Sequential Bayesian Estimation of the Probability of Detection for Tracking

Transcription

1 2th International Conference on Information Fusion Seattle, WA, USA, July 6-9, 2009 Sequential Bayesian Estimation of the Probability of Detection for Tracking Kevin G. Jamieson Applied Physics Lab University of Washington Seattle, WA, U.S.A. Maya R. Gupta Electrical Engineering Dept. University of Washington Seattle, WA, U.S.A. David W. Krout Applied Physics Lab University of Washington Seattle, WA, U.S.A. Abstract We propose a Bayesian estimation method to sequentially update the probability of detection for tracking. A beta distribution is used for the prior, which can be centered on the best a priori guess for the probability of detection. The tracker s belief about whether it detected the target at the last scan is used to update the posterior estimate of the probability of detection. The method can be applied to any tracking algorithm that requires an estimate of the probability of detection. Experiments with the probabilistic data association (PDA) tracker show that the proposed estimation method can increase the amount of time a target is tracked and decrease the localization error when compared to using a fixed value. Experiments also show that for some values of the probability of detection, using an inflated value of the probability of detection in PDA can actually lead to better performance. Keywords: Tracking, filtering, Bayesian estimation, probability of detection, probabilistic data association (PDA). Introduction The probability of detection is an important parameter in many tracking algorithms. Each scan by the tracker can result in hundreds of detected returns. One of these returns may be a detection of a target, but the overwhelming majority will be false alarms originating from clutter that is characterized by the terrain, environment, and other sources of noise. The probability of detection θ is the probability that a return from the target will exceed a set threshold. If θ is overly optimistic, the tracker will accept returns from clutter as detections of the target and potentially lose the track. Likewise, if the parameter is pessimistic, the tracker will reject returns from the target and lose the track if the target makes a maneuver. In this paper, we propose a Bayesian estimation method for the probability of detection that fuses the stream of information from the tracker and takes into account prior information. In contrast to using an assumed constant value for the probability of detection, the proposed Bayesian estimation strategy continuously updates its estimate of the probability of detection θ and feeds it back into the tracker for improved performance. This estimation strategy can be used with any tracking algorithm that uses an estimate of the probability of detection, but experimentally we focus on the probabilistic data association (PDA) tracker []. PDA was chosen because of its ubiquitous use in industry and research alike. Mušicki demonstrated the potential sensitivity of the PDA algorithm with respect to the estimate of θ [2]. We show that averaged over 0,000 runs of a simulation, the average tracking time of a target and the localization error can be improved by estimating θ rather than using a fixed inaccurate value. Section 2 will review related work on estimating the probability of detection and how our proposed strategy differs from that work. Section 3. will derive the proposed estimation method while Sections 3.2 and 3.3 will give a brief review of PDA and show how the necessary information is extracted from PDA. Sections 4 and 5 will present and discuss the results from testing the algorithm in simulation. Finally, we will conclude with closing remarks and proposed future work in Section 6. 2 Related Work In practice, the probability of detection is the probability that the amplitude of a return from the target exceeds some detection threshold. The amplitude of a return is a function of a number of well-understood but difficult to model factors that may be time varying and hard to estimate. The most influential of these factors are environmental conditions, obstructing terrain, and the target strength of the target. Some efforts have treated θ as a random variable and used the sonar equation to estimate the expected value of θ, which is then used as the estimate for the tracker [3]. While theoretically satisfying, this method requires many variables of the scenario to be known and relies heavily on accurately modeling the environment. The probabilistic data association filter with amplitude information (PDAFAI) offers a method of estimating the probability of detection by fitting distributions to the amplitudes of targets and clutter []. The probability of detection is calculated by integrating the target distribution above the detection threshold. Other work has used these same distributions to directly set the threshold and determine θ [4]. Unfortunately, these distributions are disputed because they sometimes poorly fit observed data [5]. Also, the distribution used to model the amplitudes of returns from targets typically does not address the aspect de ISIF 64

2 pendency of the target and varying target strength [6]. There have been efforts made to incorporate aspect-dependent target strength information into the target distribution [7] but these attempts can run into problems parameterizing the distributions due to a lack of relevant data. Incorrectly parameterizing the distributions can lead to performance far worse than what could have been achieved without considering amplitude information at all. While amplitude information may be a valuable feature in determining θ, the contribution of this paper is to instead consider estimating θ based on feedback from the tracker. For the PDA tracker, that means the estimate is based only on the distance from the predicted location of a target return. Combining amplitude information with the proposed method is of interest for future work. 3 Estimating θ The proposed method forms a feedback loop where the tracker output is used to estimate θ after each scan, and then the estimated θ is fed back to the tracker to use at the next scan. First, we describe in Section 3. the proposed Bayesian posterior estimate of θ given a prior and the observed data from the target after k scans. The observed data is provided by a tracker; for the experiments we use the probabilistic data association tracker. Relevant aspects of the PDA tracker are reviewed in Section 3.2. In Section 3.3 we detail how the output from PDA is used to sequentially form the posterior estimates of θ. 3. Proposed Bayesian Estimation of the Probability of Detection We model the probability of detection θ [0, ] as a random variable Θ. We assume a prior distribution P (θ), and form the kth posterior P (θ X k ) after the kth scan based on the data X k received by the tracker up through the kth scan. Then before the k + th scan, we feedback to the tracker the current best estimate of the probability of detection, which we take to be the expectation of the kth posterior, E PΘ Xk [Θ]. Let M k {0,,..., k} be the random number of target returns within k scans. It is assumed that at most one return per scan can have originated from the target. The likelihood P (M k m θ) is binomial: ( ) k P (M k m θ) θ m ( θ) k m. m We assume the distribution P (M k m X k ) is output by the tracker based on its input data X k, and we take the prior P (θ) to be a beta distribution with parameters (α, β) because it is the conjugate prior for the binomial: Θ B(α, β) θα ( θ) β, where B(α, β) denotes the beta function evaluated with α,β. 642 Using Bayes rule and the assumption that M k is a sufficient statistic for X k, the derivation in Appendix A. shows that the kth posterior of θ given M k can be written as P (θ M k m, X k ) θm+α ( θ) k m+β B(m + α, k m + β). Marginalizing out the M k, the kth posterior is P (θ X k ) P (θ M k m, X k )P (M k m X k ) θ m+α ( θ) k m+β B(m + α, k m + β) P (M k m X k ). () After the kth scan, we feedback to the tracker an estimate that is the expectation of the uncertain probability of detection given its kth posterior, E PΘ Xk [Θ]. As detailed in Appendix A.2, E PΘ Xk [Θ] θp (θ X k )dθ, α + E[M k] k + α + β. (2) Next, we compute E[M k ]. Let Y k be a binary random variable indicating whether the target was detected on the kth scan. Then the total number of detections M k in k scans can be expressed recursively, { M k, given Y k 0 M k M k +, given Y k. Thus, E[M k ] E[M k ] + P (Y k ) P (Y n ). (3) n After each scan, we get P (Y k ) from the tracker, and use it to compute the new estimated probability of detection as per (2) and (3). 3.2 Brief Review of PDA PDA uses the theory of Kalman filtering to probabilistically associate returns with the single assumed target or with clutter []. The target state x, containing position and velocity, is described by a Gaussian distribution with mean ˆx k k and covariance P k k. The target s state evolves according to the linear transformation F with Gaussian noise ν k N (0, Q). That is, at time k + the state is described by ˆx k+ k Fˆx k k, (4) P k+ k FP k k F T + Q.

3 The measurement z k at time k is related to the target state by a linear transformation H with additive noise w k N (0, R) such that z k N (ẑ k, S k ) with ẑ k Hˆx k k, S k HP k k H T + R. Using the Gaussian assumption, an elliptical validation region for measurements is defined as (z k ẑ k ) T S k (z k ẑ k ) φ, where φ is the gate size. All measurements outside this validation region are discarded. Let r k be the number of returns from the kth scan and let z (i) k denote the ith measurement. PDA assigns z (i) k the probability b i that it originated from the target and all others came from clutter. Assuming a Poisson clutter model, b 0 is defined as the probability that none of the measurements within the validation region originated from the target. Then, 4 Simulations The performance and impact of the proposed estimation strategy of θ was evaluated by running Monte Carlo simulations for a maneuvering target in a variety of different clutter densities and choices of θ. 4. Simulation Set-up The target exhibits a trajectory with constant speed and constant non-zero acceleration as shown in Figure. The θ parameter is most critical in situations with a maneuvering target because the first-order dynamics modeled by F do not predict an accelerating target, thus the tracker is more reliant on the measurements than its internal dynamics. The proposed algorithm may be applied to a number of tracking scenarios and the simulation shown here does not attempt to reflect a specific tracking environment. where e 0 b 0 e 0 + r k, (5) j e j e i b i e 0 + r k, i, 2,..., r k j e j (i) e i N (z k ; ẑ k, S k ), ( ) e 0 λ θ P G, (6) λ is the clutter density, and P G is the probability that the true target return falls within the validation region. Once the b i s are defined, ˆx k k is found to be ˆx k k ˆx k k + K r k i b i (z (i) k ẑ k) (7) where K is a gain matrix. In practice, an estimate of θ is used in (6), and one sees from (6) and (7) that an inaccurate estimate can significantly affect performance. 3.3 Calculating P (Y k ) using PDA To form the kth estimate of θ to use in (6), P (Y n ) is needed for n,..., k in (3). As reviewed in Section 3.2, PDA assumes the association events are mutually exclusive, and thus the probability that at least one of the measurements originated from the target is ( b 0 ). We estimate P (Y n ) as P (Y n ) ( b 0,n ), (8) where b 0,n is b 0 from (5) calculated at the nth scan. This risks inaccurate feedback from the tracking filter, but our simulations did not show any unstable behavior. However the implications of this feedback should be studied further. 643 Figure : The true trajectory of the target is shown as the smooth arc in the 20x20 km search space. The dots are returns from both the target and clutter for one scan. The connected dots are the previous estimated target locations. Any measurements falling within the large oval are in the validation region. Here, the clutter density is the highest considered in the experiments. The target state x tracks two-dimensional position and velocity where it evolves as described by (4) where [ ] FT 0 F 0 F T and F T with state process noise covariance defined as [ ] QT 0 Q and Q 0 Q T σν 2 T [ ] T, 0 [ T 4 /4 T 3 /2 T 3 /2 T 2 Parameter T is the duration between measurements and in our scenario is set to a constant, T. The measurement covariance matrix is defined as ].

4 [ ] σ 2 R w 0 0 σw 2, where w k N (0, σ 2 w) was added to the target s true trajectory, σ ν σ w 0.. Figure shows the case where baseline clutter has been added to the scenario. Three clutter density levels were used; a baseline amount (shown in Figure ), two-thirds this amount, and one-third this amount. The track was initialized by two point initialization from the target s true trajectory plus the added Gaussian noise. The complete simulation was 50 scans long. 4.2 Simulated θ and prior For each scenario we fixed the true value of θ and iterated over different prior values of ˆθ to test how well the algorithm adjusted to tracking using an incorrect prior. We also varied the α and β parameters from the prior beta distribution used in (2) to see the effect on performance. The values of α and β are defined as α γ ˆθ β γ( ˆθ) (9) where γ is a precision parameter and ˆθ is the prior value of the probability of detection. A higher value of γ corresponds to a stronger prior. A baseline prior method using a fixed value of ˆθ is used to compare with the estimation method. The probability of detection for the scenario was implemented such that if the true θ was p, ( p) of the target returns were removed. This approach was used rather than simulating Bernoulli random variables with some probability p because this latter method would create variance on the true probability of detection each run. Variance on the true θ could potentially create bias on the algorithm s posterior estimate of θ due to it calculating statistics on more favorable draws from of the Bernoulli random variables. 4.3 Metrics Each metric analyzed is the result of 0,000 Monte Carlo runs. Three metrics are reported Expected track time Sum squared error () from true trajectory The posterior estimate of θ The expected track time is the mean amount of time the target was tracked before the simulation ended. A lost track is declared when the distance between the estimated target position and the actual target position exceeds a fixed threshold (equal to roughly four times the steady-state standard deviation). The posterior estimate of θ and the were calculated from 0,000 completed tracks and do not include runs if any of the compared methods lost the track. 5 Results In general, the disparity between methods is largest when the scenario is hard, that is, when lower true values of the probability of detection and higher clutter densities are used. An increase in clutter density for any given true probability of detection scales each method s scores such that the rank in which the methods perform does not change but the proportion by how much does. We present results in Figures 2 and 3 for the highest level of clutter density. Figure 2 shows the results for the three metrics using the true θ, the baseline where there is a fixed prior ˆθ, and the proposed estimated θ with three different strength prior distributions centered at ˆθ. The true probability of detection used for each row s results is stated to the left of each of the rows. The x-axis of each of the plots scales over different values of the prior ˆθ used in each of the methods. When θ ˆθ, the method using the true probability of detection and the baseline prior method are equal. The columns from left to right describe the results of expected track time,, and the final posterior estimate of θ. Figure 3 maintains the layout of Figure 2 but shows the comparison of results using feedback from PDA versus those using perfect feedback where P (Y k ) takes values of only or 0 with perfect accuracy. 5. The expected track time, found in column of Figure 2, shows that the baseline using a fixed prior value ˆθ falls off sharply for values of ˆθ away from the true value of θ. The proposed estimation method with a strong prior distribution (γ 00) exhibits performance correlated with using a fixed prior but for almost all scenarios and metrics is better by a small amount. Using a more moderate prior (γ 0) the method performs better in some places but worse in others. The performance of the estimation with the most adaptable prior (γ ) is mostly invariant to the prior ˆθ because it relies almost solely on the PDA output. However, being so reliant on the PDA output leads to poor performance for that method. Perhaps surprisingly, in the third-row of Figure 2, for θ, one sees that using an inflated value of the probability of detection outperformed using the actual probability of detection in terms of expected track time and. It is our hypothesis that this occurs because setting the probability of detection higher with PDA produces a smaller gate size, which means fewer returns will fall within the gate and PDA will be less confused by false contacts. This issue is more important when θ is low because there is in fact a higher relative number of false contacts. When the true probability of detection is raised to or any performance gain over using the correct probability of detection is within the variance of the error. 644

5 θ θ θ θ Baseline PDA Feedback γ PDA Feedback γ 0 PDA Feedback γ 00 Figure 2: Comparison of using the true probability of detection, the baseline prior, and the Bayesian estimates. Column : Expected track time over the scenario of length 50. Column 2: Sum squared error. Column 3: Final posterior probability of detection. 645

6 θ θ θ Perfect Feedback γ Perfect Feedback γ 0 Perfect Feedback γ 00 PDA Feedback γ PDA Feedback γ 0 PDA Feedback γ 00 Figure 3: Comparison of using PDA feedback (solid) versus perfect feedback (dotted). Column : Expected track time over the scenario of length 50. Column 2: Sum squared error. Column 3: Final posterior probability of detection. 646

7 5.2 Sum of Squared Error When looking at the scores in the results presented in Figure 2 the reader may wonder why the method using the true probability of detection is not getting consistent, flat scores across a scaling value of ˆθ. Because the method uses the true probability of detection regardless of the varying prior ˆθ this scaling parameter should have no effect on the score. However, it is important to consider that this metric was only calculated if all methods completed the run. All methods are more likely to complete the track if the random draws from the added noise distributions are more favorable, leading to easier tracking scenarios, which partly explains why the is lower when there is a large disagreement between the true and prior value of the probability of detection. However, this effect is confounded by the idiosyncrasies of PDA. For a given prior value the methods scores can be meaningfully compared. Across varying values of the true probability of detection, we can observe in the center column of Figure 2 that the has unmistakable structure. The baseline prior method, which uses the prior value ˆθ as a constant and is markerless in the plots, performs poorly relative to the other methods at the lowest and highest bounds of the scaling prior ˆθ. We might attribute this to the prior ˆθ simply being far from the true value of the probability of detection but even when the true θ and ˆθ the baseline prior is still a significant loser. Overall, the clear winners for are the Bayesian methods using values of γ equal to 0 and Final Estimates Notably, Figure 2 shows in column three that the final estimate of θ after the fifty scans consistently underestimates θ, even when the prior beta distribution is centered at the correct value. 5.4 Effects of Imperfect Tracker Feedback We hypothesized that the proposed method would work better if the tracker feedback was better. To test this, we compared using PDA to provide its estimate of P (Y k ) to using the feedback that would be given by a perfect tracker that returns P (Y k ) if the target was in fact present and P (Y k ) 0 if the target was not actually present (the tracker in the simulation is still the standard PDA however). The results are shown in Figure 3. First, notice that in the third column of Figure 3 the dotted lines show that the perfect-feedback posterior estimates are closer to the correct θ than the PDA feedback posteriors illustrated by the solid lines. In particular, if the beta prior distribution is centered on the correct θ value, the final perfect feedback estimate is correct, unlike when using the PDA tracker feedback. Despite the difference in final estimates, given strong priors (the purple and dark blue lines) there is little difference in the track times between the perfect feedback and the PDA feedback, as shown in column one of Figure 3. On the other hand, given a weak prior (shown as the light-blue lines), one sees that expected track time can be increased up to 4% 6 Conclusions and Open Questions The goal of the proposed Bayesian estimation strategy was to accurately estimate the probability of detection to improve tracking performance. Estimating the probability of detection did improve tracking performance if a strong prior was used (the beta prior with γ 00). We hypothesize that a more highly-maneuvering target or a higher clutter density would produce stronger benefits from adapting to the data. The most surprising result is that an overestimate of the probability of detection when θ yields improved performance over using the true θ. The assumption of this paper was that using the true value of the probability of detection in PDA is better than using an incorrect one. Because we found scenarios where this assumption does not hold, perhaps instead of trying to estimate the true probability of detection, we should shift our efforts to directly model the performance of PDA as a function of the probability of detection. In this paper we formed a Bayesian estimate of θ, but actually θ enters into the PDA formula as /θ. Assume for the simplicity of the following argument that P G. We implemented PDA s e 0 by substituting the Bayesian estimate E PΘ Xk [Θ] from (2) for θ in (6) to find e 0 λ( β+k E[M k] α+e[m k ] ). Alternatively, one could form a Bayesian estimate of /θ as E PΘ Xk [/Θ], which results in e 0 λ( β+k E[M k] α+e[m k ] ). Note that an inflated value of θ results in a smaller e 0, and as we just concluded above, improves performance. Because using the expectation of the inverse of θ results in a strictly larger value of e 0, we would expect worse performance. In fact, our results with that estimator (not reported in this paper due to lack of space) confirm this hypothesis. This is further evidence to suggest that the implementation of the probability of detection in PDA may be suboptimal. The probability of detection is often used as a parameter in track-management algorithms. If θ was estimated inaccurately due to poor feedback and used in track-management algorithms, false tracks may persist longer than they would have if a fixed θ was used. While we recognize the potential to design a better measure of track quality by estimating θ, we leave the problem of how to implement it in a robust way as an open question. In the derivation of our Bayesian estimation strategy we assumed the probability of detection to be a constant, but in practice the probability of detection is likely to be timevarying. A natural extension of this work would be to model the probability of detection as a time-varying parameter based on the environment and the varying target strength. Much of the related work in estimating θ has focused on parametric models of the distribution of the amplitudes from returns. How to integrate that approach and the estimation 647

8 method presented here is an open question, but one that we believe deserves further study. Acknowledgment This work was funded by the U.S. Office of Naval Research, Contract Number N G-0460 and by an U.S. Office of Naval Research YIP Award. References [] T. Kirubarajan and Y. Bar-Shalom, Probabilistic data association techniques for target tracking in clutter, Proceedings of the IEEE, vol. 92, no. 3, pp , Mar [2] D. Mušicki and X. Wang, Reliability of PDA based target tracking in clutter, in Proceedings of the Seventh International Conference on Information Fusion, Per Svensson and Johan Schubert, Eds., Mountain View, CA, Jun 2004, vol. II, pp , International Society of Information Fusion. A. Posterior of θ given M k P (θ M k m, X k ) P (θ M k m) P (M k m θ)p (θ) P (M k m) C P (M k m θ)p (θ), ( ) k θ m+α ( θ) k m+β C B(α, β) m where the normalization constant C can be simplified: C P (M k m θ)p (θ)dθ ( ) k θ m+α ( θ) k m+β dθ θ0 B(α, β) m ( ) k B(m + α, k m + β). B(α, β) m Combining terms, P (θ M k m, X k ) θm+α ( θ) k m+β B(m + α, k m + β). A.2 The Expectation of the Posterior [3] L.D. Stone. and B.R. Osborn, Incorporating performance prediction uncertainty into detection and tracking, U.S. Navy Journal of Underwater Acoustics, Dec [4] S.B. Gelfand, T.E. Fortmann, and Y. Bar-Shalom, Adaptive detection threshold optimization for tracking in clutter, IEEE Transactions on Aerospace and Electronic Systems, vol. 32, no. 2, pp , Apr 996. [5] D.A. Abraham and A.P. Lyons, Novel physical interpretations of k-distributed reverberation, IEEE Journal of Oceanic Engineering, vol. 27, no. 4, pp , Oct [6] D.M. Drumheller, The bistatic simple integrated structure (basis) model, Tech. Rep. FRL/FR-MM/ , Naval Research Laboratory, May [7] J.W. Pitton and W.L.J. Fox, Incorporating target strength into environmentally-adaptive sonar tracking, OCEANS Europe, pp. 5, June A Appendix This appendix contains the complete derivations of the paper s results. 648 E PΘ Xk [Θ] θp (θ X k )dθ, then substituting from () θ P (M k m X k ) θm+α ( θ) k m+β dθ B(m + α, k m + β) θ m+α ( θ) k m+β dθ P (M k m X k ) B(m + α, k m + β) B(m + α +, k m + β) P (M k m X k ) B(m + α, k m + β) ( ) Γ(k + α + β) P (M k m X k ) Γ(m + α)γ(k m + β) ( ) Γ(m + α + )Γ(k m + β) Γ(k + α + β + ) m + α P (M k m X k ) k + α + β P (M k m X k )(m + α) k + α + β ( ) α + P (M k m X k )m k + α + β α + E[M k] k + α + β.