Estimation and Maintenance of Measurement Rates for Multiple Extended Target Tracking

Transcription

1 FUSION 2012, Singapore 118) Estimation and Maintenance of Measurement Rates for Multiple Extended Target Tracking Karl Granström*, Umut Orguner** *Division of Automatic Control Department of Electrical Engineering Linköping University, Sweden **Department of Electrical and Electronics Engineering Middle East Technical University, Turkey

2 Introduction 218) Extended targets ξ k give rise to sets of measurements { Z k = z j) k } Nz,k N z,k is a random number. With multiple targets, clutter and association uncertainty, it is of interest to be able to predict/estimate N z,k for each target. Gilholm et al. 2005) models the number of measurements as Poisson distributed with rate γ k j=1 e γ k p N z,k γ k ) = PS N z,k ; γ k ) = γn z,k k N z,k!.

3 Problem formulation and contribution 318) How can a single rate γ k per target be estimated? Contribution: Recursive Bayesian estimator for a single rate γ k per target.

4 Problem formulation and contribution 318) How can a single rate γ k per target be estimated? Contribution: Recursive Bayesian estimator for a single rate γ k per target. With multiple targets, distribution mixtures arise naturally. How can a mixture of γ k estimates be reduced? Contribution: Merging of weighted sum of N estimates into one estimate. Criterion to determine which estimates should be merged.

5 Bayesian estimator for γ correction 418) Conjugate prior is known to be the Gamma distribution, p γ ) Z k 1 k =GAM ) γ k ; α k k 1, β k k 1 = βα k k 1 k k 1 Γ α k k 1 ) γ α k k 1 1 k e β k k 1γ k. Measurement update known to be ) Z k p γ k =GAM ) γ k ; α k k 1, β k k 1 PS Nz,k ; γ k ) =GAM ) ) γ k ; α k k, β k k Lγ αk k 1, β k k 1, N z,k where α k k =α k k 1 + N z,k β k k =β k k and likelihood L γ is negative-binomial

6 Bayesian estimator for γ prediction 518) Suggested prediction: Exponential forgetting prediction α k+1 k = α k k η k, β k+1 k = β k k η k, η k >1. Expected value constant, and variance increased by factor η k. The effective window length is w e = η k η k 1, i.e. we only trust measurements from last w e time steps.

7 Bayesian estimator for γ results 618) γ0 + γ γ ˆγ ±ˆγ 1/ γ k True γ k k η k = 1.10, w e = γ ˆγ ±ˆγ 1/2 6 4 γ ˆγ ±ˆγ 1/ k η k = 1.25, w e = k η k = 2.25, w e = 1.8

8 Multiple extended targets 718) Let ξ k denote the augmented extended target state, ξ k = γ k, x k ), where x contains states for position, velocity, size, shape etc. Given a set of measurements Z k and a prior distribution p ξ k Z k 1), the posterior distribution is ξ k Z k 1) p ξ k Z k) =p Z k ξ k ) p =p N z,k γ k ) p Z k x k ) p γ k Z k 1) p x k Z k 1).

9 Multiple extended targets 818) The posterior distribution and predicted likelihood is p γ k Z k) p x k Z k) ) ) L γ αk k 1, β k k 1, N z,k Lx ˇxk k 1, Z k, }{{}}{{} posterior predicted likelihood where ˇx k k 1 denotes the sufficient statistics of x k. Any framework that estimates multiple states x k can be augmented to also estimate of the measurement rates γ k. e.g. the ET-PHD filter.

10 Multiple extended targets 918) Distribution mixtures often used in multi target tracking, e.g. MHT, PHD p ξ k ) = J k k j=1 ) w j GAM γ k ; α j) k k, βj) p k k x k ; ˇx j) k k ). For large J k k, mixture reduction necessary, e.g. merging. Merging for Gaussian distributed x well known. We show merging for Gamma distributed Poisson rates, using minimization of the Kullback-Leibler divergence.

11 Kullback-Leibler divergence 1018) KL-div is a measure of how similar q and p are ) px) KL p q) = px) log dx, qx) Positive, KL p q) 0. In general asymmetric, KL p q) = KL q p). Well known moment matching characteristcs. Considered optimal difference measure in max likelihood sense. Minimizing KL-div can be rewritten as maximization problem min q KL p q) = max q px) log qx)) dx.

12 Theorem 1 Merging of N gamma components 1118) Let p ) be a weighted sum of Gamma components, p γ) = N i=1 where w = N i=1 w i. Let w i GAM γ ; α i, β i ) = N i=1 q γ) = wgam γ ; α, β) w i p i γ), be the minimizer of the KL-div between p γ) and q γ) among all Gamma distributions, i.e. q γ) arg min KL p γ) q γ)). qγ) GAM

13 Theorem 1 parameters 1218) The parameter β is given by β = and the parameter α is the solution to 0 = log α ψ 0 α) + 1 w N i=1 α 1 w N i=1 w i α, i β i w i ψ 0 α i ) log β i ) log Corresponds to matching the expected values of γ and log γ, w E q [γ] = w E q [log γ] = N i=1 N i=1 w i E pi [γ] w i E pi [log γ]. 1 w N i=1 w i α i β i ).

14 Theorem 1 proof 1318) The proof is simple. Requires basic knowledge of calculus. Full details given in the paper.

15 Merging criterion 1418) Kullback-Leibler difference for components i and j defined as D KL pi, p j ) = KL pi p j ) + KL pj p i ) Due to assumed conditional independence it becomes D KL pi ξ), p j ξ) ) =D KL pi x), p j x) ) + D KL pi γ), p j γ) ) =D x KL + Dγ KL, Merge components i and j if D KL < U or if D x KL < U x) & D γ KL < U ) γ Further information about criterion in the paper.

16 Merging results 1518) 20 gamma components reduced to 7 by merging p p γ

17 Multiple targets results 1618) Three extended targets, with true rates 5, 15 and 30. Extended target PHD filter used for multiple target tracking γk, ˆγ k k γk, ˆγ k k η k = 1.25, w e = 5 k η k = 1.01, w e = 101 k

18 Summary and future work 1718) Main contributions: Recursive Bayesian estimator for a single rate γ k per target. Merging of weighted sum of N estimates into one estimate. Criterion to determine which estimates should be merged.

19 Summary and future work 1718) Main contributions: Recursive Bayesian estimator for a single rate γ k per target. Merging of weighted sum of N estimates into one estimate. Criterion to determine which estimates should be merged. Future work: Use further in implementations of PHD filter for extended targets. Include position, velocity, shape and size of target into prediction of rate.

20 The End 1818) Thank you for listening! Any questions?