Heterogeneous sensor fusion based on Copula theory and importance sampling

IET Radar, Sonar & NavigationVolume 13, Issue 11 p. 1943-1950 Research ArticleFree Access Heterogeneous sensor fusion based on Copula theory and importance sampling Kelin Lu, Corresponding Author Kelin Lu klu@seu.edu.cn School of Automation, Southeast University, Nanjing, People's Republic of China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing, People's Republic of ChinaSearch for more papers by this authorChangyin Sun, Changyin Sun School of Automation, Southeast University, Nanjing, People's Republic of China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing, People's Republic of ChinaSearch for more papers by this authorQien Fu, Qien Fu School of Automation, Southeast University, Nanjing, People's Republic of China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing, People's Republic of ChinaSearch for more papers by this authorQian Zhu, Qian Zhu China Academy of Launch Vehicle Technology, Beijing, People's Republic of ChinaSearch for more papers by this author Kelin Lu, Corresponding Author Kelin Lu klu@seu.edu.cn School of Automation, Southeast University, Nanjing, People's Republic of China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing, People's Republic of ChinaSearch for more papers by this authorChangyin Sun, Changyin Sun School of Automation, Southeast University, Nanjing, People's Republic of China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing, People's Republic of ChinaSearch for more papers by this authorQien Fu, Qien Fu School of Automation, Southeast University, Nanjing, People's Republic of China Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Nanjing, People's Republic of ChinaSearch for more papers by this authorQian Zhu, Qian Zhu China Academy of Launch Vehicle Technology, Beijing, People's Republic of ChinaSearch for more papers by this author First published: 01 November 2019 https://doi.org/10.1049/iet-rsn.2019.0132AboutSectionsPDF 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 This study deals with the problem of heterogeneous sensor fusion based on Copula theory and importance sampling. The proposed fusion algorithm is grounded on Copula statistical modelling and Bayesian probabilistic theory. The distinctive advantage of this Copula-based methodology is that it formulates the internal dependency between the local sensors' data, which is usually unknown but essential for accurate track fusion. To this end, a joint distribution of the local sensors' observations is constructed based on Copula functions, and the corresponding fusion rule is derived with a specific correlation term. In addition, a Monte–Carlo importance sampling technique is adopted to improve the computational efficiency by drawing less random samples from the local estimates to be fused. After that, a procedure of Kernel density estimation is applied to learn a Gaussian approximation of the fused density. In the end, extensive Monte–Carlo simulations are conducted to evaluate the proposed sensor fusion method in a distributed target-tracking scenario. 1 Introduction Distributed state estimation for dynamic systems is an important technique in the field of digital signal processing, which has been used in a lot of real-world applications, including cooperative target tracking [1], distributed formation of space-telescopes [2], and remote environmental monitoring [3]. Nowadays, the rapid growth of wireless sensors hardware with advanced sensing, computing, and communication abilities has facilitated the development of data fusion technologies for distributed estimation system. Sensor fusion methods [4] provide extended sensing coverage than one single sensor by broadening geographical reach and improving filtering accuracy. For instance, in a distributed tracking scenario, observations from different sensors can be directly fused together in an augmented manner [5], then the fusion node is able to estimate the state of interest with a standard Kalman filter based on a centralised measurement model. The centralised measurement fusion method mentioned above achieves an optimal estimation performance in the sense of minimum mean-squared error [6]. However, for the sake of practical limitations such as energy constraint on sensor battery or bandwidth constraint on wireless communication, the data transmission of raw measurements from sensors to the fusion node is not energy-efficient as a persistent manner [7]. The track-to-track fusion [8] architecture has a lot of advantages compared with the measurement fusion architecture including reduced bandwidth requirement and improved reliability [9]. In a typical track-to-track fusion scenario, each sensor maintains a local filter to pre-process their local measurements and transmit the local estimation results to a fusion node. The fusion node fuses the received data to obtain a global estimate, which is expected to be more accurate than those calculated by the individual sensors [10]. As a result, there is a great deal of interest in designing fusion algorithms that can fuse local estimates instead of raw measurements in a multisensory system due to their flexibility and energy-efficiency [11]. One of the main concerns for the track-to-track fusion problem is the construction of the correlation between the local estimation results to be fused [12, 13]. In detail, although the sensor measurements are usually assumed to be independent to each other, the local estimates are indeed correlated due to the common process noise model [14]. The ignorance of this correlation term between the local estimates might lead to an inconsistent fused result or even filtering divergence [15]. To address the issue of inconsistency, the exact correlation term between the local tracks is reconstructed in terms of a cross-covariance matrix [16] based on the prior filtering system model, and a one-scan track-to-track fusion algorithm is proposed by making use of a static linear estimation model [17]. While this fusion algorithm is proved to be optimal in the maximum likelihood sense [18], it is not practical in reality for the reason that the cross-covariance matrix required during the fusion process is complex and difficult to calculate [19]. Besides, the complete reconstruction of this cross-covariance matrix is not communication-efficient since it is supposed to transmit the local Kalman gains of each sensor at full rate. In addition, a previous communication between the fusion node and the local sensors can further increase the complexity of the correlation between the local estimates to be fused [20], which makes the exact reconstruction of the correlation term become implicit. Consequently, there has been a great deal of suboptimal but robust track-to-track fusion algorithms developed irrespective of a specified correlation model to be reconstructed. The representative example of such kind of fusion techniques is the covariance intersection (CI) method [21], which is able to yield a convex combination of the means and covariances of the local estimates with an unknown degree of cross-correlation based on some optimisation methods. In recent works, the ellipsoidal intersection method [22] and the inverse CI method [23] have been developed to reduce the conservative property of CI by including additional parameters to maximise the common information contained in the local estimates. At the same time, the performance improvement of these intersection-based fusion algorithms [24] is still limited due to their ignorance of the internal correlation structure between the local tracks. A new data fusion method [25] grounded on statistical modelling to reconstruct the correlations is proposed to address the inconsistency problem when the conditional independence assumption is not met; however, the proposed fusion rule is only valid for linear system. Emerging sensor fusion systems require applications among disparate data sources to increase information diversity, in which the observation models used in local sensors might be different. For track-to-track fusion from homogeneous sensors, the cross-covariance term caused by the common process noise has been theoretically established. However, existing sensor fusion methods cannot combine correlated data from disparate sources in an optimal manner as they rely on known correlations. Then, those suboptimal fusion algorithms just compute a conservative result with lower estimation accuracy. Specifically, a generalised modelling method does not exist for constructing inter-modal dependence structures and joint probability density determination among statistically dependent sensor observations. In this regard, the goal of this paper is to address the problem of heterogeneous data fusion for disparate sensor observations with unknown dependency. Specifically, it investigates a mathematical foundation based on Copula theory and Bayesian probabilistic inference to estimate the dependency between random variables for disparate data. Recently, a Copula fusion (CF) rule [25] has been proposed for homogeneous fusion, in which the unknown correlations between the local tracks are reconstructed based on Copula functions. However, the correlation coefficient of the Copula function is predetermined in an offline fashion, which becomes intractable for the heterogeneous case. In addition, the computational efficiency of the CF rule requires further improvement. To this end, the presented method consists of three parts, it first formulates a joint distribution of the heterogeneous sensors' observations with a Copula function representing the inter-modal dependency; second, the correlation coefficient of the Copula function is optimised online following the rule of minimum discrimination information; third, the Gaussian approximation of the fused density is calculated efficiently by using importance sampling and Kernel estimation techniques. The remainder of this paper is organised as follows: the problem is set up in Section 2, and the heterogeneous sensor fusion method is proposed in Section 3. A numerical simulation example is introduced in Section 4 and the conclusion is given in Section 5. 2 Problem formulation 2.1 Local filtering system Consider a point target [26] whose dynamic is given by (1) with (2) (3) (4) (5) where represents the state vector at time step k, T denotes the sampling interval, represents a discrete Gaussian noise sequence. We consider that is uncoupled across its components, i.e. and , then the covariance of the noised term is given by (6) with (7) We further assume that the target described in (1) is observed with heterogeneous sensors, including the bearing sensor and the range sensor. In this regard, the measurement equations can be written as (8) (9) where and represent the positions of the bearing sensor and range sensor, respectively. In addition, and are the corresponding measurement noises. In the end, a processor is assumed to be co-located with the sensor to generate local tracks for target tracking. To this end, the state estimate of the target, namely is generated iteratively at each time step with extended Kalman filter [27], which can be described as (10) (11) (12) (13) (14) where for the bearing sensor and for the range sensor, denotes the Jacobian matrix with respect to the specific type of measurement equation. In this work, the corresponding Jacobian matrices for the bearing sensor and range sensor are given in Appendix 1. 2.2 Sensor fusion system Consider a distributed estimation system with multiple nodes. As shown in Fig. 1, the basic sensor fusion system consists of two local sensors and a fusion node, where each sensor estimates the state of interest, i.e. , is based on its own observations . The objective of the fusion node is to fuse a global result given all the local estimates, then send this fused estimate back to each local sensor for performance improvement. In this work, it is assumed that the local sensors work in a synchronous manner, which means that both sensor 1 and sensor 2 operate with an identical sampling interval T. In addition, the out-of-sequence problem [28] is omitted for the sake of brevity, that is, the communication links between the sensors and the fusion node is without delay or data loss. Fig. 1Open in figure viewerPowerPoint Sensor fusion system with two sensors and one fusion node Denote as the historical measurements until time step k for the ith sensor. Fusing the local tracks is straightforward when the local measurements are mutually exclusive and conditionally independent, i.e. (15) Then, the local tracks can be fused based on the Bayesian fusion paradigm [18] as follows (16) where represents the fused density, is the prior estimate to be removed out. At the same time, the fusion rule (16) becomes invalid when the conditional independence assumption (15) does not hold. For example, if the sensors communicate at less than full rate, the measurements will be correlated due to the common process noise [15]. In that case, a complete formulation of the joint distribution of the measurements is necessary, and the fusion rule becomes (17) It can be seen from (17) that an exact derivation of the joint density is required during the fusion process, which is generally intractable when the measurement models of local sensors are disparate. In this work, we propose a heterogeneous fusion method grounded in the marriage of Copula theory and kernel methods, in which the Copula function models correlations between the heterogeneous sensors' data, while the kernel methods enable the density estimation where the fused density is calculated based on a set of random samples. 3 Heterogeneous sensor fusion In this section, we first introduce the data fusion rule for heterogeneous sensor fusion, where the correlation between the sensors' data is constructed based on Copula theory. Then, we describe an optimisation method to guarantee robustness of the fusion process. After that, we apply a kernel density estimator to obtain a Gaussian approximation of the fused estimate. In the end, we summarise the proposed heterogeneous sensor fusion method. 3.1 Copula fusion rule The Bayesian fusion requires to calculate the joint distribution as mentioned in (17), which is generally intractable when the sensors are heterogeneous. In this part, we propose to estimate the unknown correlation between the sensors' data with the help of Copula theory, which serves as an effective tool for connecting marginal distributions to a multivariate one with a choice of inter-modal dependency. Formally, a Copula function can be defined as (18) where denotes a Copula function, and are marginally uniform random variables in . Define as a random vector, let denote the cumulative distribution function and as the marginal distribution function for . Then, we have the Sklar's theorem [29] as follows: Sklar's Theorem: Let be any multidimensional distribution function over a random vector X, then there exists a Copula function such that (19) Conversely, for any univariate distribution function, the Copula function defines a multivariate distribution with marginals given as for . From the perspective of statistical modelling, the Sklar's theorem provides a framework to construct a joint distribution with a set of marginal; it also enables the ability to design their inter-modal dependency with the Copula functions. In addition, consider that the joint distribution function has a density , then this density function can be derived as (20) where is denoted as a Copula density function. Equation (20) states that a joint multivariate probability density function (PDF) can be formulated as a product of its marginals and a Copula density function representing their inter-modal dependency. In this regard, the joint density in (17) can be constructed as (21) with (22) (23) By substituting (21) into (17) we have (24) The derived CF rule, i.e. (24), is similar to the original Bayesian fusion rule given in (16), in which the only difference between them is the inter-modal dependency term represented by the Copula density function . Thus, the primary advantage of the proposed CF rule is that it enables to approximate the actual correlation between the sensors' data and combine the local tracks according to this estimated correlation property. There exists a range of Copula functions that can be applied to characterise various correlation properties. In this work, we propose to focus on the bivariate Gaussian Copula given as (25) with (26) where represents the correlation coefficient, is the cumulative distribution function of the standard Gaussian distribution, i.e. , and is its functional inverse. In addition, it can be observed from (22) and (23) that the exact calculation of for requires each sensor to transmit its full pedigree of measurements, which is inefficient for practical communication. To reduce the communication load, we propose to reconstruct a single pseudo-measurement with its local tracks. As a result, the term can be approximated as follows (27) with (28) for , in which n is the number of observations between two consecutive fusion steps. Instead of using the raw measurements from each sensor, now we are able to calculate the Copula density function, i.e. with only the local estimates. The derivation of (28) is given in Appendix 2. 3.2 Correlation coefficient optimisation The aim of this section is to optimise the correlation coefficient in (26) to guarantee that the two local estimates reach a consistent agreement during fusion. In order to formally state the problem of correlation coefficient optimisation, the first issue to be addressed is how to define the optimisation metric given the local tracks for . To this end, it is noteworthy to recall that given a set of points , their weighted average satisfies the property of minimising the mean squared distance from these points, namely (29) where denotes the Euclidean norm. In statistics, a typical choice to measure the distance between two PDFs and corresponds to the Kullback–Leibler divergence, which is defined as (30) From the perspective of information theory, the Kullback–Leibler divergence is interpreted as the information gain achieved when updating a prior PDF to a posterior one. Let now denote the averaged PDF that minimises the sum of the information gains from and , namely (31) Then the averaged PDF is a geometric mean of the local PDFs which satisfies the following property (32) According to (31) and (32), now we define a CF result to be consistent, when it satisfies that (33) with the optimisation metric defined as (34) In the end, it is noteworthy to point out that this choice is coherent with the rule of minimum discrimination information [30], according to which the distribution that best represents the true state is the one that produces the information gain as small as possible. In this regard, the optimisation metric is calculated with varying from to , and the optimal value is determined by a grid search over the set specified by a constant increment . As shown in Fig. 2, the optimisation process can be implemented in a parallel manner to reduce the computational time. Fig. 2Open in figure viewerPowerPoint Parallel implementation of the correlation coefficient optimisation algorithm. The unknown correlation coefficient is optimised by minimising the objective function given in (34) 3.3 Kernel density estimation In this part, we apply a procedure of kernel density estimation to obtain a Gaussian approximation of the fused density derived in (24). The fundamental idea behind this approximation procedure is to represent the fused density with a set of random samples, then calculate the corresponding statistical parameters based on these samples. Furthermore, a Monte–Carlo importance sampling technique is adopted to improve the computational efficiency by drawing smaller number of random samples from the local estimates to be fused. The importance sampling-based approach exploits the following identity (35) with (36) where denotes an importance density [31], which is supposed to be with a similar peak structure to the fused density and is both easy to sample from and evaluate for fixed parameters. A particularly effective choice for the importance density is to use the result of the generalised CI (GCI) [32, 33] in track-to-track fusion as (37) where denotes the fusing coefficient and is optimised to ensure that (38) Suppose that we first draw samples from the importance density , which can be denoted as , then the weighted set is a sample-based representation of in (24). As a result, the CF result can be treated as a Driac-mixture approximation as (39) with (40) Various density estimation techniques can be used to learn the statistical distribution of the CF result represented by . In this work, the weighted samples is resampled first by conducting systematic resampling [34] to generate a uniformly weighted set of samples , as done in the standard particle filter [35]. After that, the statistical distribution of the CF result is obtained with the kernel density estimator given as (41) where denotes a random vector following the distribution of , is a kernel function whose variance is controlled by the bandwidth h. Furthermore, we adopt the normal density as the kernel function, so that (42) where denotes the normal density function in z with mean 0 and standard deviation h. As a result, the optimal smoothing parameters are given by (43) with (44) where denotes the median of the sample in dimension j. In the end, the process of the kernel density estimation is summarised in Fig. 3. Fig. 3Open in figure viewerPowerPoint Process of the kernel density estimation 3.4 Summary of the heterogeneous sensor fusion method In this section, we summarise the proposed heterogeneous sensor fusion method as follows. At each fusion step, the pseudo-measurement is calculated from each sensor to represent the information contained in the pedigree of the local measurements. Furthermore, for the reason that the fused density cannot be calculated in a parameterised form, the fused result of the generalised CI of the local estimates is obtained as an importance density, and the corresponding importance sampling is performed to improve the computational efficiency. In the end, the uniformly weighted samples are obtained by conducting systematic resampling from the original weighted samples, and the Gaussian approximation of the fused density is constructed with the help of kernel density estimation. As shown in Fig. 4, in the proposed framework, we construct a sampling set based on the result of the generalised CI as mentioned in (37). After that, the value of the Copula fused density for each element in the sampling set is calculated according to (24)–(26), where the bivariate Gaussian Copula is calculated based on the distribution function of the pseudo-measurement given in (27) and (28). Fig. 4Open in figure viewerPowerPoint Flow chart of the proposed heterogeneous sensor fusion method 4 Numerical simulation and results In this section, the problem of heterogeneous sensor fusion is considered in a distributed target-tracking scenario to validate the performance of the proposed CF method. The dynamics of the point target follows (1) with the transition matrix and the process noise matrix given as (45) (46) where the sampling interval is set to be 0.05 s, and the process noise model is given as (47) The system state consists of the position and velocity of the target state in X, Y–axis, respectively. For the two-sensor case, the measurement equations are set to be for the first sensor, and for the second sensor. Besides, the measurement noise model is given as (48) In the simulation, we implement independent Monte–Carlo runs, and in each run, a length of 100 time steps of the iterative filtering process are simulated. The root-mean-squared error (RMSE) of the position estimate is set as the performance validation metric, which is defined as (49) where and are the true target position in X, Y–axis, respectively, and are the fused estimate in X, Y–axis, respectively. In the simulation, we start from a basic distributed target-tracking scenario with a fusion node and two heterogeneous sensors configured in the fusion system. The fusion node processes the local tracks to be fused with a period of nT, in which each sensor updates its local estimate with a sampling interval T. In addition, when the fusion process is completed, each sensor receives a feedback of the fused estimate from the fusion node, then replaces its local track with the received fused one. Under this kind of fusion configuration, when the sensors communicate with the fusion node at less than full rate, the measurements are correlated given the target state at the previous observation time step due to the existence of the common process noise [15]. To verify the performance of the proposed CF method, we compare the results with the CI (CI) method [32], which is able to combine the local tracks under unknown correlations of local estimation errors. Figs. 5-7 compare the positional RMSE for CF and CI at full rate, 1/2 rate and 1/4 rate, respectively. It is to be noted that when the communication between the local sensors and the fusion node is at full rate, the local measurements are conditionally independent given the target state. As a result, it can be seen from Fig. 5 that the estimation accuracy between CF and CI is very close to each other in the full rate case. However, if the communication frequency is decreased to reduce the communication load, then as shown in Figs. 6 and 7, the performance of CI degrades as it accounts for the unknown correlations between the local tracks in a conservative manner. At the same time, with regard to the proposed CF, the results demonstrate that it enables to guarantee consistent estimation accuracy irrespective of the communication frequency. The reason is that the Copula function provides a reliable mechanism to estimate the unknown correlation between the local sensors' data. Fig. 5Open in figure viewerPowerPoint Positional RMSE for CI and CF at full rate Fig. 6Open in figure viewerPowerPoint Positional RMSE for CI and CF at 1/2 rate Fig. 7Open in figure viewerPowerPoint Positional RMSE for CI and CF at 1/4 rate Furthermore, we extend the fusion system to make it incorporate more sensors. Here we introduce a hierarchical architecture which can be used to fuse multiple local estimates with the proposed CF in a sequential manner. As shown in Fig. 8, suppose that there are three sensors with their local tracks indexed as for , then we first fuse the estimates from the first two sensors, namely and to obtain an initial fused track . After that, we fuse with the local track from the third sensor. Consequently, this sensor fusion system is scalable as it updates the fused estimate recursively in a hierarchical manner and it is able to handle arbitrary number of heterogeneous sensors. Fig. 8Open in figure viewerPowerPoint Sequential fusion structure with three sensors In the following, we conduct simulations to analyse the performance of the sequential fusion system with varying number of sensors. In detail, we denote as the RMSE deviation, which can be defined as (50) where and represent the averaged position RMSE obtained from CF and CI, respectively. As shown in Fig. 9, the proposed CF works consistently better than CI since keeps above zero with the number of sensors increasing. In addition, it can be seen from Fig. 9 that the discrepancy of the fusion accuracy between CF and CI grows larger as the number of local tracks to be fused increases. The reason is that with more tracks incorporated during the fusion process, the complexity of the correlations between the estimates to be fused keeps increasing. Although for CI it requires no exact correlations between the sensors' data, it preserves the robustness of the estimation result at the cost of more conservative fusion manner. At the same time, for the proposed CF it adopts a Copula function to formulate the inter-modal dependency of local sensor's data, which facilitates to optimise the combination of the local estimates by taking into account their correlations. Fig. 9Open in figure viewerPowerPoint Deviation of averaged position RMSE between CF and CI with varying number of sensors 5 Conclusion In this paper, a heterogeneous sensor fusion methodology is introduced based on Copula theory and importance sampling. We first present a mathematical formulation with respect to the local observations' dependency, where the unknown correlation term is reconstructed based on Copula statistical modelling. After that, the recursive sensor fusion method is implemented by approximating the pedigree information of local measurements between two consecutive communication steps with a single pseudo-measurement, then the Gaussian approximation of the fused result is obtained with the help of importance sampling and kernel density estimation techniques. In addition, the correlation coefficient of the Copula function is optimised according to the rule of minimum discrimination information by comparing the Kullback–Leibler divergence between the fused estimate and the local ones. Extensive Monte–Carlo simulation results demonstrate that the proposed heterogeneous sensor fusion method is scalable, and it is able to provide a robust estimation result irrespective of the correlation between the local tracks over a wide range of communication rates. A challenge for future work in this field is to apply the CF method to more complex local filters, such as IMM filters and Gaussian sum filters. It is also challengeable to consider the generalisation of the CF method to the case of higher dimensional measurements. 6 Acknowledgments This work was supported by the Jiangsu Province Science Foundation for Youths (No. BK20180358), and the National Natural Science Foundation of China (U1713209, 61520106009, 61533008, and 61773031), and the Jiangsu Province '333' Project (No. BRA2018034). 8 Appendix 8.1 Appendix 1: Jacobian matrices of the measurement equations for the bearing sensor and the range sensor Assume that the Jacobian matrix is defined by (51) where for the bearing sensor and for the range sensor, then the Jacobian matrix for the bearing sensor is derived as (52) (53) (54) (55) and the Jacobian matrix for the range sensor is derived as (56) (57) (58) (59) 8.2 Appendix 2: Derivation of the pseudo-measurement for the Copula fusion rule Suppose that the system model is given as (1), then the local estimate from sensor i at time step k can be computed with extended Kalman filter as (60) with (61) Substitute (61) into (60), we have (62) which can be rewritten as (63) In this regard, we can construct the pseudo-measurement as (64) where represents the pseudo-measurement from time step k - n to k, and (65) (66) 7 References 1Gunay, M., Orguner, U., Demirekler, M.: 'Chernoff fusion of Gaussian mixtures based on Sigma-point approximation', IEEE Trans. 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