Optimal Fusion Performance Modeling in Sensor Networks

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1 Optimal Fusion erformance Modeling in Sensor Networks Stefano Coraluppi NURC a NATO Research Centre Viale S. Bartolomeo La Spezia, Italy coraluppi@nurc.nato.int Marco Guerriero and eter Willett ECE Department University of Connecticut Storrs, CT 06269, USA marco.guerriero@engr.uconn.edu willett@engr.uconn.edu Abstract This paper studies sensor network surveillance performance at the automatic tracker output. In particular, we develop a simple analytical model for tracker performance, where the interest is in a compact representation of the impact of sensor revisit time. This model, combined with a previously developed contact fusion model, allows for an analysis of two fusion architectures: a standard centralized tracking configuration and a fuse-before-track two-stage approach. We find that the latter architecture is promising, though in the specific instantiation considered here there are performance limitations when targets are in close proximity. Additionally, in the context of centralized tracking, we study optimal sensor thresholding and sensor network size for a given false track rate. Keywords: Sensor Networks, Sensor Fusion, Target Tracking, erformance Modeling. Introduction A simple approach to static fusion known as contact sifting is investigated in [], including analytical expressions for fusion performance that are validated through simulations. Alternative approaches to static fusion are investigated in [2]. Tracker performance modeling is addressed at length in [3-4]. Extensions that address target fading effects and distributed tracking architectures are in [5-6]. For our purposes here, we use a simple tracker performance model that identifies a compact relationship between scan rate and performance. Leveraging the contact fusion and tracker models, we compare the performance of centralized (single-stage) tracking with a two-stage fuse-before-track approach that concatenates contact sifting and tracking. Future work will include a simulation-based comparison of the two architectures. The paper is organized as follows. Section 2 presents a simple tracker performance model; analysis based on this model is in section 3. Contact fusion performance modeling is summarized in section 4. Section 5 provides a comparison of centralized tracking with the concatenation of contact fusion and tracking, which we term fuse-beforetrack. Section 6 addresses parametric and architectural issues for optimal centralized tracking. Section 7 includes final remarks and directions for future work. 2 Tracker Model Modeling parameters: Target: nearly constant position in 2D with maneuvering index q [m 2 s - ]; fixed target SNR; Sensor: scan every t sec; positional measurements with covariance R; surveillance region of size A [m 2 ], detection cells of size C [m 2 ], detection threshold DT; Tracker: declare track on N consecutive associated detections, terminate track on K consecutive coasts (missed updates), association probability gate G and gating parameter γ (with 2D measurements, γ = 9. 2 corresponds to = details in [3]). G The following derived quantities are of interest. Detection probability D : False alarm density [m -2 ]: False alarm rate [hr - ]: DT D = exp (3.) + SNR ( DT ) λ = exp (3.2) C 3600 λa FAR = robability of correct association: ( ) (3.3) CA = exp λv (3.4) 574

2 / 2 V = π S (3.5) S = ( ) + R (3.6) ( ) = ( + ) + Q ( + ) = ( ) ( ) ( ( ) + ) R ( ) (3.7) q 0 Q = 0 q (3.8) (3.9) time. It is easy to show (by linearity of the expectation operator) that equations ( ) hold. No active track N ( ) t N U Active track M robability of track update and miss (i.e. track coast): N U D M M K = (3.0) U D G M U CA Average track confirmation time : = (3.) D K M K ( ) t Average track hold time: τ C = N N + (3.2) D U τ H = K K + (3.3) M Track : T D robability of false update: robability of false track: M False track rate [hr - ]: τ H = (3.4) τ + τ C H ( ) FU = exp λv γ (3.5) / 2 Vγ = γπ S (3.6) = K FT FU (3.7) 3600 FTR = FT λa (3.8) An illustration of the Markov chain model that corresponds to the modeling above is given in figure 2.. Note that equations ( ) both rely on nested geometric probability distributions that is to say, the time prior to tentative track initiation has a geometric distribution, as does the total track initiation Note that the expected value of the geometric distribution with parameter p is given by /p. Fig. 2.. Markov chain model for tracker logical state, in the target present case. (A similar Markov chain applies to the target absent case.) 3 Tracker erformance Analysis We are interested to examine input and output performance curves (FAR vs. D, and FTR vs. T D, respectively) as a function of the detection threshold DT, and as a function of the number of sensors. The latter can be addressed by setting the scan rate to t = / Z, where Z is the number of equally-performing sensors and t is the single-sensor rate. arameters are set as indicated in table 3., and performance curves are in figure 3.. (Note that by object we mean either contact or track.) Table 3.. Model simulation parameters. arameter Setting Maneuverability index q 00 m 2 s - Target amplitude SNR 0dB Scan interval t 60 sec Measurement covariance 00 0 matrix R 0 00 m 2 s - Surveillance region A 0 8 m 2 Detection cell C 00 m 2 Detection threshold DT dB Track initiation N 3 Track termination K 3 Association gate γ 9.2 Gate probability G 0.99 These curves are qualitatively similar to the Monte Carlo-based tracker performance curves documented in [7]. Key conclusions: With a low constraint on false object rate, best to use few sensors; 575

3 With a larger constraint on false object rate, best to use more sensors; For any given number of sensors, sweet-spot phenomenon wherby track-level detection performance decreases at lower detection thresholds sensor (contacts) sensor (tracks) 0 sensors (contacts) 0 sensors (tracks) 00 sensors (contacts) 00 sensors (tracks) Fusion erformance Curves 2 ~ ξ D = D exp (3.2) 2 B ξ = (3.22) πr r 0 R = (3.23) 0 r 5 Centralized Tracking vs. Fuse-before- Track We are interested to compare two fusion architectures: centralized tracking and fuse-before-track. Both architectures are illustrated in figure false object rate (per hour) Fig. 3.. erformance curves for several network-size assumptions. 4 Contact Fusion Model The system operating characteristics (SOC) curve introduced in [] is a generalization of the receiver operating characteristics (ROC) curve, in which detection performance relies on a localization criterion. The principal advantages of SOC-based performance characterization are two-fold: The coupling of detection and localization provides an operationally-relevant performance assessment; A direct comparison of performance at the input and output of contact fusion is achieved, with precisely the same performance criteria. Here, we will not focus on the distinction between probability of detection and probability of localization as defined in []. Rather, we will compare single-sensor and multi-sensor SOC performance curves so as to ascertain the benefit of contact fusion in terms of target SNR. Analytical expressions for the performance of the contact sifting approach to contact fusion are as follows, where B is the size of the contact sifting cells and J is the contact sifting threshold: L Z Z ~ j ~ ( J, Z ) = D ( D ) + J j= 0 FAR j= J Z ~ j L j j D ( J Z ) Z j i ~ Z j ( ) ( λb) D j= J i= J j ( ZλB) j i! exp ( λb) (3.9) A, = exp( ZλB) (3.20) B j! Multi-sensor contacts Fig. 5.. Candidate fusion and tracking architectures. In particular, consider centralized tracking performance for a wide range of numbers of sensors (figure 5.2), for a DT range of dB. 0.3 Contact fuser sensor 0 sensors 0 2 sensors 0 3 sensors 0 4 sensors Tracker Centralized tracking Fused contacts Fusion erformance Curves Tracker Fuse-before-track false object rate (per hour) Multi-sensor tracks Multi-sensor tracks Fig Contact-level performance given by continuous lines, track-level performance given by dotted lines. For a relatively small false object rate (00 per hour), track consistently degrades with increasing number of sensors. Indeed, as noted previously, the benefit of larger sensor numbers is observed at higher false object rates; in fact, there is no discernible benefit for very larger sensor numbers. For the operating point of interest (00 false objects per hour), can 576

4 we do better than one-sensor centralized tracking? We address this issue with the parameters identified in table 3., to which we add the parameters in table 5.. Table 5.. Additional model simulation parameters. arameter Setting Contact sifting cell size B 400 m 2 Number of sensors Z 0 Contact sifting threshold J 3 Figure 5.3 shows two families of curves: single-sensor SOC curves (in blue), and fused-contact SOC curves (in dotted black); each set of curves corresponds to 0dB, db, and 2dB targets. A precise analysis is difficult; nonetheless, we argue that 0dB multi-sensor performance is comparable to db single-sensor performance. probability of localization dB target multi-sensor 0dB ~ single-sensor db number of false contacts Fig dB-2dB targets: single-sensor SOC curves (blue) and multi-sensor SOC curves (dotted black). Note that the single-sensor SOC curves are significantly impacted by the relatively small contact-sifting cells. Indeed, significantly improved SOC curves can be achieved by considering larger contact sifting cell sizes; the limitation of this approach is an inability to handle multi-target cases with targets in close proximity. Next, we consider the impact of db target performance curves on tracker performance (figure 5.4). We see that the fuse-before-track performance curve outperforms both single-sensor and ten-sensor centralized tracking, for all detection thresholds. While the fuse-before-track result is encouraging, we caution that it is based on a simple tracker performance model: validation of these results with simulated sensor data is required. Further, we emphasize again that a significant limitation of the contact-sifting paradigm is an inability to handle targets in close proximity. An algorithmic extension to include an estimate of the number of targets in each sifting cell may partially alleviate this difficulty. Alternatively, more powerful contact-fusion approaches may be considered [2]. 0.3 sensor 0 sensors 0 2 sensors 0 3 sensors 0 4 sensors 0 sensors FbT Fusion erformance Curves false object rate (per hour) Fig Centralized tracking performance curves, and fuse-before-track (FbT) result for the ten-sensor case. 6 Optimal Detection Threshold and Number of Sensors Another approach to improving centralized tracking performance that does not rely on a two-stage architecture is to optimize the local sensor detection threshold (DT) as well as the number of sensors to be processed (Z), as a function of a constraint on the false track rate. We anticipate that this will lead to a performance curve that is the envelope of the family of curves shown in figure 5.2. Assuming that the measurement covariance matrix R is diagonal with element r > 0, it can easily be shown that the solution to the discrete algebraic Riccati equation ( ) is given by the following: p 0 0 ( ) = p 2 (3.24) q + ( q) + 4rq p = (3.25) 2 Hence, it follows from equations ( ) that ( ( ( p r) )) CA = exp λ π + (3.26) For a given FTR, we wish to optimize the local sensor detection threshold (DT) as well as the sensor scan rate t (from which we infer the number of sensors). This optimization problem can be recast as the following constrained maximization problem: (, DT ) (, DT ) = α max D T, DT (3.27) s. t FTR This optimization problem does not lend itself to an analytical solution. Using the same parameter settings given in table 3., the solution to equation (3.27) leads as 577

5 expected to the envelope of the family of curves in figure 5.2; this is illustrated in figure Fusion erformance Curves sensor 0 sensors 0 2 sensors 0 3 sensors ERFORMANCE CURVE false track rate FTR (per hour) Fig erformance curve obtained solving the optimization problem (3.27). It is instructive to examine the optimal scan interval opt t and the optimal detection threshold DT opt as a function of FTR; these are illustrated in figures opt Finally, figure 5.8 illustrates the variation in both t and DT opt as a function of FTR. It is interesting to note that, with increasing FTR, the optimal T D is achieved with a reduction in both t and DT: we both increase the number of sensors and lower the detection threshold. t opt false trackrate FTR (per hour) Fig Optimal scan interval (inversely proportional to number of sensors) as a function of FTR. DT opt in db false track rate FTR (per hour) Fig Optimal detection threshold as a function of FTR. Note that our analysis has been limited to the assumptions of equally-performing sensors with identical detection thresholds. Relaxing either of these assumptions introduces the need for a more complex tracker performance model, an interesting topic of future research. 7 Conclusions The intuitive assumption that an increased number of likeperforming sensors necessarily leads to improved surveillance performance is not always supported by experimental investigations; analysis in the context of active-sonar based undersea surveillance is in [7]. This fact has led us to study the fundamental limitations of scan-based tracking performance, in particular as a function of the sensor scan rate. The resulting model, coupled with our previously developed model for static fusion (or contact fusion) [], allows for a comparison of centralized tracking and fuse-before-track architectures for automatic tracking in sensor networks. reliminary conclusions are that the fuse-before-track provides improved surveillance performance, at the cost of a limited ability to handle closely spaced targets. This limitation might be overcome by pursuing more advanced techniques to contact fusion; in this context, we believe that batch processing approaches will prove to be most competitive [2]. Alternatively, as discussed here, improved surveillance performance with a centralized tracker can be achieved by an optimized selection of sensor detection threshold and the number of sensors, as a function of the allowable false track rate. In future work, we plan to study more closely our analytical modeling for the fusion process, particularly the modeling mismatch between probability of localization at the contact-fusion output and the probability of detection at the second-stage (tracker) input. Further, we plan to assess the preliminary conclusions documented here through model validation with simulated multi-sensor data. 578

6 References [] S. Coraluppi, M. Guerriero, and. Willett, Contact Fusion in Large Sensor Networks: Operational erformance Analysis, submitted to the NATO RTO SET anel Symposium on Sensors and Technology for Defence Against Terrorism, April 2008, Mannheim, Germany. [2] M. Guerriero, S. Coraluppi, and. Willett, Analysis of Scan and Batch rocessing Approaches to Static Fusion in Sensor Networks, in roceedings of the SIE Conference on Signal and Data rocessing of Small Targets, March 2008, Orlando FL, USA. [3] S. Blackman, Multi-Target Tracking with Radar Application, Artech House, 986. [5] S. Coraluppi, Advances in Multisensor Tracker Modeling, in roceedings of the 2006 SIE Conference on Signal and Data rocessing of Small Targets, April 2006, Orlando FL, USA. [6] W. Blanding,. Willett, Y. Bar-Shalom, and S. Coraluppi, Multisensor Track Termination for Targets with Fluctuating SNR, in roceedings of the IEEE International Conference on Acoustics, Speech, and Signal rocessing (ICASS), April 2007, Honolulu HI, USA. [7] S. Coraluppi and C. Carthel, erformance Limits of Real-Time Contact-Based Tracking, in roceedings of OCEANS 2007, June 2007, Aberdeen, Scotland. [4] Y. Bar-Shalom and X. Li, Multitarget-Multisensor Tracking, YBS ublishing, FTR = 0 0, T D = FTR = 0, T D = FTR = 0 2, T D = 2 FTR = 0 3, T D = 0 8 DT opt in db FTR = 0 4, T D = 8 FTR = 0 5, T D = FTR = 0 6, T D = 0 FTR = 0 9, T D = 4 FTR = 0 7, T D = FTR = 0 0, T D = 5 FTR = 0 8, T D = t opt Fig Relationship between optimal revisit time and optimal detection threshold. 579