Estimation and Maintenance of Measurement Rates for Multiple Extended Target Tracking

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1 Estimation and Maintenance of Measurement Rates for Multiple Extended Target Tracing Karl Granström Division of Automatic Control Department of Electrical Engineering Linöping University, SE-58 83, Linöping, Sweden Umut Orguner Department of Electrical and Electronics Engineering Middle East Technical University 653, Anara, Turey Abstract In Gilholm et al. s extended target model, the number of measurements generated by a target is Poisson distributed with measurement rate γ. Practical use of this extended target model in multiple extended target tracing algorithms reuires a good estimate of γ. In this paper, we first give a Bayesian recursion for estimating γ using the well-nown conjugate prior Gamma-distribution. In multiple extended target tracing, consideration of different measurement set associations to a single target maes Gamma-mixtures arise naturally. This causes a need for mixture reduction, and we consider the reduction of Gamma-mixtures by means of merging. Analytical minimization of the Kullbac-Leibler divergence is used to compute the single Gamma distribution that best approximates a weighted sum of Gamma distributions. Results from simulations show the merits of the presented multiple target measurement-rate estimator. The Bayesian recursion and presented reduction algorithm have important implications for multiple extended target tracing, e.g. using the implementations of the extended target PHD filter. Index Terms Poisson distribution, Poisson rate, Gamma distribution, conjugate prior, Bayesian estimator, extended target, PHD filter, mixture reduction, Kullbac-Leibler divergence. I. INTRODUCTION In target tracing, the assumption is often made that a target gives rise to at most one measurement per time step, see e.g. []. However, in extended target tracing this assumption is relaxed, and the extended targets are modeled as potentially giving rise to more than one measurement per time step. In an extended target tracing scenario it is therefore of interest to model the number of measurements that each target gives rise to. One such model is given by Gilholm et al. [], where the measurements are modeled as an inhomogeneous Poisson point process. At each time step, a Poisson distributed random number of measurements are generated, distributed around the target. Mahler has given an extended target Probability Hypothesis Density PHD filter under this model [3], and a Gaussian Mixture implementation of this filter, called the Extended Target Gaussian Mixture PHD ET-GM-PHD filter, has been presented [4], [5]. A Gaussian inverse Wishart implementation of [3], called the Gaussian inverse Wishart PHD GIW-PHD filter is presented in [6]. The measurement model [] can be understood to imply that the extended target is sufficiently far away from the sensor for the measurements to resemble a cluster of points, rather than a geometrically structured ensemble, see e.g. [4], [5] for simulation examples. However, the ET-GM-PHD filter and GIW-PHD filter have also been applied successfully to data from laser range sensors, which give highly structured measurements, see [5] [7]. In the extended target PHD filter the Poisson rate γ is modeled as a function of the extended target state x, i.e. γ γ x. In the ET-GM-PHD filter γ is approximated as a function of the extended targets state estimates [5]. It has been noted that having a good estimate ˆγ of the true parameter is important when multiple targets are spatially close [5]. More specifically, under the assumption that the true rate is constant and eual for all targets, the true parameter must lie in the interval ˆγ ˆγ γ ˆγ + ˆγ for the estimated cardinality to be correct [5]. However, in the most general case the rates are neither constant over time, nor eual for all extended targets. It might also be the case that the true function γ is difficult to model, or even time-varying. All of these issues raise the need for a method to estimate individual Poisson rates for multiple extended targets. In this paper we consider multiple extended targets under the measurement model []. The set of extended targets at time t is denoted { X x i } Nx,. At each time step, the number of measurements generated by the i:th target is a Poisson distributed random variable with rate γ i. The measurement set at time t, denoted { } Z z j Nz,, 3 j is the union of all target measurements and the clutter measurements. The number of clutter measurements generated at each time step is assumed to be Poisson distributed with rate λ. Let Z denote all measurement sets up to, and including, time t. We assume the existence of an underlying multiple extended target tracer that estimates the target states x i, e.g. [4], [5]. The set of measurements that are used to update the state of the i:th target at time t is denoted Z i. 7

2 The first objective of this wor is to estimate the set of measurement rates γ i, given seuences of measurement sets { } Z,i Z i,..., Zi, i,..., n x,. 4 To this end, in Section II we give a recursive Bayesian estimator for γ i, with exponential forgetting for the prediction step. We also show how the predicted lielihood is affected when the measurement rates γ i are estimated in addition to estimating the target states x i. In the multiple target case under clutter and missed detections, there might be multiple alternative measurement sets corresponding to different association hypotheses Z i, Z i,..., Z i N i 5 that are used to update the i:th target state at time t. In this case, the state densities of the targets are represented by mixture densities. As time progresses, the number of mixture components grow. To obtain computationally tractable algorithms, hypothesis reduction must be performed, e.g. via pruning or merging. The second objective of this wor is to show how a mixture of γ estimates can be reduced. In Section III, we consider merging a weighted sum of measurement rate estimates by minimization of the Kullbac-Leibler divergence, and we also give a criterion that is used to determine whether or not two components should be merged. The proposed Bayesian estimator and merging method is evaluated in Simulations in Section IV, and the paper is finalized with concluding remars in Section V. II. BAYESIAN RECURSION FOR γ In this section, we consider recursive estimation of the i:th target s measurement rate γ i from the seuence of measurement sets Z,i. We also show how estimating the measurement rate affects the resulting extended target predicted lielihood. Since we consider only the i:th target, from this point on in this section, we suppress the superscript i. A. Measurement update and prediction The conjugate prior to the Poisson distribution is well nown to be the Gamma distribution, see e.g. [8]. Assume that at time t the prior distribution for the Poisson rate γ is a Gamma distribution, p γ Z GAM γ ; α, β 6 βα Γ γ α e β γ. 7 α Let the :th measurement set Z contains N z, elements, where N z, is Poisson distributed with rate γ, p N z, γ PS N z, ; γ 8 γn z, e γ N z,!. 9 The posterior distribution is p γ Z GAM γ ; α, β PS Nz, ; γ βα γα +N z, e β +γ Γ α Nz,! GAM γ ; α + N z,, β + a b c Γ α α + N z, β Γ α β + α +N z, N z,! GAM γ ; α, β L γ α, β, N z,, d where the predicted lielihood L γ is a negative binomial distribution, see e.g. [8]. In case the true parameter is nown to be constant over time, the posterior distribution can be predicted as p γ Z p γ Z. However, in the general case γ may change over time. We propose to use exponential forgetting with a forgetting factor η for the prediction of γ, α + α η, β + β η, where η >. This prediction has an effective window of length w e /η η η. Using exponential forgetting prediction with window length w e approximately means that we only trust the information that was contained in the measurements from the last w e time steps. The expected value and variance of γ are E [γ ] α β, Var γ α β. Note that the prediction corresponds to eeping the expected value constant while increasing the variance with a factor η >. B. Extended target predicted lielihood The measurement update and corresponding predicted lielihood is an important part of any framewor for multiple target tracing under uncertain association and clutter. Let ξ denote the augmented extended target state, ξ γ, x. 3 Given a set of measurements Z and a prior distribution p ξ Z, the posterior distribution is p ξ Z p Z ξ p ξ Z 4a p Z ξ p γ Z p x Z. 4b Note that there is an implicit assumption here that the prior distribution p ξ Z can be factorized as p γ Z p x Z. This assumption neglects the dependence between the number of measurements and any extension parameters that are included in x. However the probability density over the number of measurements, conditioned 7

3 on the target extension, is unnown in most applications, and we believe that this assumption is valid in most cases. Assume also that the measurement lielihood can be decomposed as p Z ξ p Z, N z, ξ p N z, γ p Z x. 5 The validity of this assumption is also dependent on the considerations mentioned above. The posterior distribution and predicted lielihood is p γ Z p x Z }{{} posterior L γ α, β, N z, Lx ˇx, Z, 6 }{{} predicted lielihood where ˇx denotes the sufficient statistics of x. Thus, any extended target tracing framewor that estimates the states x can be augmented to also include estimates of the measurement rates γ. In the results section below we give an example where we integrate γ estimation into the ET-GM-PHD filter [4], [5]. The posterior distribution for γ and the corresponding predicted lielihood L γ are given in d. The details for the posterior distribution and predicted lielihood for x, as well as the full filter recursion, can be found in [4], [5]. III. MULTI-TARGET MIXTURE REDUCTION A straightforward way to model uncertainty in multiple target tracing is to use mixtures of distributions, see e.g. the Multi-hypothesis Tracing filter [9], or the Gaussian Mixture PHD-filters [4], [5], []. Let p be a mixture of distributions, J p ξ w j p j ξ 7 j J j w j GAM γ ; α j, βj p x ; ˇx j where each distribution p j is called component. A common choice is to model the state x as Gaussian distributed, see e.g. [4], [5], [9], [], which would give a Gamma Gaussian GG distributed extended target ξ. In Koch s random matrix framewor [], the extent is modeled as an inverse Wishart distributed random matrix X, and the inematic parameters are modeled as a random vector x. In this case we have ξ γ, x, X, and 7 would be a mixture of Gamma Gaussian inverse Wishart distributions. A natural conseuence of the tracing framewors [4], [5], [9], [] is the increasing number of mixture components, or hypotheses. To eep the target tracing implementation at a tractable level, the mixture must be reduced regularly, which is typically performed via pruning or merging. The output of Position, velocity and acceleration. mixture reduction is an approximate mixture, J p ξ w j p j ξ 8 j J j w j GAM γ ; α j j, β p x ; ˇx j where J < J and the difference between p and p is small by some measure. Here we address mixture reduction via component merging. One approach to merging is to successively find component pairs that are close by some merging criterion, and merge them, see e.g. [4], [5], []. Different methods for merging of Gaussian mixtures are given in e.g. [] [6], a method for merging of Gaussian inverse Wishart mixtures is given in [7]. In Section III-A we give a theorem which is used to find the Gamma distribution that minimizes the Kullbac-Leibler divergence between w and the sum p Σ i L w i p i, where w Σ i L w i and L {,..., J }. When the extended targets are modeled with a mixture 7, the merging criterion should consider both parts of the components, i.e. the distributions of both γ and x. Different merging criteria for Gaussian distributions are given in e.g. [4], [5], [], [] [6]. In Section III-B we give a merging criterion for mixtures of Gamma distributions. A. Merging N Gamma components The Kullbac-Leibler divergence KL-div, px KL p px log dx, 9 x is a measure of how similar two functions p and are. The KL-div is well-nown in the literature for its momentmatching characteristics, see e.g. [8], [9], and for probability distributions it is considered the optimal difference measure in a maximum lielihood sense [3] [5]. Note that minimizing the KL-div between p and w.r.t. can be rewritten as a maximization problem, min KL p max, px log x dx. Theorem : Let p be a weighted sum of Gamma components, p γ w i GAM γ ; α i, β i w i p i γ, where w N w i. Let γ wgam γ ; α, β be the minimizer of the KL-div between p γ and γ among all Gamma distributions, i.e. γ arg min KL p γ γ. 3 γ GAM 7

4 Then the parameter β is given by β w and the parameter α is the solution to α N w, 4 i αi β i log α ψ α + w w i ψ α i log β i α i log w i. 5 w β i Proof: Given in Appendix A. Remars: The expression for β 4 corresponds to matching the expected values under both distributions and p, w E [γ] w i E pi [γ]. 6 The expression for α 5 corresponds to matching the expected values of the logarithm under both distributions and p, w E [log γ] w i E pi [log γ]. 7 A value for the parameter α is easily obtained by applying a numerical root finding algorithm to 5, e.g. Newton s algorithm, see e.g. []. B. Merging criterion for Gamma components In this section we derive a criterion that is used to determine whether or not two Gamma components should be merged. When reducing the number of components, it is preferred to eep the overall modality of the mixture. Thus, if the initial mixture p has M modes, then the reduced mixture p should have M modes. The optimal solution to this problem is to consider every possible way to reduce J components, compute the corresponding KL-div:s, and then find the best trade-off between low KL-div and reduction of J. For J components, there are B J different ways to merge, where B i is the i:th Bell number []. Because B i increases rapidly with i, e.g. B 5 5 and B 5975, the optimal solution can not be used in practice. Instead a merging criterion must be used to determine whether or not a pair of Gamma components should be merged. As merging criterion the KL-div could be used, however because it is asymmetrical, KL p KL p, 8 it should not be used directly. Instead we use the Kullbac- Leibler difference KL-diff, defined for two distributions p γ and γ as D KL p γ, γ KL p γ γ + KL γ p γ 9a p γ γ p γ log dγ + γ log dγ. γ p γ 9b Let p γ and γ be defined as p γ GAM γ ; α, β, γ GAM γ ; α, β. The KL-div between p and is KL p γ γ α log β log Γα + α ψ α log β α α log β + log Γα α α ψ α log β + β β β α Γα log + log β α Γα + α α ψ α log β + α β β 3a 3b 3a, 3b and the KL-div between and p is defined analogously. The KL-diff between p and becomes D KL p γ, γ α α ψ α ψ α + log β β α + β β α. β β C. Merging of extended target components 3 When merging is used to reduce an extended target mixture 7, the merging criterion should be defined over both γ and x. For example, the following merging criterion could be used D KL p i ξ, p j ξ < U, 33 where D KL is the KL-diff between two extended target components. Owing to the assumed conditional independence of the distributions over γ and x in 4, the KL-diff can expressed as a sum D KL p i ξ, p j ξ D γ KL i, j + Dx KL i, j, 34 where D γ KL i, j D KL p i γ, p j γ is given in 3 and DKL x i, j D KL p i x, p j x. Thus, the following merging criterion could alternatively be used D γ KL i, j < U γ & D x KL i, j < U x, 35 where & is the logical and operator. In case x is Gaussian distributed, possible merging criterions DKL x i, j are given in e.g. [], []. 73

5 .8 33 γ + γ 8 γ ˆγ ±ˆγ / log b γ a True γ γ ˆγ ±ˆγ / b η., w e 6 γ ˆγ 4 ±ˆγ / log a Fig.. KL-diff for two Gamma distributions, when α aα a and β bβ. When a b, the expected value is approximately the same for both distributions, and the the ratio of the variances is /b. This explains the elongated shape of the KL-diff along a b c η.5, w e d η.5, w e.8 Fig. 3. Single target results. a: The true rate γ varied between γ 5 and γ + γ 35. b, c and d: The solid line shows the average estimation error, the gray area is the average estimation error ± one standard deviation, and the dashed lines are the bounds ± ˆγ, c.f.. A higher η gives a lower average estimation error, however the estimation error also has much larger standard deviation Fig.. Merging of Gamma components, using the merging method and criterion of Section III. The reduced mixture has 7 components and has preserved the overall modality. A. Merging criterion γ IV. RESULTS Letting α aα and β bβ, the KL-diff simplifies to D KL p γ, γ α a log b + ψ α ψ aα + α b a, 36 b i.e. it becomes independent of the specific value of β. It can be shown that, for given a and b, a larger α means a larger KL-diff. For α, the KL-diff is shown in Figure. B. Comparison of merging algorithms An intensity p γ with Gamma components was reduced using the merging method and criterion presented in Section III. The Gamma mixture parameters were sampled uniformly from the following intervals, w i [.5,.95], α i [5, 5], β i [5, 5], 37 i.e. α i and β i were sampled such that the expected value and variance of γ belongs to [, 5] and [, ], respectively. p p The original mixture and the approximation are shown in Figure. The reduced mixture has 7 components, and manages to capture the overall modality of the original mixture. C. Single target results A single target with time varying γ was simulated for 5 time steps, the true measurement rate varied with time as shown in Figure 3a. The estimation error, averaged over 4 Monte Carlo runs, is shown in Figures 3b, 3c and 3d, for η., η.5 and η.5, respectively. With a higher η, the estimate responds faster to changes in the true parameter, at the expense of being more sensitive to noise. As with any prediction and correction recursion, setting the parameter reuires a trade off between noise cancellation and tracing capabilities. D. Multiple target results The Bayesian γ estimator was integrated into the Gaussian Mixture Probability Hypothesis Density ET-GM-PHD filter [4], [5]. A scenario with three targets was simulated for time steps, the true Poisson rates were set to γ 5, γ 5 and γ 3 3. Estimation results for η.5 are shown in Figure 4a. The estimates ˆγ i αi β i 38 are a bit noisy, however they remain within the bounds given by γ i ± γ i, 39 i.e. the true mean ± one standard deviation. With η. the estimation error is much smaller, see Figure 4b. However, as discussed previously, with a low η the response to changes in the true parameter would be slower. 74

6 γ, ˆγ γ, ˆγ a η.5, w e b η., w e Fig. 4. Three targets with true rates γ 5, γ 5 and γ 3 3, shown as dar gray lines. The estimates ˆγ i, shown as blac dots, remain within the bounds γ i ± γ i, i.e. the true mean ± one standard deviation, shown as light gray areas. The same seuence of measurement sets is used in a and b, with different η. V. CONCLUDING REMARKS This paper presented a Bayesian estimator for the rate parameter γ of a Poisson distribution. The conjugate prior of γ is the Gamma distribution, and, using exponential forgetting prediction, it is possible to trac a rate γ that changes over time. To manage multiple targets with different rates, a mixture of Gamma distributions is utilized. Mixture reduction is addressed, where components are merged via analytical minimization of the Kullbac-Leibler divergence between a weighted sum of Gamma distributions and the single Gamma distribution that best approximates the sum. A simulation study was used to show the merits of the Poisson rate estimation framewor. In future wor, we intend to integrate the rate estimation fully into the Gaussian mixture and Gaussian inverse Wishart extended target PHD filters. Having a good estimate of the measurement rate could have important implications for the performance, especially during the measurement partitioning step. Future wor also includes improving upon the exponential forgetting prediction. The number of measurements generated can be affected by the extended target s position, as well as its shape and size. Including the estimated position, size and shape in the prediction step could possibly improve tracing of Poisson rates that change over time. APPENDIX A PROOF OF THEOREM First we derive an expected value which is needed in the proof of Theorem. A. Expected value of logarithm Let y be a uni-variate random variable. The moment generating function for y is defined as µ y s E y [e sy ], 4 and the expected value of y is given in terms of µ y s as E [y] dµ y s ds. 4 s Let y log γ, where γ GAM γ ; α, β. The moment generating function of y is µ y s E [γ s ] 4a γ s βα Γ α γα e βγ dγ 4b βα Γ s + α Γ α β s+α GAM γ ; s + α, β dγ 4c Γ s + α Γ α β s. 4d The expected value of y is E [y] E [log γ] 43a d Γ s + α ds Γ α β s 43b s d dsγ s + α Γ s + α d β s Γ α β s + Γ α ds s s 43c ψ α log β 43d where ψ is the digamma function a..a. the polygamma function of order. B. Proof Proof: We have given as γ arg min arg max arg max KLp pγ logγdγ w i p i γ log γ dγ, where the i:th integral is p i γ log γ dγ p i γ [α log β log Γα + α + log γ βγ] dγ α log β log Γα + α E i [log γ] β E i [γ] 44a 44b 44c α log β log Γα + α ψ α i log β i β α i β i

7 Taing the derivative of the objective function with respect to β, euating the result to zero, and solving for β, we get α β N w w. 46 i αi β i Now, we tae the derivative of the objective function with respect to α and euate the result to zero to obtain w i log β ψ α + ψ α i log β i 47 w log β wψ α + w i ψ α i log β i. 48 Inserting β and rearranging the terms we obtain log α ψ α + w w i ψ α i log β i α i log w i. 49 w β i [] D. J. Salmond, Mixture reduction algorithms for target tracing in clutter, in Proceedings of SPIE Signal and Data Processing of Small Targets, Orlando, FL, USA, Jan. 99, pp [3] J. L. Williams and P. S. Maybec, Cost-Function-Based Gaussian Mixture Reduction for Target Tracing, in Proceedings of the International Conference on Information Fusion, Cairns, Queensland, Australia, Jul. 3. [4] A. R. Runnalls, Kullbac-Leibler approach to Gaussian mixture reduction, IEEE Transactions on Aerospace and Electronic Systems, vol. 43, no. 3, pp , Jul. 7. [5] D. Schieferdecer and M. F. Huber, Gaussian Mixture Reduction via Clustering, in Proceedings of the International Conference on Information Fusion, Seattle, WA, USA, Jul. 9. [6] D. F. Crouse, P. Willett, K. Pattipati, and L. Svensson, A Loo At Gaussian Mixture Reduction Algorithms, in Proceedings of the International Conference on Information Fusion, Chicago, IL, USA, Jul.. [7] K. Granström and U. Orguner, On the Reduction of Gaussian inverse Wishart mixtures, in Proceedings of the International Conference on Information Fusion, Singapore, Jul.. [8] C. M. Bishop, Pattern recognition and machine learning. New Yor, USA: Springer, 6. [9] T. Mina, A family of algorithms for approximate Bayesian inference, Ph.D. dissertation, Massachusetts Institute of Technology, Jan.. [] J. Stoer and R. Bulirsch, Introduction to Numerical Analysis, nd ed. New Yor: Springer-Verlag, 993. [] G.-C. Rota, The number of partitions of a set, The American Mathematical Monthly, vol. 7, no. 5, pp , May 964. ACKNOWLEDGMENT The authors would lie to than the Linnaeus research environment CADICS and the frame project grant Extended Target Tracing 6--43, both funded by the Swedish Research Council, and the project Collaborative Unmanned Aircraft Systems CUAS, funded by the Swedish Foundation for Strategic Research SSF, for financial support. REFERENCES [] Y. Bar-Shalom and T. E. Fortmann, Tracing and data association, ser. Mathematics in Science and Engineering. San Diego, CA, USA: Academic Press Professional, Inc., 987, vol. 79. [] K. Gilholm, S. Godsill, S. Masell, and D. Salmond, Poisson models for extended target and group tracing, in Proceedings of Signal and Data Processing of Small Targets, vol San Diego, CA, USA: SPIE, Aug. 5, pp [3] R. Mahler, PHD filters for nonstandard targets, I: Extended targets, in Proceedings of the International Conference on Information Fusion, Seattle, WA, USA, Jul. 9, pp [4] K. Granström, C. Lunduist, and U. Orguner, A Gaussian mixture PHD filter for extended target tracing, in Proceedings of the International Conference on Information Fusion, Edinburgh, UK, Jul.. [5], Extended Target Tracing using a Gaussian Mixture PHD filter, IEEE Transactions on Aerospace and Electronic Systems,. [6] K. Granström and U. Orguner, A PHD filter for tracing multiple extended targets using random matrices, IEEE Transactions on Signal Processing. [7] K. Granström, C. Lunduist, and U. Orguner, Tracing Rectangular and Elliptical Extended Targets Using Laser Measurements, in Proceedings of the International Conference on Information Fusion, Chicago, IL, USA, Jul., pp [8] A. Gelman, J. B. Carlin, H. S. Stern, and D. B. Rubin, Bayesian Data Analysis, ser. Texts in Statistical Science. Chapman & Hall/CRC, 4. [9] Y. Bar-Shalom and X. Rong Li, Multitarget-Multisensor Tracing: Principles and Techniues. YBS, 995. [] B.-N. Vo and W.-K. Ma, The Gaussian mixture probability hypothesis density filter, IEEE Transactions on Signal Processing, vol. 54, no., pp , Nov. 6. [] J. W. Koch, Bayesian approach to extended object and cluster tracing using random matrices, IEEE Transactions on Aerospace and Electronic Systems, vol. 44, no. 3, pp. 4 59, Jul