A Maximum Likelihood Approach to Joint Registration, association and Fusion for Multi-Sensor Multi-Target Tracking

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1 12th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009 A Maximum Likelihood Approach to Joint Registration, association and Fusion for Multi-Sensor Multi-Target Tracking Siyue Chen and Henry Leung Electrical and Computer Engineering University of Calgary Calgary, Alberta, Canada chens, leungh@ucalgary.ca Éloi Bossé Defense R&D Canada-Valcartier Valcartier, Québec, Canada Eloi.Bosse@drdc-rddc.gc.ca Abstract In this paper, we propose a maximum likelihood ML) approach to address the joint registration, association and fusion problem in multi-sensor and multi-target surveillance. In particular, an expectation maximization EM) algorithm is employed here. At each iteration of the EM, the extended Kalman filter EKF) is incorporated into the E-step to obtain the fusion results, while the registration parameters are updated in the M-step. Association of sensor measurements to the targets are also computed as the missing data in the E-step. The main advantage of the proposed method compared to the conventional approaches is that the mutual effects of registration, association and fusion are taken into the consideration when formulating the multi-sensor, multi-target tracking problem. The simulation results demonstrate that the performance of the proposed method in terms of mean square error MSE) is close to the posterior Cramer-Rao bound PCRB), and is better than one of the conventional approaches that perform registration, association and fusion separately. Keywords: sensor registration, data association, data fusion, target tracking, expectation maximization, extended Kalman filter. 1 Introduction Traditionally, data registration 1, 2, association 3 and fusion 4, 5 are treated as three independent processes in multi-sensor, multi-target surveillance. Sensor registration is usually performed first to transform the local measurements into a common coordinate. Association and fusion are then applied to build the correspondence between sensor measurements and targets and to estimate the target states. Most of fusion algorithms consider either registration error or association error, but not both. In practice, these two processes should not be treated separately. Data association error can deteriorate the sensor registration as poor association generates wrong multi-sensor data pairs for registration, while registration error can produce biases on the data which result in wrong association. It is thus reasonable to develop an approach that performs registration, association and fusion simultaneously. Since registration error or association error usually results in a poor fusion result, fusion performance can be used as a measure to evaluate whether registration and association are carried out properly. Based on this idea, we propose to treat registration, association and fusion as a joint process. More precisely, a joint likelihood function is formulated to model the multi-sensor, multi-target tracking problem by taking the effects of registration and association into account. The maximum likelihood ML) method is then employed to solve this joint optimization problem. However, it is usually difficult to obtain analytical expressions of the unknown parameters and target states by setting the derivative of the likelihood function to zero. Furthermore, the computation complexity of ML increases exponentially with respect to the number of snapshots and the targets. Developed by Dempster et al. 6, the expectation maximization EM) algorithm is a general method of finding the ML estimate of the parameters of an underlying distribution from the given observed data set when the data is incomplete or has missing values. It is particularly useful in situations where optimizing the likelihood function is analytically intractable, but the likelihood function can be simplified by assuming the existence of missing data 7, 8. In our proposed method, the true target states and the measurementto-target association is treated as the missing data. The likelihood function is then reduced to a Gaussian model with unknown parameters of registration and initial target states. In the E-step, an approximated expectation of the log-likelihood function with the complete data is computed based on the current parameter estimates. Meanwhile, the target states are estimated using an extended Kalman filter EKF). In the M-step, a new ISIF 686

2 parameter estimate is computed by finding the value that maximizes the likelihood function found in the E- step. The EM algorithm has the advantages of achieving global convergence in most instances 9, 10. This paper is organized as follows. The joint registration, association, and fusion problem is formulated in Section 2. In Section 3, the proposed EM-EKF method is described. To evaluate the performance of the proposed method, the posterior Cramer-Rao bound PCRB) is derived in Section 4. In Section 5, the simulation results are reported. At last, the concluding remarks are given in Section 6. 2 Problem Formulation In multi-sensor, multi-target tracking, the dynamic equation related to the state vector of the tth target can be described as x t k)=fx t k 1) + wk), t=1, 2,...,N t, 1) where N t denotes the number of targets, x t k) = x w ẋ w y w ẏ w T contains the spatial position and velocity of the target t in a common coordinate, F is the transition matrix given by F = 1 T T , 2) and wk) is a zero mean, white Gaussian noise with a known covariance matrix Q = Ewk)w T k). The measurement equation of target t for sensor s can be written as y s,t k)=h s x t k))+ η s +vk),, 2,...,N s, 3) where N s denotes the number of sensors, v k is also a white, zero-mean Gaussian process with a known covariance Υ = Evk)vk) T, h s ) is a known nonlinear measurement function, and η s is the systematic bias of sensor s. In order to track the targets properly, the measurements have to be assigned to their associated targets. The situation is further complicated by the spurious measurements and the missed detections. In order to deal with these problems, a fictitious measurement 11, 12, y s,0 k), is added to ys T k)=y s,1 k) y s,2 k)... y s,is k) so that ys T k)= y s,0 k) y s,1 k)... y s,is k) with y s,0 k) representing the missed detection. Here, I s represents the number of measurements at sensor s. We then introduce a measurement-to-target association process Φ s,k to represent the true but unknown origins of the measurements at sensor s. We define the association Φ s,k specifically as an N t +1) I s +1)matrixwiththe t, i) elementφ s,k t, i) fort =0,...,N t, i =0,...,I s such that { 1, if ys,i k) originats from target t, Φ s,k t, i)= 0, otheriwse. 4) It is noted that each column of Φ s,k corresponds to the measurement y s,i k) and describes the targets that contribute to the ith measurement; each row of Φ s,k corresponds to the tth target and describes the measurements that might originate from this target. To avoid ambiguity for matrix Φ s,k, we assume that Φ s,k 0, 0) = 0. If Φ s,k 0,i)=1forsome1 i I s, then the ith measurement represents a false alarm. Likewise, if Φ s,k t, 0) = 1 for some 1 t N t, then the tth target is active but has a missed detection. Since it is assumed that each target is assigned to only one measurement or declared as missed detection at sensor s,aconstraint is applied to Φ s,k.thatis I s i=0 Φ s,k t, i)=1, t =1,...,N t. 5) Using the above modeling assumption, the unknown parameters include the sensor bias η s,andapriori information on the initial target states. The parameter vector is given by ρ = η s, X 0,Σ X0 ). The complete data through time K is Z K =Z 1..., Z K ), where Z k =X k,y k,φ k ), X k =x 1 k)... x Nt k)), Y k =y 1 k)... y Ns k)), and Φ k =Φ 1,k... Φ Ns,k). The incomplete data is Y K = Y 1,...,Y K ), and the missing data is X K =X 1,...,X K )andφ K = Φ 1,...,Φ K ). Given these, the complete-data loglikelihood function thus can be written as 687 L ρ Z { K ) = log px 0 ) } K k=1 px k X k 1 )py k X k,φ k ). Assuming a Gaussian initial state density, we have 6) px 0 )= 2π) m 2 Σ X0 1 2 exp { 1 2 X 0 X 0 ) T Σ 1 X 0 X 0 X 0 ) }. 7) Meanwhile, the conditional probability density function PDF) for the target state can be expressed by px k X k 1 ) = 2π) m 2 Q 1 2 exp { 1 2 X k FX k 1 ) T Q 1 X k FX k 1 ) }. 8) To derive py k X k,φ k ), Φ s,k is first partitioned into rows, i.e., Φ s,k = φ 0 s,k, φ 1 s,k,..., φ Nt s,k T,where φ t s,k = Φ s,k t, 0) Φ s,k t, 1)... Φ s,k t, I s ) T. Hence, each φ t s,k corresponds to the tth target. The individual targets are assumed to be independent to each other. The log-likelihood function of Y k

3 canthenbewrittenas N t log py k X k,φ k )= log py k x t k), φ t 1,k,..., φ t N s,k). t=0 9) If we assume that the individual measurements are conditionally independent, given the associations, 9) can be further written as 12 N t N s log py k X k,φ k )= φ t s,k) T Uy s k) x t k)), where 1 t=0 10) Uy s k) x t k)) = log py s,0 k) e s,0,x t k)),...,log py s,is k) e s,is,x t k)) T. 11) The likelihood that the measurement y s,is k) isoriginated from target t, with the known state x t k), t = 1,...,N t,attimeinstantk,is py s,i k) e s,i,x t k)) = 1 P s d 1 ui) P s d py s,ik) x t k)) ui), 12) where PD s denotes the detection probability of sensor s, and ui) is an indicator function, defined by { 0, if i =0, ui)= 13) 1, otherwise, where ui) = 0 indicates the event of missed detection. If y s,i k) is originated from target t, wehave py s,i k) x t k)) { = 2π) l 2 Υ 1 2 exp 1 2 y s,ik) h s x t k)) η s ) T Υ 1 y s,i k) h s x t k)) η s ) }. 14) In addition, the likelihood that the measurement is spurious, i.e., there are no target associated with yi,k s,can be expressed by 11 ) uis) 1 pyi,k t s =0)=, 15) where Ψ s is the volume 2 of the field of view of sensor s. Overall, the log likelihood function in 6) can be expressed by L ρ Z k ) = 1 2 log Σ X X 0 X 0 ) T Σ 1 X 0 X 0 X 0 ) K 2 log Q K k=1 1 2 X k FX k 1 ) T Q 1 X k FX k 1 ) K k=1 φ 0 s,k )T 0, logψ s,...,log Ψ s T + K Nt k=1 t=1 φ t s,k )T Uy s k) x t k)). 16) 1 Here, py s,i k) e s,i,x tk)) represents the conditional PDF py s,i k) φ t s,k = e s,i,x tk)), where e s,i is the vector with a 1 in the ith measurement, i {0, 1,...,I s} and 0 elsewhere. 2 The PDF of a measurement if false alarms can be assumed to be approximately) uniform in the validation region 4 Ψ s 688 Our objective is to estimate the unknown parameters ρ, the missing data X k and Φ k simultaneously so that the log-likelihood function in 16) is maximized. 3 EM-EKF for Joint Registration, Association and Fusion To obtain an ML estimate of the above parameters and states, the EM algorithm is employed here because it can reduce the computational complexity and guarantees the convergence to a stationary point while continuously increasing the likelihood function. The EM algorithm consists of the following two steps: E-Step: E L ρ, ˆ ρ r) )=E L ρ Z K ) Y K, ρ r) M-Step: ρ r+1) =max ρ E L ρ ˆ ρ r) ), 17) where r corresponds to the rth iteration of the algorithm. The first step involves calculating of the conditional expectation based on the current estimate of the parameters ˆ ρ r) and the measurements. The secondstepprovidesanupdatedparameterestimateby maximizing the likelihood function. By iterating these two steps, the parameters are determined while the likelihood function can also be increased. 3.1 E-Step We define the conditional mean and the covariance functions as ˆx t k K) = Ex t k) Y 1,...,Y K, P t k K)=covx t k) Y 1,...,Y K andp t k,k 1 K)= covx t k),x t k 1) Y 1,...,Y K, respectively. Assuming that h s ) is a smooth function with respect to the independent variables, it can be approximated by the first order Taylor series expansion, i.e., h s x t k)) h s ˆx t k K))+H T s k)x t ˆx t k K)), 18) where H s k) = hsxtk)) xtk)=ˆx tk K). Furthermore, let s denote the conditional expectation of the 1st line of 16) as Item 1, the conditional expectation of the 2nd and the 3rd line as Item 2, and that of the 4th and the 5th lines as Item 3 and Item 4, respectively. By doing so, E L ρ, ˆ ρ r ) can be written as E L ρ,ˆ ρ r ) = Item 1 + Item 2 + Item 3 + Item 4. 19) The problem now becomes how to derive these 4 items. Item 1 and Item 3 are relatively straightforward. That is, Item 1 = 1 Nt { 2 t=1 log Σr) x t0) 1 Nt 2 t=1 Tr Σ 1r) x P t0) t0 K) } +ˆx t 0 K) x r) t 0))ˆx t 0 K) x r) t 0)) T, 20)

4 where Tr denotes the trace of a matrix, and K N s Item 3 = φ s 0,k) T 0, log Ψ s,...,log Ψ s T. k=1 For the other two items, we have 21) the forward and backward filtered outputs can be formulated as where ˆx t k K)= P t k K) P t k k) 1ˆx t k k) + P t k k +1) 1ˆx t k k +1), 32) Item 4 = K N t N s φ t s,k) T E Uy s k) x t,k ), 22) k=1 t=1 and Item 2 as in 24), where ξ k = K k=1 Pt k K)+ˆx t k K)ˆx t k K) T) K k=1 Pt k,k 1 K)+ˆx t k K)ˆx t k 1 K) T) F T K k=1 F P t k 1,k K)+ˆx t k 1 K)ˆx t k K) T) + K k=1 F P t k 1 K)+ˆx t k 1 K)ˆx t k 1 K) T) where F T. 25) From the above derivations, computing E L requires evaluating the conditional expectation of the system state and its covariances. For a nonlinear dynamic system, this can be carried out by using the EKF method. Given the previous state estimate ˆx t k 1 k 1) and the state covariance P t k 1 k 1), there are two steps to obtain the estimate x t k k). That is, Prediction ˆx t k k 1) = Fˆx t k 1 k 1), 26) P t k K)= P t k k) 1 + P t k k +1) ) It should be noted that validation has to be performed before applying EKF. Measurement validation is based on the fused state estimates at the previous iteration. That is, ŷ s k K) Y s k)) T Λk) 1 ŷ s k K) y s k)) <G, 34) r) ŷ s k K)=h s Fˆx t k K)) + ˆ η s, 35) Λ k = H s k)p t k K)H T s k)+υ, 36) and G is a gating threshold from a χ 2 -distribution that corresponds to a given gate probability p G for a certain degree of freedom. If a measurement, saying y s,i k), is found to be invalid for any target, the value of Φ s,k t, i) is set as 0. If there is no measurement at sensor s validated to target t, the dummy measurement y s,0 k) is assigned to target t. After measurement validation and data fusion using EKF, the element of the association matrix Φ s k is thus determined by P t k k 1) = FP t k 1 k 1)F T + Q, 27) Updating φ s t,k =max φ s t,k N t N s φ s t,k) T EUY s,k x t,k ) 37) t=0 ˆx t k k)= Sk)=H s k)p t k k 1)H T s k)+υ, 28) Kk)=P t k k 1)H T s k)sk), 29) ˆx t k k 1) + Kk) y s k) h s ˆx t k k 1)) ˆ η r) 30) P t k k)=p t k k 1) Kk)H T s k)p tk k 1). 31) Because the conditional expectation is based on all the measurements, the above EKF only gives an approximated evaluation based on the current observation. This approach is called the pseudo EM algorithm 14, 15. Its main advantage lies in the potential ability for on-line implementation. In fact, the exact evaluation of E L based on Z k is possible if we employ a fixed interval smoother 16 instead of the above forward filter. The smoother uses the forward and backward Kalman filter banks to obtain a smooth estimate. Assume that the state and the corresponding estimation error covariances can be obtained from the forward and backward filters, i.e., ˆx t k k), ˆx t k k +1)andP t k k), P t k k + 1), respectively. The smoother that combines 689 subject to the constraint in 5). 3.2 M-Step The parameters of this system are x t 0), Σ xt0),and η s. Their estimates are iteratively updated by taking the corresponding partial derivative of the expected log likelihood, setting to zero and solving for the solution. That is, x r+1) t 0) = ˆx t 0 K), 38) Σ r+1) x t0) = P t0 K), 39) and the derivation of the registration parameters are given in 40), as shown in the next page. 4 Performance Analysis To evaluate the proposed method, it is beneficial to obtain a lower bound of the estimation performance. In this section, we derive a PCRB for both the target state estimates and the registration parameter estimates. IfwehaveΘ=ˆx t k) η s T,thePCRBisgivenby E Θ) Θ) T Θ) Θ) J 1, 41)

5 Item 2 = K 2 N t t=1 log Q 1 2 N t t=1 K Tr Q 1 E x t k) Fx t k 1))x t k) Fx t k 1)) T Z K k=1 }{{} ξ k 24) K Nt η s r+1) k=1 t=1 = Is K k=1 Nt t=1 where Θ) is an estimator of Θ based on the observation Y k,andj is the Fisher information matrix FIM) with the i,j)th element defined by J i,j = E 2 log py k, Θ), 42) Θ i Θ j where py k, Θ) denotes the joint PDF of Y k, x t k), and η s. Based on 6), py k, Θ) can be expressed by py k, Θ) = px t k) x t k 1)) py k x t k),φ k ). 43) Basedon16),wehave i=1 Φs k t, i)υ 1 y s,i k) h s ˆx t k N))) log py k, Θ) = 1 2 x tk) Fx t k 1)) T Q 1 x t k) Fx t k 1)) + N s φ t s,k Uy sk) x t k)). 44) with some constants, such as log 2π, log Q, skipped. The derivatives of log py k, Θ) with respect to the target states and the calibration parameters are computed as follows: D 11 k)=e 2 log py k, Θ) 2 = F T Q 1 F, 45) x t k 1) D 12 k)=d 21 k)=e 2 log py k, Θ) =0, 46) x t k 1) η s D 13 k)=d 31 k)=e 2 log py k, Θ) = F T Q 1, x t k 1) x t k) 47) D 22 k) =E 2 log py k,θ) { 2 η s Ns = E φ t s,k 0 Υ 1... Υ 1 } T, D 23 k)=d 32 k)=e = E φ t s,k 2 log py k,θ) η s 0 hsx tk)). hsx tk)) ) T Υ 1 ) T Υ 1, 48) 49) 690 Is i=1 Φs k t, i)υ 1. 40) D 33 k)=e E φ t s,k 2 log py k,θ) 2 x tk) hsx tk)) hsx tk)) = Q ) T Υ 1 h sx tk)). ) T Υ 1 h sx tk)). 50) As shown in 41), the inverse of J gives the PCRB on the estimation error of the target states and the registration parameters. The recursive calculation of J 17 can be computed as follows J 11 k+1) = D 11 k) D T 13k)D 11 k)+d 11 k) 1 D 13 k), 51) J 12 k +1)= D 23 k) D T 13 k)j 11k)+D 11 k) 1 J 12 k)+d 12 k), 52) J 21 k +1)=J 12 k +1) T, 53) J 22 k +1)= J 22 k)+d 22 k) J 12 k)+d 12 k) T J 11 k)+d 1 1k) 1 J 12 k)+d 12 k). 54) 5 Experimental Results In this section, we evaluate the proposed algorithm using computer simulations. In our simulation scenario, two radars are used for tracking eight targets with constant velocities. The sampling interval is 2 seconds. The target positions are initialized as: -50km, 21km), -50km, 23.5km), -50km, 25.8km), -50km, 27.9km), -50km, 31km), -50km, 34.5km), -50km, 36.8km), -50km, 38.9km). The velocities of these targets are the same as 0.2km/s, 0km/s). The process noise covariances are also the same for all targets, i.e., Q = diag0.0005km 2, km 2 /s 2, km 2, km 2 /s 2 ). The two radars are located at 0km, 0km) and 20km, 5km), respectively. Both the measurement noise covariances are equal to diag0.001km 2, rad 2 ). The sensor biases are η 1 = 0.12km 0.01rad T and η 2 =0.16km 0.013rad T. The detection probabilities of the two sensors are both The clutter densities of the two sensors are both 0.1returns/km 2.

6 Association error rate of the joint method Association error rate of M D assignment with registration error Association error rate 100%) Discrete time instant Figure 1: The association error rates of the proposed joint method and the S-D assignment method a) The number of the time instants, K = 150, is used here. Figure 1 shows the data association performance of the proposed approach. The association results by applying the S-D assignment approach 11 are also plotted for comparison. To reduce the number of candidate associations for the multidimensional assignment problem, we make an assumption that a target must be detected by all the two sensors to be considered in the association process. It is shown in the figure that the data association performance of the proposed algorithm is better than that of the S-D assignment method. If sensorregistrationis not consideredin the data association process, the registrationerror will deteriorate the data association performance. To illustrate the registration performance of the proposed method, Figure 2 shows the convergence of the estimated registration parameters. Apparently, the estimated registration parameters converge rapidly with the EM iterations. To compare with sensor registration without simultaneous data association and registration, we first apply the S-D assignment method for association and then employ the ML registration and fusion method in 13. The maximum likelihood registration and fusion algorithm in 13 uses all the pre-associated measurements for registration and fusion. Figure 3 shows the performance comparison of registration by using the evaluation criteria of mean square error MSE) between the true registration parameters and the estimated ones. As we discussed before, the association error deteriorates the registration performance if the association and registration are not conducted in a simultaneous fashion. Our simulation confirms this observation. The state estimation performance of the proposed method is evaluated here. We perform a total of 50 Monte-Carlo runs to obtain the MSEs. Figure 4 and 5 show the MSEs of the position and velocity estimates of a target by the two methods. It is observed that the estimation performance of the proposed method is better than the method that performs registration, association and fusion separately. In our opinion, it is mainly b) c) d) Figure 2: Estimation of the sensor biases by EM-EKF: a) η 1 1); b) η 1 2); c) η 2 1); and d) η 2 2). 691

7 a) a) b) b) Figure 4: The MSE results of the target position estimation in a) x-axis; b) x-axis. because the effect of association error and registration error on data fusion can be reduced by performing data association, registration and fusion simultaneously. c) d) Figure 3: The MSE results of estimating a) η 1 1); b) η 1 2); c) η 2 1); and d) η 2 2). 6 Conclusions In this paper, a joint registration, association and fusion approach based on EM-EKF is proposed. Since association and registration are the two processes depending on each other, the proposed method performs these functions in a simultaneous fashion. More specifically, it uses EM and EKF to estimate the target states, measurement-to-target association and registration parameters simultaneously. Simulations show that the proposed EM-EKF is convergent and outperforms one method that carry out registration and association separately. References 1 Dana, M.P., Registration: a prerequisite for multiple sensor tracking, Multitarget-Multisensor Tracking: Advanced Applications, Y. Bar-Shalom, Ed. Artech House, Norwood, MA, Zhou, Y., Leung, H. and Yip, P.C., An exact maximum likelihood registration algorithm for data fusion, IEEE Trans. Signal Processing, vol. 45, no. 6, pp ,

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