Distributed GM-CPHD Filter Based on Generalized Inverse Covariance Intersection

In this paper, we propose a distributed Gaussian mixture cardinalized probability hypothesis density (GM-CPHD) filter based on generalized inverse covariance intersection that fuses multiple node information effectively for multi-target tracking applications. Covariance intersection (CI) is a well-known fusion method that produces a conservative estimate of the joint covariance regardless of the actual correlation between the different nodes. Inverse covariance intersection (ICI) is the updated version to obtain fusion results that guarantee consistency and less conservative than CI. However, the ICI is not extended to multi-sensor multi-target tracking system yet. Since the ICI formula can be re-structured as naïve fusion with covariance inflation in Gaussian pdf, this method was applied to the GM-CPHD with generalization. The formula for random finite set (RFS) fusion was derived in the same way as the conventional generalized covariance intersection (GCI) based fusion. The simulation results for multi-target tracking show that the proposed algorithm has smaller optimal sub-pattern assignment (OSPA) errors than naïve fusion and the GCI-based fusions.


I. INTRODUCTION
Multi-target tracking (MTT) is the research that tracks multiple targets with their states, such as position and velocity. MTT problem can be solved by finite set statistics (FISST) without complicated data association, which is based on a random finite set (RFS) [1]. The probability hypothesis density (PHD) filter is the first RFS filter that propagates the filter's first-order moment [2]. To improve the cardinality estimation performance of the PHD filter, a cardinalized PHD (CPHD) filter is introduced, which jointly estimates intensity and cardinality distribution [3]. Unlike the suboptimal filters like PHD and CPHD, the first optimal multi-target Bayesian filter is proposed using labeled multi-Bernoulli RFS, which is called δ-generalized labeled multi-Bernoulli filter (δ-GLMB) [4]. In addition, labeled multi-Bernoulli (LMB) and Gibbs sampling based LMB sensor. Especially, average consensus drew attention in the distributed fusion research area, which is flexible and scalable [13], [17]- [20]. The linear arithmetic average (AA) and the log-linear geometric average (GA) are widely used among average consensus. According to the previous works, AA performs better than GA when the detection probability is low, and targets are close [17], [21]. On the opposite, GA performs better than AA in high clutter density and high false alarm rate.
CI is a well-known fusion method that produces a conservative estimate of the joint covariance regardless of the actual correlation between the different nodes. Ellipsoidal intersection (EI) [31] was proposed to achieve a less conservative result than CI, but it does not guarantee consistency [32]. To obtain a fusion result that guarantees consistency and less conservative than CI, inverse covariance intersection (ICI) was proposed [32]. It was proved that the covariance of the ICI is smaller than CI [32], [33]. As CI has been extended to GCI, both the EI and ICI are also extended to a generalized ellipsoidal intersection [34] and nonlinear inverse covariance intersection (NICI) [35] to fuse arbitrary pdfs, respectively. To unify the terms, we will call the NICI as generalized inverse covariance intersection (GICI). Besides, to apply the generalized fusion to MS-MT, fusion formulation on RFS must be defined which was done in CI [27]. However, EI and ICI are not extended to the MS-MT system yet.
In this paper, we propose a distributed GM-CPHD filter based on GICI. To extend GICI to the MS-MT system, we suggest a new approach to GICI, which is different from the conventional GICI [35]. The suggested method uses the covariance inflation technique to naïve fusion, which will be discussed in the next section. Notice that it is based on covariance inflation of the CI fusion without feedback which is different from the information sharing of the federated Kalman filter [36], [37]. By this approach, we can formulate GICI based GM-CPHD filter.
This paper is organized as follows. In section II, the MS-CPHD filter and GCI fusion are explained briefly. The proposed GICI based GM-CPHD filter is described in section III. The simulation results are presented in section IV, and the conclusion is written in section V.

II. BACKGROUND
When tracking multi-target, which estimates both the state vectors and the cardinality, RFS-based trackers are preferred to the multiple independent single-target trackers since they jointly solve both problems [22]. In addition, a mathematical tool called FISST is widely used to formulate multi-target tracking using Bayesian filtering problems in the RFS framework. This section reviews the background of FISST [1]. Also, based on FISST, CPHD filter [3], and GCI fusion for GM-CPHD [27] are reviewed briefly.

A. FINITE SET STATISTICS
A multi-object density function f (X) is a real-valued function of an RFS X = {x 1 , · · · , x n }, and a multi-object density function characterizes the RFS. The set integral of the multiobject density function is defined as [1], Besides, probability generating functional (PGFL) G[h] and probability hypothesis density D(x) are defined as where h is the test function. PGFL uniquely determines the multi-object density, and it can be interpreted as one of the transforms to solve problems in MTT easily. Otherwise, PHD is the first-order moment of multi-object density. By integrating the PHD function in some regions, we can obtain the expected number of targets in the region.

B. CPHD FILTER
CPHD filter is defined by the multi-object distribution with an independent, identically distributed (i.i.d.) process. Let p(n) be the cardinality distribution of the point process |X| = n, then [27] f (X) n! · p(n) · f (x 1 ) · · · f (x n ).
The PGFL of an i.i.d. process is The intensity function of an i.i.d. process is found with VOLUME 9, 2021 which is the product of the single-object density f (x), and the expected number of objects ∞ n=1 n · p(n).

C. GCI FUSION FOR CPHD FILTER
Assume two Gaussian distributions x ∼ N (x a ,P a ) and where subscript a and b are the node numbers. Then, fusion by CI can be applied as where ω ∈ (0, 1) is a weight. GCI is the generalized rule of CI formula (8) to fuse multiobject densities with arbitrary density, which is also known for Chernoff fusion, In reverse, CI is a special case of GCI when distributions to be fused are Gaussian [38]. GCI is proved to minimize the Kullbeck-Leibler divergence(KLD) [22], Now, consider the local multi-object densities of the CPHD filter that should be fused are where s(x) means local density. By applying GCI, the results of the fused multi-object density and cardinality are [27],

D. GAUSSIAN MIXTURE IMPLEMENTATION
For Gaussian mixture form of the local density If the Gaussian components are well-separated as Following approximation can be applied where κ(ω, P) The approximation is from pseudo-Chernoff fusion [30] and is widely used in GM-GCI approaches.
By using (18), fused multi-object density (13) can be obtained in closed form,

III. DISTRIBUTED FUSION BASED ON GENERALIZED NAÏVE FUSION WITH COVARIANCE INFLATION
To apply ICI to multi-object density fusion, naïve fusion with covariance inflation is proposed in this section.

A. THREE COMMON DISTRIBUTED FUSION RULES
Let us consider the fusion of two nodes Since EI is proven to be inconsistent [26], it is not dealt with in this paper.
Firstly, naïve fusion [39], which is optimal when there is no correlation, Meanwhile, CI is the fusion rule that keeps consistency under unknown correlation of the nodes where ω CI ∈ (0, 1) is a weight and often obtained by min ω (det (P CI )) or min ω (trace (P CI )). Since the optimization procedure for calculating ω CI is computationally intensive, ω CI can be obtained by various fast covariance intersection methods which provide a closed-form solution without optimization [40]- [43].
The thing to note (25)(26)(27)(28) is, by changing covariance P A and P B to P A,CI and P B,CI respectively, the CI formula is transformed to naïve fusion.
Recently, ICI is proposed to obtain fusion results, which guarantee consistency and less conservative than CI [32]. It is proved that the covariance of the ICI is smaller than CI [32], [33]. The ICI formula can be written as [32],

B. INVERSE COVARIANCE INTERSECTION FROM THE PERSPECTIVE OF COVARIANCE INFLATION
To apply ICI to the MS-MT, we changed the structure of the original ICI formula (29)-(32) as, where ω ICI ∈ (0, 1) is a weight and obtained by min ω (det (P ICI )) or min ω (trace (P ICI )). In the extreme case ω ICI = 0, it becomes x ICI = x A , P ICI = P A . In the opposite case ω ICI = 1, it becomes x ICI = x B , P ICI = P B . Since they are not fusion and only result from one node, we defined ω ICI in the open interval (0, 1).
As shown in (35)(36), the ICI formula can also be transformed to naïve fusion by changing the nodes' covariance.
Since ω ∈ (0, 1) it is clear that P A, CI > P A , P B,CI > P B , (37) P A, ICI > P A , P B,ICI > P B , where P x > P y means P x − P y is positive definite. (37)(38) means that CI and ICI can be represented by naïve fusion with inflated covariance.
Since (35)(36) have a minus operator that seems mathematically unstable, we will prove the positive definiteness (35)(36). The equation (26) of the [44] is By substituting A = X, U = Y, B = V = I, the inverse of a sum of matrices can be represented as Substitute X = P A , Y = ω 1−ω P A P −1 B P A and remove subscript ICI from ω ICI for simplicity, Hence, For the positive definite matrix P B , its inverse P −1 B is also positive definite since eigenvalue P −1 B is inverse to that of P B which is also positive. Hence, for x T P A x > 0 and y T P −1 B y > 0, set y = P A x then It means P A,ICI it is also positive definite and P B,ICI can be proved positive definite in the same way. To sum up, CI and ICI can be represented in naïve fusion with inflated covariance, which is also positive definite.

C. GENERALIZED INVERSE COVARIANCE INTERSECTION FOR MULTI TARGET TRACKING
As GCI is defined on the fusion of arbitrary pdfs, naïve fusion is defined as [38], Since GICI proposed in [35] includes convolution operation, it cannot be easily applied to the fusion of the finite sets. VOLUME 9, 2021 Therefore, we will propose the GICI rule for the GM-CPHD by inflating the naïve fusion formula's covariance. The GM-naïve-CPHD formulas listed in (45-51) are the same from sections II-C, D except for erasing weight component of the CI.
Then, a closed-form solution is (51)

D. COVARIANCE INFLATION FOR GICI
In the GM form of multi-object distribution (48-51), GICI can be proposed by substituting single object density covariance to inflated covariance using (35)(36). The pseudocode for distributed GICI fusion is listed in Table 1. When extending naïve fusion to GICI, there is a question that inflating single object density is appropriate since the pdf in GM-CPHD is assumed to be a Gaussian mixture. The covariance of the Gaussian mixture can be represented as standard mixture merging [19], Since P (a) j it is the error covariance of the state and T is the covariance generated by the state of the targets, the latter one is much bigger than the former one in the general case. It is the same context in the assumption on pseudo-Chernoff fusion (16,17), where the targets are well-separated. However, the coefficient becomes zero when fusing targets far apart, as shown in (51). We have to focus on fusing targets with a similar state, so inflating only a single object density is reasonable and proved in the simulation result.

IV. SIMULATIONS
In the simulation, two-dimensional multi-target tracking examples with the most widely used linear Gaussian measurement model and a nonlinear measurement model are considered. The state is 2D-position and velocity x k = [x kẋk y kẏk ] T . In fig. 1, eight targets move from circle to triangle through solid black lines, which is the trajectory same as in [11]. The red circle is the position of the sensors, which are only relevant to the nonlinear measurement model. Fig.2 is another scenario that targets 1 and 2 encounters in the 40s, and targets 1 and 3 meet at the 80s. All the fusion weight ω is obtained by fast covariance intersection [42], which just calculates traces of the matrices that do not increase the overall computational burden. Total simulation time is 100 s with time interval 1s, and a GM-CPHD filter is used to track each sensor node's targets.
Since it is evident that the performance is proportional to the number of sensors, we mainly analyze two sensor fusions. The system model is a constant velocity model, The probability of detection is 0.9, and the probability of survival is 0.99. The Poisson average rate of uniform clutter per scan is 5, and the birth densities are located at (±400m, ±400m), as shown in the solid black circles in Fig. 1. All simulations are done with 100 Monte Carlo simulations, and ground truth is changed by process noise for each ensemble. The pruning threshold is 10 −5 , and the merge threshold is 2.
In the following analysis, five implementations of GM-CPHD filter are compared: single sensor [45], naïve fusion, conventional GCI [22], proposed GICI, and centralized fusion [10]. Tracking performance is measured by optimal sub-pattern assignment (OSPA) distance [46]. The parameters for OSPA distance are set as cut-off parameter c = 100 and order parameter p = 1.

A. LINEAR GAUSSIAN MEASUREMENT MODEL
Firstly, a linear Gaussian measurement model is selected, where σ v = 10m, and the model is irrelevant to the sensor position. Fig. 3 is a cardinality estimate comparison for five methods. The solid black line is the true cardinality, + is the mean cardinality, and dot lines are 3σ standard deviations (std) of cardinality. Since it is hard to distinguish the mean cardinality performance with the cardinality estimate figure, only GICI is plotted representatively, and it will be compared in other OSPA error figures. Standard deviations are also hard to distinguish between 3 distributed fusions, but 1 sensor shows the biggest value, and centralized fusion shows the most negligible value as expected.  Figs. 6 and 7 are simulation results for track 2. In track 2, two targets are meet exactly in the 40s and 80s. However, since the crossing targets' velocities are not the same at that time, significant performance degradation is not shown. Besides, there is a relatively bigger degradation on naïve and VOLUME 9, 2021   GCI than GICI and centralized fusion. In fig. 7, the box plot shows that GICI shows almost the same performance as the centralized fusion.
Figs. 8 and 9 are results for track 1, and figs.10 and 11 are for track 2. As shown in the linear measurement case, the GCI and GICI showed better performance than the 1 sensor and naïve fusion. However, GCI and GICI show similar performance.   Also, additional simulations were performed using 5 snesors as shown in fig. 12. The result figs. 13 and 14 can be compared with the figs. 8 and 9, which are the results for the two sensors. As the number of sensors increases, centralized OSPA definitely decreases, whereas distributed fusion decreases the error less. In distributed fusions, the error peaks in figs. 8 and 13 are similar. It means that the cardinality error does not improve when the number of targets changes in the case of distributed fusion. In the section where the cardinality does not change, the localization error of fig. 13 is    lower than that of fig. 8. However, the cardinality error is similar to the case of two sensors, so the performance is not significantly improved when viewed as a box plot average. On the other hand, as shown in fig. 14, it can be seen that even in 5 sensors, GICI has slightly superior performance over GCI. Table 2 is the computational time for Monte Carlo 100times with Intel Core i7-9700K CPU 3.60GHz processor and 16GB RAM. It is assumed that the local nodes calculations are done in parallel in distributed fusion. The most time-consuming calculations are matrix inversion and determinant [23], and the proposed method have one more step to inflate the covariance than the naïve fusion.

V. CONCLUSION
In this research, we proposed the distributed GM-CPHD filter based on GICI and verified performance with multi-target tracking simulations. To apply the GICI fusion to MS-MT, the GICI formula was re-structured as naïve fusion with covariance inflation. For the positive definite covariance of the node, we proved the inflated covariance is also positive definite. The formula for RFS fusion was obtained in the same way as the previous GCI-based algorithms.
As a result, the proposed algorithm showed a smaller OSPA error than naïve fusion and conventional GCI-based algorithms. The performance improvement was shown clearly in the nonlinear measurement case and more robust when targets were crossing. Computational time becomes larger than the naïve fusion since the added step with matrix inversion is time-consuming. In future work, the GICI fusion will be applied to other RFS filters, such as MB filter.