Scholar 25Liu, L., Ji, H.B., Fan, Z.H.: 'Improved iterated-corrector PHD with Gaussian mixture implementation', Signal Process., 2015, 114, pp. 89– 99CrossrefWeb of Science®Google Scholar 26Mahler, R.: ' Approximate multisensor CPHD and PHD filters'. Proc. 13th Conf. Information Fusion, Edinburgh, Scotland, July 2010, pp. 1– 8Google Scholar 27Liu, L., Ji, H.B., Fan, Z.H.: 'A cardinality modified product multi-sensor PHD', Inf. Fusion, 2016, 31, pp. 87– 99CrossrefCASWeb of Science®Google Scholar 28Mahler, R.: ' The multisensor PHD filter, I: general solution via multi-target calculus'. Proc. SPIE Int. Conf. Signal Processing, Sensor Fusion, Target Recognition, Orlando, Florida, May 2009, pp. 1– 12Google Scholar 29Nannuru, S., Blouin, S., Coates, M. et al: 'Multisensor CPHD filter', IEEE Trans. Aerosp. Electron. Syst., 2016, 52, (4), pp. 1834– 1854CrossrefWeb of Science®Google Scholar 30Saucan, A.A., Coates, M.J., Rabbat, M.: 'A multisensor multi-Bernoulli filter', IEEE Trans. Signal Process., 2017, 65, (20), pp. 5495– 5509CrossrefWeb of Science®Google Scholar 31Vo, B.N., Singh, S., Doucet, A.: 'Sequential Monte Carlo implementation of the PHD filter for multi-target tracking', IEEE Trans. Aerosp. Electron. Syst., 2005, 41, (4), pp. 1224– 1245CrossrefWeb of Science®Google Scholar 32Vo, B.N., Ma, W.K.: 'The Gaussian mixture probability hypothesis density filter', IEEE Trans. Signal Process., 2006, 54, (11), pp. 4091– 4104CrossrefWeb of Science®Google Scholar 33Schuhmacher, D., Vo, B.T., Vo, B.N.: 'A consistent metric for performance evaluation of multi-object filters', IEEE Trans. Signal Process., 2008, 56, (8), pp. 3447– 3457CrossrefWeb of Science®Google Scholar 34Balzano, L., Nowak, R.: ' Blind calibration of sensor networks'. Proc. Sixth Int. Symp. Information Processing Sensor Networks, New York, USA, April 2007, pp. 79– 88Google Scholar Volume14, Issue9September 2020Pages 1451-1461 FiguresReferencesRelatedInformation