Probability of detection and clutter rate estimation in target tracking systems: generalised maximum likelihood approach

IET Radar, Sonar & NavigationVolume 13, Issue 11 p. 1963-1973 Research ArticleFree Access Probability of detection and clutter rate estimation in target tracking systems: generalised maximum likelihood approach Asem Al-Hasaeri, Asem Al-Hasaeri Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorAleksandra Marjanović, Aleksandra Marjanović Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorPredrag Tadić, Predrag Tadić orcid.org/0000-0002-8845-997X Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorSanja Vujnović, Corresponding Author Sanja Vujnović svujnovic@etf.bg.ac.rs Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorŽeljko Đurović, Željko Đurović Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this author Asem Al-Hasaeri, Asem Al-Hasaeri Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorAleksandra Marjanović, Aleksandra Marjanović Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorPredrag Tadić, Predrag Tadić orcid.org/0000-0002-8845-997X Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorSanja Vujnović, Corresponding Author Sanja Vujnović svujnovic@etf.bg.ac.rs Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this authorŽeljko Đurović, Željko Đurović Signals and Systems Department, School of Electrical Engineering, University of Belgrade, Bulevar kralja Aleksandra 73, 11120 Belgrade, SerbiaSearch for more papers by this author First published: 01 November 2019 https://doi.org/10.1049/iet-rsn.2019.0064AboutSectionsPDF 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 onFacebookTwitterLinkedInRedditWechat Abstract The efficacy of any moving target tracking technique depends on the understanding the background, or clutter, and the parameters that describe the detection properties of the objects. The nature of these quantities is statistical and not only are they unknown as a rule in practice, but they are also variable over both time and space. The study proposes a method for estimating the time-varying probability of detection of each tracked object individually, and the density of false alarms in the immediate vicinity of the current position of an object. The method is based on the generalised maximum likelihood approach, assuming tracking of a solitary target. The applicability and constraints of the proposed solution are illustrated by simulations. 1 Introduction Moving target tracking became a topic of particular interest back in the 1970s, when pioneering papers reported initial results of upgrades of moving target tracing filters and the development of techniques for the association of observations during tracking in a cluttered environment [1-5]. The interest in this topic has been growing over the past 15 years. There seem to be two reasons for the increased output of scientific results. One is that the applicability of the developed algorithms has expanded considerably. The focus has shifted from solely military systems to the civilian sector, including surveillance systems for traffic, pedestrian hubs, subway stops, and the like [6]. Such systems also attracted special attention of biological research, where cells of a certain type or even molecules need to be identified, recorded and tracked [7, 8]. It should be noted that radar signals are no longer the only type of input data for tracking systems. Now there are also thermal imaging cameras, TV cameras, sonars, microscope imaging etc. The other important reason for the increasing interest is the novel theoretical framework in which such systems are developed. Previous algorithms were based on a more traditional Bayesian probabilistic approach, such as multiple hypothesis tracking (MHT) and All-Neighbours Data Association, including their versions: Probabilistic Data Association (PDA) and Joint Probabilistic Data Association (JPDA) [9-12]. However, over the past ten years, the theory of random finite sets has provided a basis for the development of new tracking filters, such as probability hypothesis density (PHD) and cardinalised probability hypothesis density (CPHD) filters [7, 13, 14]. Literature sources describe a large number of different structures of target tracking systems, various solutions for target state estimation filters, and a number of algorithms for measurement-to-track association. However, all of these concepts share one trait – their efficacy largely depends on the knowledge of the target detection probability and density of encountered false alarms. Except in very rare cases, these parameters are unknown and there have been many attempts to estimate them. A valuable result in this regard is reported by Yildrim et al. [15], where an expectation maximisation (EM) algorithm is used to estimate parameters in Gaussian multiple target tracking (MTT) models, which include target detection probability and false alarm density. The research reported in that paper developed both batch and on-line procedures based on experimental experience and concluded that the sequential Monte Carlo (SMC) online EM procedure is highly suitable if the computational budget is available, albeit with bias in the estimation of unknown parameters. An application described by Schlangen et al. [7] is very interesting because the clutter rate of a PHD filter is estimated as part of image analysis in single-molecule localisation microscopy. The results are very satisfactory, keeping in mind that the molecule detection probability is not estimated and that a wide surveillance region for estimating the unknown false alarm density parameter is observed simultaneously. The results reported by He et al. [12] should also be mentioned because a JPDA filter for data association is used to estimate both target detection probability and false alarm density. The simulation results showed that the proposed algorithm has a number of advantages compared to the standard JPDA filter, demonstrated by a smaller mean optimal sub-pattern assignment (OSPA) distance. The research by Chen et al. [16] is remarkable because it assumes a non-homogenous clutter background. They developed two techniques. The first is a generalised maximum likelihood (GML) estimator based on predictive measurement set likelihood functions, whereas the second uses PHD filters in combination with the normal-Wishart mixture function. Simulations showed that the developed clutter rate estimators improved PHD filter performance. Mahler et al. [14] claim that clutter rate parameters and a detection profile are essential for PHD and CPHD filter efficiency. They developed two versions of these filters, which can adaptively estimate the parameters. Closed-form solutions were derived using Beta and Gaussian mixtures. In that paper as well, the contribution of the estimators was demonstrated by an increase in the OSPA miss distance, clutter rate estimation, and cardinal statistics. The numerical complexity of the algorithm is very high, the false alarm density is estimated in a very large region whose size is two orders of magnitude larger than the standard deviation of the measurement noise of the target model, and the detection probability is undeterminable for particular targets. The paper highlights estimator sensitivity to initial filter values. A careful review of the available literature leaves the impression that the estimation of these parameters is extremely complex. However, the same literature makes room for improvement of the existing methods. If the requirements in designing these estimators, the limitations arising from their practical implementation and the motives underlying the main driving force of research in this field were to be summarised, the list would be too long. Therefore, we have been inclined to single out the motives representing the main driving force of the research presented in this paper: The method for estimating probability of target detection and false alarm density must be simple enough to be implemented in real time; This method must be applicable when the time window, within which the observations are obtained, is short, because real applications often involve tracking targets that remain in the observational area for a short time; This method should provide a good compromise between the quality of the probability of moving target detection and false alarm density estimates, because it is well known in the literature that the quality of the second estimate undermines the quality of the first one; This method should be functional even when the probability of target detection is significantly smaller than 1 and the density of false alarms is as large as possible; The new method should enable the local estimation of false alarm density, i.e. in the vicinity of the moving target; It would be practical if the new method could assign probability of detection to each target being monitored; The new method should offer a good compromise between the ability of tracking non-stationary parameters and and the estimation quality; The new method should not be overly susceptible to initial values of the estimator thereby not requiring too much a priori knowledge. Also, it should not require precise adjustment of a large number of parameters affecting the quality of the whole algorithm; The new method should have a general form and its applicability should not depend on the reference system in which the observation is carried out, its dimensionality, the structure of the filter used for tracking kinematic characteristics of the target and the structure of measurement association algorithm. In other words, the new method should be a system capable of supporting different multi target filtering and tracking systems. The new method, presented in this paper, has the following distinctive qualities that can be considered as contributions to the field: (i) the proposed method is numerically simple and can be easily implemented in real-time operation; (ii) the method simultaneously estimates the detection probability for each individual target and (iii) density of false alarms in the immediate vicinity of each of the target's position; (iv) a compromise between estimator variance and the ability to track non-stationary parameters and is easily achieved; (v) it shows good results even in the case when the probability of target detection is significantly smaller than 1; (vi) its implementation requires a minimum of a priori knowledge about the nature of the object being monitored as well as its surroundings. Many authors reported progress, originating from the fact that the number of measurements in a certain local volume of the monitored space is a function of both the unknown false alarm density and unknown number of targets, along with their unknown detection probability. This is why many researchers are trying to decouple the estimation of these two parameters by estimating the clutter rate in a much wider region, assuming that it is homogeneous. As a consequence, the number of included false targets increases, relative to the number of actual targets, such that the relative effect of the actual targets is reduced. On the other hand, keeping in mind the increasingly demanding and complex environments in which MTT systems are used, it is clear that the clutter-rate homogeneity assumption is often not sustainable. In addition, given the relations of the Kalman filter that needs to be adapted to the fact that a certain measurement is associated to a track with only a certain probability, there is a definite need to associate a detection probability with each target. Finally, available estimators are highly demanding, either because of the number of needed mathematical operations in real time or the complexity of expressions that have to be computed. This paper proposes an approach to the estimation of the probability of detection of a single target as it moves in a cluttered environment, along with the estimation of the density of false alarms in its immediate vicinity. A major contribution of this research is reflected in the fact that the estimation of false alarm density is not based on a set of measurements from a wide region in which the target moves, but on measurements from a gate located around the predicted target position. As such, it is not necessary to assume that this parameter is uniform in the wide surveillance region, which is generally not a sustainable assumption. The procedure is based on GML principles and is numerically much simpler than the previously described approaches, because each scan comprises no more than two particular hypotheses. The procedure can be generalised for multiple targets, whereby a detection probability and false alarm density in the immediate vicinity are associated with each track. Another advantage of the proposed technique is that it does not require a specific form of data association, so that it can be used in parallel with any moving target tracing algorithm or a general estimator of the detection profile and clutter environment. Section 2 describes the structure of the proposed approach. Section 3 discusses its properties, introduces improvements, and shows simulation results. Finally, Section 4 describes how the method can be extended to handle multiple targets, discusses constraints, and contains the most important conclusions. 2 Structure of the proposed estimator The following assumptions are made in order to apply a two-step GML approach to the estimation of target detection probability and false alarm density, which are very important statistical parameters of an MTT system. Let Kalman filter be based on the moving target model: (1) where is the track state vector, is the stochastic process of the model error, and and are state and input matrices, which can be time dependent in the general case. Let us assume that the observation relation is (2) where is the measurement vector, is the measurement matrix, and is the stochastic process of measurement noise. Based on the well-known Kalman filter relation, one of the outputs of each scan is the state vector prediction of the next scan. Let us denote this prediction by and the prediction error covariance matrix by . Further, in order to apply one of the association techniques, let there be a gate around the track prediction . The dimensions of the gate are , where is the gate parameter, is the ith element on the diagonal of the covariance matrix , and n is the dimension of the space in which the gate is located. Given that the target detection probability is <1 and that the target moves in a cluttered area, there can be zero, one or more observations in each scan within the gate. For illustration purposes, Fig. 1 shows track position predictions in three consecutive scans, around which there are gates. It is apparent that scan detected only one observation in the gate, denoted by , that there are three observations in scan k, and that scan detected two measurements within the gate. The gate size in the figure is identical from scan to scan, but it generally depends on the value of the diagonal in the matrix , so that it need not be stationary. Fig. 1Open in figure viewerPowerPoint Gate functions and observations in three consecutive scans The following is proposed in order to construct hypotheses whose likelihood can be maximised to estimate unknown parameters. Two cases are differentiated. If the number of incoming observations in the kth scan is , there can only be one partial hypothesis: : no target detected and zero false alarms. If the number of incoming observations included in the gate of the kth scan is , then there can be two partial hypotheses: : no target detected and there are false alarms, and : target detected and there are false alarms. If the last N scans are considered by associating the partial hypothesis, then integral hypotheses are formed. It should be noted that contrary to other MHT approaches, the number of hypotheses in this case is considerably smaller. The savings can easily be discerned by looking at the kth scan in Fig. 1. Namely, in general, four partial hypotheses could be defined for the scenario shown in the figure: (i) the zero hypothesis, with all incoming observations being false alarms; (ii) the first hypothesis – that observation originates from the target and that the other two are false alarms; (iii) the second hypothesis – that observation originates from the target and the first and third are false alarms; and (iv) the third hypothesis – that observation originates from the target and that the first and second are false alarms. However, since we will use the GML approach to compute the likelihood of each hypothesis separately, it is obvious that the first and second hypotheses above will be much less likely than the third, because the statistical distance from the predicted position is smallest for , such that they will not affect the algorithm. Hence, each partial hypothesis that assumes target detection will also assume that it corresponds to the observation with the least statistical distance from the predicted position. The paper proposes an approximate GML method, comprised of the following steps: Step 1: Determine an integer variable N that represents the length of the time interval, or the number of scans, based on which the unknown target detection probability and false alarm density in the immediate vicinity of its current position will be estimated. Step 2: Symbolically, each integral hypothesis that covers scans from to kth can be represented as a series of N binary numbers: where variable assumes a zero value if all the observations in the jth scan within the gate are declared as false alarms, and the value 1 if in the given scan the observation statistically closest to the prediction is declared as the actual target. Step 3: Form integral hypotheses in the manner described in the previous section and associate the likelihood function with each. Step 4: Compute the parameter pair for each integral hypothesis, thus maximising the likelihood of hypothesis . Step 5: Among all the optimal values of likelihood, select the highest and determine the optimal estimators from the chosen hypothesis. This approach is known in the literature as minimum description length [17]. In approximate terms, it is equivalent to the GML rule [18]. In order to implement such a methodology, it is necessary to define an expression for the likelihood of the integral hypothesis. Consequently, the following common assumptions are made: Assumption 1.Prediction , which is the result of applying the Kalman filter, is an unbiased state estimate of . Assumption 2.The prediction error covariance matrix , computed by Kalman filter, is a true measure of prediction uncertainty. Assumption 3.The vector process is Gaussian, where the mathematical expectation is equal to the true state , with the corresponding covariance matrix . Assumption 4.The false alarm frequency is Poisson's stochastic process with a mean value of , where is the false alarm density and is the volume of the gate. Assumption 5.The volume of the gate is sufficiently small, such that the spatial distribution of false alarms can be considered uniform. Assumption 6.The observation sequence N is sufficiently short, such that the probability of target detection can be considered constant. Consequently, the likelihood of the ith integral hypothesis can be computed as follows: (3) where is the likelihood of the partial hypothesis in the jth scan and, depending on whether or 1, (4) Here is the statistical distance between prediction position and the statistically closest observation in scan j: (5) Then the likelihood of integral hypothesis can be determined as (6) Here, is the position of the observation in the jth scan, which is statistically closest to the predicted target position, and is the corresponding probability density function: (7) The following equalities need to be solved in order to maximize the above likelihoods of unknown parameters: (8) The results are simple and they provide estimates of detection probability and false alarm density: (9) This estimator will be described in the next section and modifications will be proposed to overcome the noted shortfalls of the approach. 3 Estimator features and improvement 3.1 Analysis of estimator features A large number of Monte Carlo simulations were undertaken to test the properties of the proposed estimation technique. This included 20,000 reruns. The length of the estimation sequence was . Fig. 2a shows the quality of target detection probability estimation. Parameter , which refers to the width of the gate, was varied on the horizontal axis. The false alarm density was constant in all the experiments and amounted to . The vertical axis showed the average value of target detection probability estimation. The resulting graphic is very interesting because it leads to the conclusion that target detection probability estimation is highly dependent on the size of the gate. The plot in Fig. 2b shows the estimation variance. The results were largely as expected. Namely, as the target detection probability increased, the variance decreased. However, it is interesting that this logic changed somewhat as the gate size decreased. Specifically, as the size of the gate decreased, the probability of an object measurement being outside the gate increased. This favours the estimation of low detection probabilities, such as ∼<0.6. The very high variance of this estimator was surprising. For example, for detection probabilities and gate size , the standard deviation of the estimators was of the order of 0.2, which is high compared to the quantity that is being estimated. However, it is good to know that the standard deviation decreases considerably as the detection probability increases. For instance, where and , the standard deviation is about 0.1 and becomes acceptable. Fig. 2Open in figure viewerPowerPoint Statistical parameters as a function of gate width at a fixed false alarm density, (a) Averaged estimation of target detection probability, (b) Standard deviation of estimation The estimation of false alarm density for different values of the gate size and real target detection probabilities was analysed in a similar manner. Fig. 3a corroborates the estimator quality, or that the false alarm density estimation bias depends on the size of the gate. What is interesting is that as the size of the gate decreases, the estimate increases, and vice-versa. The reason for this is likely that an estimate of false alarm density is the quotient of the estimated number of false alarms in the gate and the volume of the gate. The volume of the gate increases with the power 3 of parameter and the estimated number of false alarms by a slower function of this parameter, resulting in the dependency shown in the figure. It is also important to note that the optimal that ensures unbiased estimation of both parameters, and , is dependent on true and unknown value of . Fig. 3b shows the variance of false alarm density estimates. It is evident that as the gate increases, the variance decreases, as expected. On the other hand, it is apparent that this variance, too, is quite large. The standard deviation of false alarm density estimation, , for gate size , is very high and amounts to ∼. Fig. 3Open in figure viewerPowerPoint Statistical parameters as a function of gate width at a fixed false alarm density, (a) Averaged false alarm density estimation, (b) Standard deviation of estimation Fig. 4 shows how the quality of detection probability estimation changes as a function of false alarm density. The estimation quality was analysed for two fixed values of false alarm density – and . It is obvious at first glance that the detection probability estimation bias is greater when the density of false alarms is high, which is as expected. Another possible conclusion is that the difference in the quality of estimation of is more apparent at lower detection probabilities or, in other words, that at higher detection probabilities the false alarm density has a slightly smaller effect on its estimation. Fig. 4Open in figure viewerPowerPoint Quality of detection probability estimation as a function of false alarm density at (full line) and (dashed line) Additional Monte Carlo simulations were undertaken to obtain a clearer picture of the estimator quality. The width of the gate was fixed at , the actual detection probability was and the actual false alarm density . The length of sequences N was varied. Fig. 5a shows detection probability estimation as a function of the length of sequence N. It is apparent that for a sequence length of , the relative detection probability estimation bias is ∼ 2% and that it additionally decreases as N increases. In parallel, Fig. 5b also shows the variation in the variance of this estimation. The plot indicates that the estimation quality improves considerably as parameter N increases. The standard deviation of this estimation was 0.23 for and then dropped to 0.16 at , which is ∼20% of the estimated quantity and not negligible. Fig. 5Open in figure viewerPowerPoint Statistical parameters as a function of the length of sequence N at fixed parameters Kg = 2.4, Pd = 0.8 and (a) Averaged estimation of target detection probability, (b) Standard deviation of the corresponding estimation A similar conclusion follows from the plots in Fig. 6, which shows the false alarm density estimation and corresponding variance, respectively. In this scenario, the false alarm density bias is positive at smaller lengths of sequence N, but it decreases as this parameter increases, as does the estimator variance. It is interesting that the bias increases as the false alarm density decreases. It is difficult to efficiently estimate such a low false alarm density, especially if estimated locally, in a limited observation space. The conclusion is that the standard deviation of the estimation of this parameter is high, relative to the absolute value of the parameter being estimated. The results suggest that the variance of the proposed estimators can be decreased if the number of consecutive scans used for the estimation is increased. However, this would preclude the possibility of matching the dynamics of unknown parameter estimation. Fig. 6Open in figure viewerPowerPoint Statistical parameters as a function of the length of sequence N at fixed parameters Kg = 2.4, Pd = 0.8 and (a) Averaged false alarm density estimation, (b) Corresponding standard deviation of estimation To illustrate the quality (or lack thereof) of such an estimator, Fig. 7 depicts the estimation of these parameters in 100 scans, where the actual values were and . The plots show to what extent the estimators defined by relation (9) are not usable. Namely, a more detailed analysis shows that only 31% of 100 estimates fall within the confidence interval . The situation is even worse when it comes to density of false alarms because 26% of the estimates are in the interval . Bearing in mind that all serious data association techniques assume the knowledge of these two parameters, such large estimation errors can impose significant limitation on the association quality. Fig. 7Open in figure viewerPowerPoint Tracking parameters' estimates at fixed parameters Kg = 2.8, Pd = 0.5 and (a) Detection probability estimation, (b) False alarm density estimation Taking into account the conclusions drawn above about the properties of the proposed estimator, it follows that it has serious shortfalls, primarily reflected in a large bias and large variance. Two improvements are proposed below. 3.2 Estimator bias reduction The first considered step was to introduce bias compensation for both estimators, keeping in mind a nearly deterministic dependency on three parameters: gate size , estimated target detection probability and estimated false alarm density . The proposed improvement is of the form (10) where and represent the new corrected estimates and and are estimates given by (9). Proper selection of parameters and which appear in (10) can improve the estimators' qual