Optimal Signal Detection for False Track Discrimination

Transcription

1 Optimal Signal Detection for False Track Discrimination Thomas Hanselmann Darko Mušicki Dept. of Electrical an Electronic Eng. Dept. of Electrical an Electronic Eng. The University of Melbourne The University of Melbourne Parkville, VIC 3 Parkville, VIC 3 Australia Australia t.hanselmann@ee.mu.oz.au.musicki@ee.mu.oz.au Abstract Optimal signal etection for false track iscrimination is etermine by simulation using the Integrate Probabilistic Data Association (IPDA) algorithm. The IPDA algorithm is an efficient probabilistic ata association algorithm with estimates of target existence probabilities that can be use to istinguish true an false tracks. The rate of confirme false tracks is hel constant an the optimal signal etection probability is etermine by the maximum confirme true tracks for a given signal to noise ratio. This allows the etermination of the optimal etection probability which are reporte for a range of signal to noise ratios. Keywors: Optimal signal etection probability, tracking, IPDA, track existence, false track iscrimination. Introuction In tracking, ata association is the most essential part because it greatly etermines the performance of a tracking system. Measurements of uncertain origin have to be associate with tracks of real targets, as well as false tracks which o not follow targets. The tracking system has to establish tracks using available measurements. However, as the existence of the targets behin the tracks is unknown, this problem setup is of stochastic nature. Therefore, it is of avantage to introuce probabilities of existence of tracks or equivalently targets, estimating the target track probability of an assume track to originate of a true, real existing target. This probability is calle probability of target existence. This probabilistic iscrimination of tracks is integrate with a probabilistic ata association algorithm (PDA) an is calle integrate PDA (IPDA). The probability of existence is recursively estimate base on new measurements an their likelihoos of belonging to the track uner investigation. Base on the Bayes-Chapman- Kolmogorov equation, posterior probabilities are calculate from the prior probabilities an the new measurements. IPDA nicely treats also the possibility of not having a target as well. This is one in conjunction with a Markov moel for target existence propagation. There are two moels: Markov chain One an Markov chain Two. The first one is a special case of the secon where target existence is equal to the target being etectable, whereas in the secon moel they are istinct. In the esign of tracking systems for raar or sonar, the probability /5/$. of etection is a central 5 parameter. IEEE The probability of etection shoul be low enough to capture true targets but high enough to prevent overwhelming numbers of false etection. There has been some investigation into the etermination of the optimal etection threshol for target tracking applications. Li an Bar-Shalom have evelope a hybri conitional average algorithm [] to etermine a temporal an spatially constant probability of etection. Similarly, Hong an Evans have evelope a theoretical approximation framework to estimate the probability of etection [], assume to be constant over space an time. Willett et al. [3] has introuce some feeback from the PDA filter to the etector to allow for a temporary an spatially varying threshol, integrating Bayesian etection base on prior probabilities of expecte location an covariance of target measurements. While the first two publications are ealing with no a priori knowlege (all measurements are treate equally), the latter one is changing the tracking setup by incorporating knowlege base on measurements. This may be beneficial in some applications but in general varying clutter istributions might introuce some bias. A more etaile iscussion of the first two publications is one in section 5. In this paper Monte Carlo simulations, base on the stanar IPDA algorithm [4], are use to etermine the optimal probability of etection, P, as a temporally an spatially constant threshol. P is a function of the signal to noise ratio an the clutter ensity (the probability of false alarm). To compare the true tracks statistics the confirmation threshol is chosen such that the rate of confirme false tracks scans is about one in 5, which seems reasonable for practical applications. Notation an mathematical basics for IPDA. Data preprocessing an etection thresholing Fig. 3 shows the signal flow in the experiment. The IPDA algorithm oes the tracking part an elivers probabilities of target existence which are then use in true track etection moule to obtain confirme tracks. The probability of etection P is a consequence of signal processing which generates measurements when the

2 signal amplitue is higher than the threshol in a har limiter. Thus, the probability of target etection is correlate with the probability of false alarm or clutter measurement generation. Fig. shows the classification mae in the threshol etection unit into noise (= negatives ) an measurements (= positives ) with respect to the amplitue threshol X t corresponing to a probability of etection P. Base on the threshol an the two pfs, four probabilities can be calculate. The two accoring to the positive classification, which generates measurements for the tracking stage, are of particular interest. These are the probability of true accepte track measurements, i.e. the probability of etection, P r(t P ) = P (TP=true positives), an the probability noise signals regare as track measurements, i.e. the probability of false alarm, P r(f P ) = P fa (FP=false positives). Fig. : Analog amplitue pfs for target signal an noise signal, pf(x noise) an pf(x signal&noise), respectively.. Tracking The measurements will then be valiate if they are insie some gating winow etermine by the IPDA algorithm on a track by track basis. These valiate measurements are then associate with corresponing tracks. Base on a Markov chain moel an Bayesian inference the probability of target existence is calculate for these tracks which are then classifie as confirme or terminate tracks, epening whether the probability of target existence excees a confirmation threshol T c or falls below a termination threshol T t, respectively. In an ieal situation the remaining tracks stay unconfirme an over time their probability of existence will waner towars one or zero epening on whether it is a true or false track. In practice some of the confirme tracks are false tracks that got misclassifie ue to the overlapping probability istributions, see Fig. 4. A receiver operating curve (ROC) with the confirmation threshol as threshol parameter can be use to etermine the performance. Fig. shows the ROC with two amplitue threshol values X t an X t with corresponing etection probabilities P an P. The area uner the bent curve is the figure of merit an the larger it is the better the performance. Ieally the probability of confirme false tracks, P r(cf T ), shoul be zero an the probability of confirme true tracks, P r(ct T ), shoul be one. The figure of merit is then, whereas the iagonal line correspons to a figure of merit of.5, inicating a track confirme as a false track woul be as likely as a track confirme as true track, an hence a ranom classification, as obtaine by throwing a fair coin, woul take place. The following notation is use: SNR B SNR P P fa ρ fa V rc signal to noise ratio in B signal to noise ratio probability of etection probability of false measurement clutter ensity resolution cell volume Fig. : Receiver Operating Curve (ROC). This gives the relations: SNR = SNR B () P fa = P +SNR () ρ fa = P fa /V rc (3) V rc = x y (4) x = 3σ x (5) y = 3σ y (6) σ x = σ y = 5m (7) where the relation () is given in [5, 6]. x an y are approximate sie lengths of the resolution cell area, assuming equal istributions x an y within the cell area, yieling a variance of x y an, respectively. Equating these expressions with the zero-mean Gaussian observation noise variances σx an σy, yiels expressions (5) an (6). Fig. 4 shows how tracks are confirme base on the Monte Carlo simulation. The number of confirme false tracks is evaluate an the confirmation threshol T c is varie until the sum of confirme false tracks ivie by the number of scans an runs is roughly one out of 5, see Fig. 9. The target motion moel is: x k+ = F x k + ν k (8) x k = [x, ẋ, y, ẏ] T (9) [ ] [ ] FT T F =, F F T = () T where T is the sampling time, (x, y) an (ẋ, ẏ) are the posi-

3 where ˆx k k = ˆx k k, ˆP k k = ˆP k k an ata association probabilities are given by: Fig. 3: Processing flow from raw ata to true track etection with the IPDA algorithm. β (k) = P P G δ k () β i (k) = P V P k G ˆm k, i =,.., m k () δ k { P P G : m k = δ k = ) () P P G ( V k ˆm k Λ i k : m k Λ i k = N (z P k; i ẑ k, Ŝk), { G i =,.., m k (3) k ˆm = : m k = m k P P G P {χ Z k } : m k (4) Fig. 4: True track (TT) an false track (FT) histogram statistics as a function of the probability of track existence P {χ} at some time t. tion an velocity coorinates, respectively. ν k is zero-mean white Gaussian noise with known variance E[ν k νl T ] = Qδ(k, l) () [ ] [ ] T QT 4 T 3 Q = q, Q Q T = 4 () T with q =.75. The observation moel is given by: T 3 T z k = Hx k + w k (3) [ ] H = (4) w k is zero-mean white Gaussian noise with known variance: E[w k wl T ] = Rδ(k, l) (5) [ ] σ R = x σy (6) The Markov chain moel one for the probability of target existence conitione on the set of observations Z k of all observations zk i, i =,.., m k mae at previous times k =,.., l is as follow: [ P {χk Z k } P {χ k Z k } ] = [.98. ][ ] P {χk Z k } P {χ k Z k (7) } with initial probability of P {χ Z } = P {χ } =. use in the two-stage track initialization run at each scan. State an covariance estimates ˆx k k an ˆP k k are given by the stanar PDA algorithm: m k ˆx k k = β i (k)ˆx i k k (8) i= m k ˆP k k = β i (k)( ) ˆP i k k + (ˆx i k k ˆx k k)(ˆx i k k ˆx k k) T (9) i= where m k is the number of valiate observations z i k ; P an P G are the probability of etection an gating valiation, respectively. The probability of target existence upate is etermine by: P {χ k Z k } = δ k δ k P {χ k Z k } P {χ k Z k }(5) in conjunction with the Markov upate (7). Estimation an preiction are obtaine by the stanar Kalman filter: Ŝ k = H ˆP k k H T + R (6) K k = ˆP k k H T Ŝ k (7) ˆx i k k = ˆx k k + K k ( z i k H ˆx k k ) (8) ˆP k k = (I K k H) ˆP k k (9) ˆx k+ k = F ˆx k k (3) ˆP k+ k = F ˆP k k F T + Q (3) 3 Experiment The objective of the experiment is to etermine the probability of etection for ifferent signal to noise ratios. The experiment consists of optimizing the performance evaluation of confirme tracks for a given signal to noise ratio, or equivalently, for a pair of (P, P fa ), using the IPDA algorithm with Markov chain one with transition probabilities as given above. An automatic initialization of tracks is use that is base on two successive scans (two point ifferencing). At every scan new tracks are initialize for all possible measurement combinations of any two measurements from ifferent scans, if the absolute measurement ifferences of such a combination fall below a certain threshol (maximum target spee). All of these tracks are assigne an initial probability of target existence. During a simulation run this probability is normally increase for true tracks by the valiation of later measurements, while for false tracks it will ecrease on average. If it rops below a termination threshol the track will be terminate. If the probability of target existence excees a confirmation threshol, the track becomes a confirme track.

4 Depening on whether a track stems from a true target or just measurement noise, they are classifie as true or false tracks, respectively. This leaves four possibilities, confirme true an false tracks (CTT an CFT) an terminate true an false tracks (TTT an TFT). Naturally, the goal is to have as many confirme true tracks with a low limite number (ieally zero) of confirme false tracks an no terminate true tracks. Because of the stochastic nature of the measurements, the associate pfs on the numbers of true an false tracks have some variance, are overlapping an are functions of unerlying parameters. This paper concentrates on the probability of etection, P, an the clutter measurement ensity, ρ fa. The initial target existence probability, P {χ } =., an the termination threshol, T t =., were chosen to be of reasonable value from experience base on experimental optimization [7] but hel constant here. The search space of the probability of etection, P, is limite to the interval [P min, ] with P min =.65 because for lower P a multistage initialization woul be neee to capture true tracks reliably with the automatic track initialization proceure. 3. Algorithm The algorithm is state in the following pseuo coe. For each signal to noise ratio a probability of etection P was teste by etermining the corresponing false alarm clutter ensity. Values that gave too heavy or almost no clutter were iscare to save calculation time. Then the optimal confirmation threshol Tc opt was etermine such that the rate of confirme false tracks was hel approximately constant at a rate of one in 5. Base on those values, the confirme true track statistics were recore an the optimal probability of etection etermine as the probability corresponing to the maximum number of confirme true tracks. For SNR B = : 3 : 9, SNR = SNR B ; For P = :. : P min, P fa = P +SNR ; ρ fa = P fa /V rc ; if ρ fa > 3 4 % too heavy clutter continue; en if ρ fa < 6 % almost no clutter continue; en en en P opt % etermine optimal Tc opt, such that % rate of CF T percent % T opt c = Tc opt (SNR, P ), SNR) % Recor CT T (P, T opt c (SNR) = arg max P {CT T (P, T opt, SNR)} c Further a limit for T cmin =.5 has been set ue to the inaccuracies cause by the reuce number of Monte Carlo runs ( to 5) to evaluate temporary points CT T (P, T c, SNR) in the optimization process. Linear regression was use to preict the next optimization point within a neighborhoo of the current point. For lower clutter ensities ρ fa < 4 5 m, Monte Carlo runs were use. The temporary reuction in Monte Carlo runs was use to save time as the whole simulation took several ays on a Pentium IV with 3GHz, using MATLAB Final results for Fig. 6 were run with 5 runs with parameter values obtaine by algorithm Results The simulation was one for five ifferent signal to noise ratios, from 9B to B in steps of 3B. The relationships (-4) were use to fin the optimal pairs of etection probability, P, an clutter ensity, ρ fa. Fig. 5 shows the main result, the probability of etection as a function of the signal to noise ratio P (SNR). With optimal parameters P an T c achieve by algorithm 3., the confirme true track statistics, using 5 runs to obtain smoother statistics, is given by Fig. 6. Clearly, for low signal to noise ratio (SNR=9B) an the associate low probability of etection, true tracks cannot be capture efficiently anymore. This is partly ue to the lack of a multistage initialization process. Fig. 7 shows the total sum of confirme true tracks as a function of the signal to noise ratio, where the summation is taken over all the scans () an runs (5). Fig. 8 shows a problematic issue ue to the exponential epenency on the signal to noise ratio, the result for ρ fa is much more sensitive to the Monte Carlo noise than it is for P. See also Fig., where it is evient that varying P slightly, gives a far larger eviation in ρ fa, especially for high SNRs. Therefore, a refine optimization might be one on a finer gri to try to obtain more accurate values. P opt (SNRB ) SNR B Fig. 5: Optimal probability of etection P opt (SNR B ) as a function of the signal to noise ratio. Thanks to the unknown reviewer for pointing out that rather then refining the gri, importance sampling woul be a more efficient way to save computational power, see e.g. [8].

5 4 B 8B 5B B 9B 9 x 5 8 number of confirme true tracks within 5 runs opt ρ (SNRB fa )/m Scans Fig. 6: Confirme true track statistics for ifferent SNRs for optimal P opt. Fig. 8: Optimal clutter ensity for a given signal to noise ratio. SNR B. x sum(ctt).6.4. Confirme false track rate SNR B Fig. 7: Sum of confirme true track scans. 5 Comparison with other work In [] a technique to etermine the etection threshol (P ), base on a hybri approach that takes care of both continuous an iscrete uncertainties. The continuous-value uncertainties are ue to target motion an their measurements an the iscrete uncertainties stem from the origin of the measurements ue to imperfect target an false etections. This le to the introuction of a hybri conitional average (HYCA) algorithm [9, ]. A cumulative probability of track loss, P T L (k) is use to specify the probability of track loss at a time τ < k. Some results for ifferent P with non-optimal CFAR are reporte. Base on P T L (k) the average track life-time τ can be calculate. It was observe that for a given CFAR higher SNRs lea to a longer track life-time but that the stability of PDA is far more sensitive to P fa, especially if P fa is not very small. This observations lea to the efinition of a track quality measure base on position estimation which was minimize in orer to obtain an optimal etection threshol, P. The obtaine optimal values of the track-quality measure turne out to be stable (not varying with time) which only makes sense when the track-life time is long enough to avoi target track loss. Interestingly, as in our simulation, that optimal P fa (or corresponing ρ fa, see Fig. 8) values are in a similar range an o not influence the optimization criterion a lot. Therefore, a compromise CFAR setting base on a reasonable P fa can be obtaine for a range of SNRs. The authors of [] also emphasize SNR B Fig. 9: Confirme false track rate as a function of the signal to noise ratio. It is kept approximately constant at 5. the use of other optimization criteria for etection threshol selection, such as the probability of confirming a true target track or a false track an the probability of track loss. The first objective has been aresse in this paper while the secon one was aresse in []. In the latter an optimization criterion of J = c P T L +c K k= f( P (k k) with weighting constants c an c balancing between the cumulative track loss probability an a function of the approximative covariance matrix, which is base also on a HYCA approach. The avantage of the HYCA approach is to avoi the costly Monte Carlo simulations, on the other han there are some questions on how accurately these various moelings are. The current research begun in this paper tries to establish links between various performance measures an theoretical moels with simulation results obtaine by Monte Carlo methos. Sometimes, theoretical values might ivert substantially from experimentally etermine values, as was notice in [7] where initial track existence probabilities an confirmation an eletion threshols base on theoretical values for IPDA from [] lea to excessive number of false confirme tracks. 6 Conclusions In this paper the optimal probability of etection, P was etermine for ifferent signal to noise ratios an a confirme false track scan rate of about one in 5 by Monte

6 . x 4.8 B 8B 5B B 9B The Commonwealth Government of Australia an with aitional support by the Australian Research Council Linkage grant. Many thanks go also to Erik Blasch for the helpful suggestions to improve the paper. ρ fa Fig. : Regions where a finer gri might be use after values for P opt (re circles) have been etermine by algorithm 3.. Carlo simulation for the IPDA algorithm. While some parameters like P can be etermine relative accurately, other more sensitive parameters like ρ fa or T c for low SNRs, woul nee far more Monte Carlo runs to smooth out the statistics reliably. But then again, these parameters are not of primary interest, whether there are exactly one in 5, one in 4 or one in 6 confirme false track scans oes not really matter in practice. Outlook: If the signal to noise ratio is known, or otherwise it coul be estimate in the Detection Threshol unit as ŜNR = E[X X>X t ] E[X X<X where X t] t is etermine by P, the clutter ensity ρ fa can be estimate an use in more avance IPDA algorithms, like IPDA-MAP [7] for improve tracking performance. These algorithms then coul be use to get a better estimate of P. This shoul certainly be the case if ρ fa is slowly varying because then it coul be smoothe by a low-pass filter for a better average estimate. The spatial istribution coul be taken into account as well. When there are significant ifferent clutter ensities at ifferent locations, it woul be worthwhile to have also ifferent optimal P opt fe back to the ecision threshol unit. The simulation results obtaine off-line in this paper coul give a static guie of how to choose the optimal P opt which than coul be aapte on-line by performance analysis of the on-line IPDA algorithm, as outline in Fig.. P opt References [] X. R. Li an Y. Bar-Shalom. Detection threshol selection for tracking performance optimization. IEEE Transactions on Aerospace an Electronic Systems, 3(3):74 749, July 994. [] Sun-Mog Hong an R. Evans. Optimization of waveform an etection threshol for range an range-rate tracking in clutter. submitte for publishing to IEEE Trans. on Aerospace an Electronic Systems. [3] Peter Willett, Ruixin Niu, an Yaakov Bar-Shalom. Integration of bayesian etection with target tracking. IEEE Transactions on Signal Processing, 49():7 9, January. [4] D. Mušicki, R.J. Evans, an S. Stanković. Integrate probabilistic ata association (ipa). IEEE Transactions on Automatic Control, 39(6):37 4, 994. [5] C. Rago, P. Willett, an Y. Bar-Shalom. Detection-tracking performance with combine waveforms. IEEE Transactions on Aerospace an Electronic Systems, 34():6 64, April 998. [6] H.L. Van Trees. Detection, Estimation an Moulation Theory, Part III. John Wiley, John Wiley, New York, 97. [7] D. Mušicki an R. Evans. Clutter map information for ata association an track initialization. IEEE Transactions on Aerospace an Electronic Systems, 4(): , April 4. [8] M. Arulampalam, Sanjeev, R. Evans, an Khale Ben Letaief. Importance sampling for error event analysis of hmm frequency line trackers. IEEE Transactions on Signal Processing, 5():4 44, February. [9] X. R. Li an Y. Bar-Shalom. Stability evaluation an track life of the paf for tracking in clutter. IEEE Transactions on Automatic Control, 36(5):64 69, May 99. [] N. Li an X. R. Li. Tracker esign base on target perceivability. IEEE Transactions on Aerospace an Electronic Systems, 37():4 5,. Fig. : Moifie processing flow from raw ata to true track etection with avance IPDA algorithms, e.g. IPDA- MAP. Acknowlegment This research has been supporte by the Cooperative Research Centre for Sensor Signal an Information Processing uner the Cooperative Research Centre scheme fune by