Multi-Target Tracking in Distributed Sensor Networks using Particle PHD Filters
aa r X i v : . [ c s . M A ] D ec SUBMITTED FOR PUBLICATION 1
Multi-Target Tracking in Distributed SensorNetworks using Particle PHD Filters
Mark R. Leonard,
Student Member, EURASIP, and Abdelhak M. Zoubir,
Member, EURASIP,
Abstract
Multi-target tracking is an important problem in civilian and military applications. This paper investigates multi-target trackingin distributed sensor networks. Data association, which arises particularly in multi-object scenarios, can be tackled by varioussolutions. We consider sequential Monte Carlo implementations of the Probability Hypothesis Density (PHD) filter based onrandom finite sets. This approach circumvents the data association issue by jointly estimating all targets in the region of interest.To this end, we develop the Diffusion Particle PHD Filter (D-PPHDF) as well as a centralized version, called the Multi-SensorParticle PHD Filter (MS-PPHDF). Their performance is evaluated in terms of the Optimal Subpattern Assignment (OSPA) metric,benchmarked against a distributed extension of the Posterior Cram´er-Rao Lower Bound (PCRLB), and compared to the performanceof an existing distributed PHD Particle Filter. Furthermore, the robustness of the proposed tracking algorithms against outliersand their performance with respect to different amounts of clutter is investigated.
Index Terms
Multi-target tracking, distributed target tracking, Particle Filter, PHD Filter, robustness, Posterior Cram´er-Rao Lower Bound
I. I
NTRODUCTION T HE problem of multi-target tracking (MTT) is becoming increasingly important in many military and civilian applicationssuch as air and ground traffic control, harbor surveillance, maritime traffic control, or video communication andsurveillance [1]–[3]. Distributed sensor networks offer a desirable platform for MTT applications due to their low cost andease of deployment, their lack of a single point of failure, as well as their inherent redundancy and fault-tolerance [4]. Acomprehensive overview of the state-of-the-art of distributed single-target tracking (STT) is given in [5]. Distributed versionsof the Kalman Filter [5], [6] and its nonlinear, non-Gaussian counterpart, the Particle Filter (PF) [7], have been well-studied.However, they cannot be applied directly to MTT as they do not account for the problem of data association. Although thereare methods such as the Joint Probabilistic Data Association Filter (JPDAF) [8] or the Multiple Hypothesis Tracker (MHT)[9] that address this problem in STT algorithms, the resource constraints in sensor networks might pose a challenge on findingsuitable distributed implementations [10]. The Probability Hypothesis Density (PHD) filter [11], [12], in contrast, resorts tothe concept of random finite sets (RFSs) to circumvent the problem of data association altogether.
The authors are with the Signal Processing Group, Institute of Telecommunications, Technische Universit¨at Darmstadt, Darmstadt 64283, Germany (e-mail:[email protected], [email protected])Manuscript submitted November 27, 2018
UBMITTED FOR PUBLICATION 2
PSfrag replacements
Target 1Target 2Target 3 y [ m ] x[m] -10 -10-20 -20 -30 -5 -5-15 -15-25-25 (a) PSfrag replacements
Target 1Target 2Target 3 y [ m ] x[m] -10 -10-20 -20 -30 -5 -5-15 -15-25-25 (b) Fig. 1: (a) Distributed sensor network with 1-coverage of the region of interest and 3 exemplary target tracks. (b) Exampleof tracking 3 targets with the Diffusion Particle PHD Filter (D-PPHDF). The small colored dots represent the target locationestimates obtained by the respective node with the same color. The light gray dots show the collective measurements obtainedby all nodes in the network.In this work, we investigate distributed MTT in a sensor network with 1-coverage of the region of interest (ROI), i.e., thesensor nodes have non- or barely overlapping fields of view (FOVs) and are distributed such that maximum area coverage isattained [13]. An exemplary network layout with these properties is depicted in Figure 1a). Autonomous distribution algorithmsfor realizing such a topology have been studied in our previous work [14]. The nodes in the network communicate with theirneighbors in order to collaboratively detect and track targets in the ROI. In addition, all of the sensors are equipped with asignal processing unit, allowing them to form decisions without a fusion center. That way, the network can autonomously reactto events such as the detection of an intruder without relying on a network operator. For the sake of simplicity, the networkis considered to be static. However, the consideration of mobile sensor nodes would enable reactions such as target pursuit orescape.Since the FOV and communication radius of each node are limited, a target is only seen by a subset of the network, whichchanges as the target moves through the ROI. Hence, at each time instant, there is an active and an inactive part of the network.The goal, thus, is to detect and observe the target in a distributed and collaborative fashion as it travels across the ROI, ratherthan reaching a network-wide consensus on its state and have the estimate available at each node.In the sequel, we develop a distributed Particle PHD filter called Diffusion Particle PHD Filter (D-PPHDF), which usesneighborhood communication to collaboratively estimate and track a single-sensor PHD at each node in the active subnetwork.In addition, we formulate the Multi-Sensor Particle PHD Filter (MS-PPHDF), a centralized extension of the D-PPHDF. Theperformance of both algorithms is evaluated in terms of the Optimal Subpattern Assignment (OSPA) metric [15], which iscalculated for the joint set of target state estimates of the active subnetwork. Furthermore, a distributed version of the PosteriorCram´er-Rao Lower Bound (PCRLB) [16]–[18]—again averaged over the active subnetwork—is introduced and used as a
UBMITTED FOR PUBLICATION 3 benchmark. Moreover, we investigate the robustness of the proposed tracking algorithms against outliers and examine theirperformance under different amounts of clutter.Other distributed solutions for MTT in a multi-sensor setup using the PHD filter have been studied, e.g., in [19], [20], [21].Contrary to our approach, they either assume overlapping FOVs or employ a pairwise communication scheme. The commonidea, however, is to extend the single-sensor PHD filter to the multi-sensor case through communication between multiplenodes, or nodes and a fusion center. A more rigorous approach for MTT with multiple sensors is to use a multi-sensor PHDfilter [22], [23], which seeks to estimate and track a single multi-sensor PHD instead of multiple single-sensor PHDs. In thiswork, we compare our methods to the approach in [20] (adapted to our scenario), which is also based on single-sensor PHDs.The consideration of methods based on a multi-sensor PHD will be the focus of future work.The paper is organized as follows: Section 2 presents the considered state-space model and recapitulates the theory ofRFSs as well as the PHD and the PHD filter. The problem of distributed MTT is addressed in Section 3. Here, we will firstdetail our modification of Adaptive Target Birth (ATB) before formulating the D-PPHDF and investigate its computationalcomplexity and communication load. In Section 4, the MS-PPHDF is developed and analyzed in terms of computationalcomplexity and communication load. Section 5 is dedicated to simulations. First, the Distributed Posterior Cram´er-Rao LowerBound (DPCRLB) is introduced. Then, we present the simulation setup and discuss our results. Finally, a conclusion is givenin Section 6. II. M
ODELS AND T HEORY
A. State-Space and Measurement Model
A linear state-space model is considered for each target at time instant i ≥ . The target state vector s tgt ( i ) = [ x tgt ( i ) , ˙ x tgt ( i )] ⊤ contains the target location vector x tgt as well as the velocity vector ˙ x tgt . For the sake of simplicity, we restrict ourselves to a2D-environment. The target state evolves according to the state equation [24]: s tgt ( i ) = F ( i ) s tgt ( i −
1) + G ( i ) n tgt ( i ) . (1)The matrices F and G as well as the vector n tgt will be explained shortly. Node k obtains a measurement z k of the targetlocation as given by the measurement equation [24]: z k ( i ) = H k ( i ) s tgt ( i ) + ν tgt k ( i ) , k ∈ M (2)with M = { m ∈ { , . . . , N } | k x m ( i ) − x tgt ( i ) k ≤ R sen } denoting the set of all nodes m that are located such that theEuclidean distance k x m ( i ) − x tgt ( i ) k between their location x m and the target location x tgt is not greater than their sensingradius R sen . Note that N is the total number of nodes in the network. Furthermore, n tgt ( i ) ∼ N ( , , Q ( i )) and ν tgt k ( i ) ∼N ( , , R k ( i )) denote the state and measurement noise processes, respectively, with the zero-mean vector , = [0 , ⊤ . Bothnoise processes are spatially and temporally white, as well as uncorrelated with the initial target state s tgt (0) and each other UBMITTED FOR PUBLICATION 4 for all i . For the sake of simplicity, we choose a time-invariant measurement noise covariance matrix R k ( i ) = R k = σ r I , (3)where σ r is the variance of each component of the measurement noise and I n denotes the identity matrix of size n .In target tracking, the model matrices are usually chosen to be time-invariant and given by [24] F = I ∆ i I , I , G = ∆ i I ∆ i I , Q = σ q I , (4)where , is the × zero matrix. Furthermore, ∆ i is the time step interval in seconds with which the state-space modelprogresses. In addition, σ q denotes the variance of a state noise component. We assume that the sensor nodes only obtaininformation on the location of a target. One common set of measurements that is often found in applications at sea is thecombination of distance and bearing measurements from which an estimate of the target location can easily be calculated. Sincewe are not interested in the exact nature of the measured location information but rather in how this information is processedby different tracking algorithms, we formulate our measurement model based on the local target location estimates at eachnode. This gives us a general model that is applicable to a wide variety of application irrespective of the exact measurementquantities. Thus, we obtain a general measurement matrix H k of the form H k = (cid:20) I , (cid:21) . (5) B. Random Finite Sets (RFSs)
A random finite set (RFS) is an unordered finite set that is random in the number of its elements as well as in their values[25]–[27]. Therefore, RFSs are a natural choice for representing the multi-target states and measurements in MTT: the stateand measurement vectors of all targets are collected in corresponding RFSs [28], [29]. Given the realization Ξ i − of the RFS Ξ i − at time instant i − , the multi-target state of our tracking problem can be described by the RFS Ξ i according to Ξ i = S i (Ξ i − ) ∪ B i , (6)where the survival set S i (Ξ i − ) denotes the RFS of targets that already existed at time step i − and have not exited the ROI,i.e., the region covered by the sensor network, in the transition to time step i . In addition, the birth set B i is the RFS of newtargets that spontaneously appear at the border of the ROI at time instant i [1], [12], [29]. Note that the statistical behavior of Ξ i can be described by the conditional probability f i | i − (Ξ i | Ξ i − ) .The multi-target measurement model is given by the RFS Σ i as Σ i = Θ i (Ξ i ) ∪ C i (Ξ i ) , (7)where Θ i (Ξ i ) is the RFS of measurements generated by Ξ i . In addition, the RFS C i (Ξ i ) represents clutter or false alarms.Given a realization Σ i of Σ i , the statistical behavior of the RFS Σ i is described by the conditional probability f i (Σ i | Ξ i ) . UBMITTED FOR PUBLICATION 5
C. The Probability Hypothesis Density (PHD)
In analogy to the single-target case, the optimal Bayesian filter for MTT recursively propagates the multi-target posterior f i | i (Ξ i | Σ i ) over time, according to f i | i (Ξ i | Σ i ) = f i (Σ i | Ξ i ) f i | i − (Ξ i | Σ i − ) R f i (Σ i | Ξ) f i | i − (Ξ | Σ i − ) µ s ( d Ξ) (8) f i | i − (Ξ i | Σ i − ) = Z f i | i − (Ξ i | Ξ) f i − | i − (Ξ | Σ i − ) µ s ( d Ξ) , (9)where µ s is a dominating measure as described in [29]. This approach requires the evaluation of multiple integrals, whichmakes it even more computationally challenging than its single-target counterpart. A common solution is to find a set ofstatistics, e.g., the moments of first or second order, which yield a good approximation of the posterior, and propagate theminstead [1].The Probability Hypothesis Density (PHD) D i | i ( s ( i ) | Σ i ) is an indirect first-order moment of f i | i (Ξ i | Σ i ) [30]. It is givenby the following integral [12], [31]: D i | i ( s ( i ) | Σ i ) = X s tgt n ( i ) ∈ Ξ i Z δ ( s ( i ) − s tgt n ( i )) f i | i ( s ( i ) | Σ i ) d s ( i ) , (10)where R f ( Y ) δY denotes a set integral.The PHD has the following two properties [30]:1) The expected number of targets ˆ N tgt ( i ) at time step i is obtained by integrating the PHD according to ˆ N tgt ( i ) = Z D i | i ( s ( i ) | Σ i ) d s ( i ) . (11)This is in contrast to probability density functions (PDFs), which always integrate to 1.2) Estimates of the individual target states can be found by searching for the j ˆ N tgt m highest peaks of the PHD, where ⌊·⌉ denotes rounding to the nearest integer.Because of these two properties, the number of targets as well as their states can be estimated independently at each time stepwithout any knowledge of their identities. That way, the data association issue is avoided. However, this also means that PHDFilters cannot deliver the continuous track of a specific target. If continuous tracks are required, an additional association stephas to be performed. Two possible association algorithms for track continuity can be found in [11]. D. The PHD Filter
The PHD Filter is an approach for recursively propagating the PHD D i | i ( s ( i ) | Σ i ) at time step i given measurements upto time step i over time. If the RFS Ξ is Poisson-distributed, then its PHD is equal to its intensity function and is, hence, a UBMITTED FOR PUBLICATION 6 sufficient statistic [12]. In this case, the PHD recursion is given by the following prediction and update equations [12]: D i | i − ( s ( i ) | Σ i − ) = b i ( s ( i )) + Z p S ( s ( i − f i | i − ( s ( i ) | s ( i − D i − | i − ( s ( i − | Σ i − ) d s ( i − (12) D i | i ( s ( i ) | Σ i ) = " − p D + X z ∈ Σ i p D f i ( z | s ( i )) λ FA c FA ( z ) + p D R f i ( z | s ( i )) D i | i − ( s ( i ) | Σ i − ) d s ( i ) D i | i − ( s ( i ) | Σ i − ) (13)Note that b i ( s ( i )) is the PHD of the birth set B i of new targets appearing at time step i . In addition, p S ( s ( i − denotes theprobability that a target survives the transition from time step i − to i . The probability of survival depends on the previousstate s ( i − because a target that is close to the border of the ROI and has a velocity vector pointing away from it isunlikely to be present at time step i . Furthermore, f i | i − ( s ( i ) | s ( i − and f i ( z | s ( i )) denote the transition probability and thelikelihood, respectively. The probability of detection p D is constant over time and the tracker’s field of view (FOV) since it isassumed that all targets can be detected if the ROI is covered. The term λ FA c FA ( z ) represents Poisson-distributed false-alarmsdue to clutter, where λ FA is the false alarm parameter, which is distributed according to its spatial distribution c FA ( z ) .III. D ISTRIBUTED M ULTI -T ARGET T ARGET T RACKING
In this section we introduce the Diffusion Particle PHD Filter (D-PPHDF), a distributed Particle Filter implementation ofthe PHD Filter for performing MTT in a sensor network without a fusion center. Before diving into the algorithm, we brieflyreview the concept of Adaptive Target Birth (ATB) and discuss the modification we applied in the D-PPHDF.
A. Adaptive Target Birth (ATB)
Standard formulations of the PHD Filter consider the PHD b i ( s ( i )) of the birth set B i to be known a priori [32]. For typicaltracking applications such as air surveillance, this is a reasonable assumption since new targets should appear at the border ofthe ROI given continuous observation. An alternative is to make the target birth process adaptive and measurement-driven assuggested in [32], [33]. To this end, the PHD—and consequently the set of particles and weights approximating it in a ParticleFilter implementation—is split into two densities corresponding to persistent objects, which have survived the transition fromtime step i − to i , and newborn objects, respectively.In [32], [33], the PHD of newborn objects is approximated by randomly placing N P new particles around each targetmeasurement, with N P denoting the number of particles per target. We improve upon this approach by only consideringmeasurements with no noticeable impact on any persistent particle weight, as these may indicate the appearance of a newtarget. That way, the number of newborn particles is further reduced and a possible overlap between persistent and newbornPHD is avoided. With the transition to time step i + 1 , the newborn particles become persistent. Furtherore, we perform theATB step towards the end of each iteration of the algorithm and only consider the particles representing the persistent PHDin the prediction, weighting, and resampling steps. Hence, the update equation (13) does not have to be modified as in [32],[33].While ATB delays the tracking algorithm by one time step, it is much more efficient as it only places new particles inregions in which a target is likely to be found. In addition, there is no need for an explicit initialization step since the firstincoming target will trigger the deployment of a newborn particle cloud around its corresponding measurement. UBMITTED FOR PUBLICATION 7
B. The Diffusion Particle PHD Filter (D-PPHDF)
The proposed Diffusion Particle PHD Filter (D-PPHDF) is an extension of the single-sensor Particle PHD Filter (PPHDF)[11], [28], [34] for the multi-sensor case. Furthermore, it relies on ATB for a more efficient target detection. The communicationscheme we employ to exchange measurements and estimates between nodes is inspired by the two-step communication usedin the context of Diffusion Adaptation [35]. However, the algorithm does not rely on least-mean-squares or any other kind ofadaptive filter. First, each node k in the active part of the network obtains an intermediate estimate of the states of the targetspresent, i.e., of the PHD of persistent targets—represented by the set n s pk, pers ( i ) , w pk, pers ( i ) o N k, pers ( i ) p =1 of persistent particles withcorresponding weights—based on neighborhood measurements. In other words, every active node runs a separate PPHDF withaccess to measurements from its neighborhood N k , defined as N k = { l ∈ { , . . . , N } (cid:12)(cid:12) k x l − x k k ≤ R com } , k = 1 , . . . , N., (14)where R com denotes the communication radius. Second, each active node combines the intermediate estimates from itsneighborhood to a final, collaborative estimate. To this end, the persistent particle sets of all neighbors are merged into acollective set n s pk, coll ( i ) , w pk, coll ( i ) o N k, coll ( i ) p =1 of persistent neighborhood particles and corresponding weights before the clusteringstep, with N k, coll ( i ) denoting the number of collective persistent neighborhood particles. In the sequel, we will look at theindividual steps of the D-PPHDF in more detail: • Merging : The sets n s pk, coll ( i − , w pk, coll ( i − o N k, coll ( i − p =1 and n s pk, new ( i − , w pk, new ( i − o N k, new ( i − p =1 consist of thecollective persistent neighborhood particles and newborn particles of node k , s pk, coll ( i − and s pk, new ( i − , respectively,at time step i − with their respective weights w pk, coll ( i − and w pk, new ( i − . These sets are merged to become thetotal set n s pk, tot ( i ) , w pk, tot ( i ) o N k, tot ( i ) p =1 of particles and weights of node k at time step i . Here, N k, tot ( i ) is the total numberof particles of node k at time step i , which is given by N k, tot ( i ) = N k, coll ( i −
1) + N k, new ( i − , (15)with N k, coll ( i − and N k, new ( i − denoting the respective number of persistent neighborhood and newborn particles atthe previous time step. Note that since the sets of particles and weights represent PHDs, merging the sets corresponds tosumming these PHDs. • Predicting:
Each particle is propagated through the system model to become a persistent particle. The system model isassumed to be the same for each target and given by Equation (1). Since the process noise is captured by the spread ofthe particle cloud, the respective term can be removed from the equation, yielding s pk, pers ( i ) = F s pk, tot ( i ) , p = 1 , . . . , N k, tot ( i ) . (16)The corresponding weights are multiplied with the probability of survival p S , which is assumed to be constant for the UBMITTED FOR PUBLICATION 8 sake of simplicity , according to w pk, pers ( i | i −
1) = p S w pk, tot ( i ) , p = 1 , . . . , N pers ( i ) . (17)The prediction of particles and weights corresponds to the second term in Equation (12). • Measuring & Broadcasting (1):
The sensor nodes obtain measurements of the targets and forward them to their neighbors. • Weighting:
The persistent particle weights of node k are updated by applying a weighting step corresponding to Equation(13) iteratively for each neighbor. Using the product operator, this weighting step can be compactly denoted as w pk, pers ( i ) = Y l ∈N k " − p D + X z j ∈ Σ li w pk,j, update ( i ) w pk, pers ( i | i − , (18)with w pk,j, update ( i ) = p D f i ( z j | x p ( i )) λ FA c FA ( z j ) + L ( z j ) , (19)where Σ li is the set of measurements obtained by node l and L ( z j ) is calculated as L ( z j ) = N pk, pers ( i ) X q =1 p D f i ( z j | x q ( i )) w qk, pers ( i | i − . (20)Note that f i ( z j | x p ( i )) is the likelihood and x p ( i ) is the location vector of particle p .Afterwards, each node k obtains the set Σ ki, cand of candidate measurements, i.e., measurements that are not responsible forthe highest weighting of any persistent particle, to be used in the ATB step later on. The set Σ ki, cand is found according to Σ ki, cand = Σ ki \ n z m p (cid:12)(cid:12)(cid:12) m p = arg max j w pk,j, update ( i ) , p = 1 , ..., N k, pers ( i ) o . (21) • Resampling:
Each node k calculates its own expected number of targets ˆ N k, tgt ( i ) from its total persistent particle massaccording to ˆ N k, tgt ( i ) = N k, tot ( i ) X p =1 w pk, pers ( i ) . (22)Consequently, the number of persistent particles of node k is updated as N k, pers ( i ) = ˆ N k, tgt ( i ) N P . (23)Furthermore, the set of persistent particles of node k has to be resampled by drawing N k pers ( i ) particles with replacementfrom it. Note that the probability of drawing particle p is given by w pk, pers ( i )ˆ N k, tgt ( i ) since the weights do not sum to unity. Then, A constant probability of survival p S is a reasonable assumption if the targets move relatively slowly with respect to the observation time and the size ofthe ROI. UBMITTED FOR PUBLICATION 9 the weights are reset to equal values as w pk, pers ( i ) = ˆ N k, tgt ( i ) N k, pers ( i ) , p = 1 , . . . , N k, pers ( i ) . (24) • Broadcasting (2):
Every node k transmits its set of resampled persistent particles and weights n s pk, pers ( i ) , w pk, pers ( i ) o N k, pers ( i ) p =1 to its neighbors. • Clustering:
Each node k forms a collective set of persistent neighborhood particles s pk, coll ( i ) and corresponding weights w pk, Nh ( i ) according to n s pk, coll ( i ) , w pk, coll ( i ) o N k, coll ( i ) p =1 = [ l ∈N k n s pl, pers ( i ) , w pl, pers ( i ) o N l, pers ( i ) p =1 , (25)with N k, coll ( i ) = X l ∈N k N l, pers ( i ) (26)denoting the number of collective persistent neighborhood particles of node k . As in the merging step, this correspondsto summing the corresponding PHDs to obtain an updated single-sensor PHD with a probability distribution reflecting theinformation of the entire neighborhood of node k . Note that the PHDs might not be independent if a target is detected bymore than one neighbor. However, this is not a problem since merging the particle sets simply results in the respectivetarget being represented by more particles. Hence, node k will be able to estimate the corresponding location moreaccurately.The estimated target states are found by clustering the collective persistent particles. Since the expected number of targets ˆ N l, tgt ( i ) , l ∈ N k might be different for each neighbor, we resort to hierarchical clustering of the single-linkage type [36].Here, the sum of the expected number of targets over the neighborhood can serve as an upper bound for the number ofclusters. Note, however, that if two targets are close to each other, clustering algorithms might not be able to resolve bothtargets correctly. • Roughening:
A roughening step is performed to counter sample impoverishment [37]. To this end, an independent jitter s j ( i ) is added to every resampled particle. Each component s jc ( i ) , c = 1 , . . . , d of the jitter with dimensionality d issampled from the Gaussian distribution N (0 , ( σ jc ( i )) ) . The component-wise standard deviation of the jitter is given by σ jc ( i ) = KE c N k, coll ( i ) − /d , (27)where E c is the interval length between the maximum and minimum samples of the respective component. To avoidevaluating E c separately for each particle cluster, it is assigned an empirically found constant value. Note that d = 4 since the dimensionality of the jitter vector s j ( i ) and the particle state vector s p ( i ) have to coincide. In addition, K is atuning constant, which controls the spread of the particle cloud. • Adaptive Target Birth: N P new particles are placed randomly around each candidate measurement z j ∈ Σ ki, cand leading Since the noise variances as well as the network topology are fixed, the true value of E c will not change significantly over time and between clusters, sothis is a valid simplification. UBMITTED FOR PUBLICATION 10 to a total number of N k, new ( i ) = N P · | Σ ki, cand | newborn particles for node k . Every newborn particle is associated with aweight that is chosen according to w pk, new ( i ) = p B N pk, new ( i ) , p = 1 , . . . , N k, new ( i ) , (28)where p B is the probability of birth. Depending on the application, p B can depend on time as well as on the locationof the respective particle. For simplicity, the probability that a new target enters the ROI is assumed to be equal for alllocations in the birth region over time. The target birth process corresponds to the first term in Equation (12).Figure 1b) shows an example of tracking three targets, which move along the deterministic tracks depicted in Figure 1a),using the D-PPHDF. Note that each small colored dot corresponds to a target location estimate obtained by the respective nodewith the same color while the light grey dots represent the collective measurements from all nodes. From this illustration, thefollowing properties of the D-PPHDF are apparent: First, the algorithm only delivers separate location estimates – representedby the small colored dots – for each time instant rather than continuous tracks, which – as mentioned before – is a commonproperty of PHD filters. Second, the network as a whole would be able to correctly track all three targets, while a single nodeonly obtains the locally relevant subtracks of the targets in its vicinity. Third, the employed two-step communication schemeis able to extend the vicinity of a node far beyond its own sensing radius of R sen = 6 m. This can, for instance, be seen fromthe fact that the lime-green node located at [ − , − is able to obtain location estimates of target 2, which enters the ROIfrom the south. Finally, Figure 1b) also illustrates the resolution problem of clustering. When targets 1 and 2, which enter theROI from the north and the south, respectively, cross paths, the nodes in their vicinity see them as just one target. This leadsto an aggregation of target location estimates around [9 , .The pseudo-code of the D-PPHDF is given in Table I. C. Computational Complexity and Communication Load
In this section we take a look at the computational complexity and the communication load the D-PPHDF imposes on eachnode in the active subnetwork. The following steps are performed at every time instant i but time dependency is omitted forsimplicity. Note that each of the steps scales with the number of active nodes when considering the computational complexityof the network as a whole. • Prediction : The prediction step described by Equations (16) and (17) is performed for each particle at every active node.Hence, it scales with the number of particles N k, tot and the dimensionality d of the particle vectors. In order to obtain atractable expression for the computational complexity, we assume each node to have the same number of particles N tot . ⇒ O ( N tot d ) • Weighting : Each particle is updated in the weighting step given by Equations (18)-(21). The weight update as well as thedesignation of candidate measurements for ATB depends on the neighborhood size |N k | of node k and the number ofmeasurements (cid:12)(cid:12) Σ l (cid:12)(cid:12) of each of its neighbors l . For tractability reasons, we assume each node to have the same number ofneighbors N nb and to obtain the same number of measurements N meas . ⇒ O ( N tot N nb N meas ) UBMITTED FOR PUBLICATION 11 input: d, E c , K, n, N, N P , p B , p S , λ FA , c FA initialize: n s pk, coll (0) , w pk, coll (0) o N k, coll (0) p =1 = n s pk, new (0) , w pk, new (0) o N k, new (0) p =1 = ∅ . while i ≤ n do for k = 1 , . . . , N do Merge the sets of collective persistent and newborn particles with corresponding weights: n s pk, tot ( i ) , w pk, tot ( i ) o N k, tot ( i ) p =1 = n s pk, coll ( i − , w pk, coll ( i − o N k, coll ( i − p =1 ∪ n s pk, new ( i − , w pk, new ( i − o N k, new ( i − p =1 . for p = 1 , . . . , N k, tot ( i ) do Predict the new state of each particle and update the weight with the probability of survival p S : s pk, pers ( i ) = F s pk, tot ( i ) w pk, pers ( i | i −
1) = p S w pk, tot ( i ) . Update the weights using neighborhood measurements: w pk, pers ( i ) = Y l ∈N k " − p D + X z j ∈ Σ li w pk,j, update ( i ) w pk, pers ( i | i − ,w pk,j, update ( i ) = p D f i ( z j | x p ( i )) λ FA c FA ( z j ) + L ( z j ) , L ( z j ) = N k, pers ( i ) X q =1 p D f i ( z j | x q ( i )) w qk, pers ( i | i − . end for Form the set of candidate measurements for ATB: Σ ki, cand = Σ ki \ n z m p (cid:12)(cid:12)(cid:12) m p = arg max j w pk,j, update ( i ) , p = 1 , ..., N k, pers ( i ) , j = 1 , ..., (cid:12)(cid:12) Σ li (cid:12)(cid:12) ∀ l ∈ N k o . Calculate the estimated number of targets: ˆ N k, tgt ( i ) = N k, tot ( i ) X p =1 w pk, pers ( i ) . Resample ˆ N k, pers ( i ) = ˆ N k, tgt ( i ) N P particles and reset the weights: w pk, pers ( i ) = ˆ N k, tgt ( i ) N k, pers ( i ) , p = 1 , . . . , N k, pers ( i ) . Merge the sets of persistent neighborhood particles and weights: n s pk, coll ( i ) , w pk, coll ( i ) o N k, coll ( i ) p =1 = [ l ∈N k n s pl, pers ( i ) , w pl, pers ( i ) o N l, pers ( i ) p =1 . Use single-linkage clustering to identify ˆ N tgt ( i ) clusters and find the set of estimated target states n ˆ s lk ( i ) o ˆ N tgt ( i ) l =1 by calculating the centroids. Add an independent jitter to each particle using a component-wise standard deviation of: σ jc ( i ) = KE c N k, coll ( i ) − /d . Place N P new particles randomly around each candidate measurement z j ∈ Σ ki, cand . Set the weights as: w pk, new ( i ) = p B N k, new ( i ) , p = 1 , . . . , N k, new ( i ) . end for i ← i + 1 end while return TABLE I: The Diffusion Particle PHD Filter.
UBMITTED FOR PUBLICATION 12 • Resampling : The estimation of the number of targets and the resampling step in Equations (22)-(24) are linear in thenumber of particles used for the calculation [38]. For the sake of simplicity, we assume each active node to have the sameestimate of the number of targets N tgt . ⇒ O ( N tot + N active N tgt N P ) • Clustering : The complexity of single-linkage clustering is cubic in the number of particles, i.e., in the number of neighbors N nb of each node, the estimated number of targets N tgt , the number of particles per target N P , and the dimensionality d of the particles [39]. ⇒ O (( N nb N tgt N P d ) ) • Roughening : Roughening (Equation (27)) is performed for every collective particle and is linear in the dimensionality ofthe particles. ⇒ O ( N nb N tgt N P d ) • Adaptive Target Birth : The birth process depends on the number of particles per target N P as well as the number ofcandidate measurements N cand , which is assumed equal for each active node to ensure tractibility. ⇒ O ( N P N cand ) As far as the communication load is concerned, the D-PPHDF requires the broadcasting of measurements, i.e., 2 scalars permeasurement, over the neighborhood in the first broadcasting step. In the second step, the sets of particles and weights, i.e., 5scalars per particle, are transmitted. Clearly, the communication load strongly depends on the number of nodes in the network,or more precisely the number of active nodes and the size of their respective neighborhood. As an extension of the D-PPHDF,one could think of changing the second broadcasting step and transmit Gaussian Mixture Model representations—instead ofthe actual particles and weights—that will be resampled at the receiver node (see e.g., [40]). That way, communication loadcould be reduced to transmitting only a few scalars in the second broadcasting step at the cost of estimation accuracy andadditional computational complexity. However, a thorough treatment of this extension is beyond the scope of this work.IV. C
ENTRALIZED M ULTI -T ARGET T RACKING
Having presented the D-PPHDF as a distributed solution for MTT in a sensor network, we propose the centralized counterpartto our approach in the sequel.
A. The Multi-Sensor Particle PHD Filter (MS-PPHDF)
The proposed Multi-Sensor Particle PHD Filter (MS-PPHDF) is a centralized, multi-sensor PPHDF that relies on a fusioncenter with access to the measurements of all nodes in the network. It is based on the formulation of the single-sensor PPHDFin [11], [28], [34] but with an extended measurement set comprising the measurements of the entire network. Hence, onemight obtain more than one measurement per target—a change to the typical assumption in target tracking that each targetproduces at most one measurement [8]. To account for this change, we add a pre-clustering step before the weighting step andnormalize the weight update accordingly. A similar partitioning of the measurement set is used in extended target tracking,where a sensor can receive multiple target reflections due to the target’s physical extent [41], [42].
UBMITTED FOR PUBLICATION 13
In the following, we will look at the individual steps of the algorithm in more detail: • Merging : The sets { s p pers ( i − , w p pers ( i − } N pers ( i − p =1 and { s p new ( i − , w p new ( i − } N new ( i − p =1 consist of the persistent andnewborn particles, s p pers ( i − and s p new ( i − , respectively, at time step i − with their respective weights w p pers ( i − and w p new ( i − . These sets are merged to become the total set { s p tot ( i ) , w p tot ( i ) } N tot ( i ) p =1 of particles and weights at time step i . Here, N tot ( i ) is the total number of particles at time step i , which is given by N tot ( i ) = N pers ( i −
1) + N new ( i − , (29)with N pers ( i − and N new ( i − denoting the respective number of persistent and newborn particles at the previous timestep. • Predicting:
As in the D-PPHDF, each particle is propagated through the system model according to s p pers ( i ) = F s p tot ( i ) , p = 1 , . . . , N pers ( i ) = N tot ( i ) (30)to become a persistent particle. The corresponding weights are multiplied with the probability of survival p S as w p pers ( i | i −
1) = p S w p tot ( i ) , p = 1 , . . . , N pers ( i ) . (31) • Measuring:
The sensor nodes obtain measurements of the targets. • Pre-Clustering:
Since there might be more than one measurement per target, the measurements of the entire network arepre-clustered before the weighting step and each measurement is assigned a label C ( z ) that reflects the cardinality of itsown cluster. This can be done, for instance, using single-linkage clustering [36]. The clustering is based on the distancebetween measurements, i.e., spatially close measurements are assumed to stem from the same target. Hence, when twoor more targets are too close to each other, cardinality errors may occur. • Weighting:
All available target measurements, which comprise the set Σ i , are used to update the persistent particle weightsaccording to w p pers ( i ) = " − p D + X z j ∈ Σ i w pj, update ( i ) w p pers ( i | i − , (32)with w pj, update ( i ) = p D f i ( z j | x p ( i ))( λ FA c FA ( z j ) + L ( z j )) C ( z j ) , (33)and L ( z j ) = N pers ( i ) X q =1 p D f i ( z j | x q ( i )) w q pers ( i | i − . (34)Note that—in contrast to the D-PPHDF—the weighting step is applied only once using the entire set of measurements.Therefore—and since there might be more than one measurement per target—we have to ensure that the weight updateterms w pj, update —and consequently the particle weights—still sum to the number of targets present. This is done by UBMITTED FOR PUBLICATION 14 normalizing Equation (33) with C ( z j ) , i.e., the cardinality of the cluster to which the current measurement z j belongs.Afterwards, we form the set Σ i, cand of candidate measurements for the ATB step according to Σ i, cand = Σ i \ n z m p (cid:12)(cid:12)(cid:12) m p = arg max j w pj, update ( i ) , p = 1 , ..., N pers ( i ) o . (35) • Resampling:
The expected number of targets ˆ N tgt ( i ) is calculated from the total persistent particle mass as ˆ N tgt ( i ) = N tot ( i ) X p =1 w p pers ( i ) . (36)Consequently, the number of persistent particles is updated according to N pers ( i ) = ˆ N tgt ( i ) N P . (37)Furthermore, the set of persistent particles is resampled by drawing N pers ( i ) particles with probability w p pers ( i )ˆ N tgt ( i ) . Then, theweights are reset to equal values as w p pers ( i ) = ˆ N tgt ( i ) N pers ( i ) , p = 1 , . . . , N pers ( i ) . (38) • Clustering:
In contrast to the D-PPHDF, there is only one estimate of the expected number of targets. Hence, we can use k -means clustering [43] to find the estimated target states by grouping the resampled particles into ˆ N tgt ( i ) clusters andcalculating the centroid of each cluster. • Roughening:
Roughening is performed analogously to the D-PPHDF. • Adaptive Target Birth: N P new particles are placed randomly around each candidate measurement z j ∈ Σ i, cand yieldinga total number of N new ( i ) = N P · | Σ i, cand | newborn particles. The corresponding weights are chosen according to w p new ( i ) = p B N new ( i ) , p = 1 , . . . , N new ( i ) , (39)where p B is the probability of birth.The pseudo-code of the MS-PPHDF is given in Table II. B. Computational Complexity and Communication Load
In this section we analyze the computational complexity and the communication load of the MS-PPHDF. The followingsteps are performed at every time instant i but time dependency is omitted for simplicity: • Prediction : The prediction step described by Equations (30) and (31) is performed for each of the N tot particles and islinear in the dimensionality d . ⇒ O ( N tot d ) • Pre-Clustering : The pre-clustering step relies on single-linkage clustering. The complexity is therefore cubic in the totalnumber of measurements N meas . [39] ⇒ O ( N meas ) UBMITTED FOR PUBLICATION 15 input: d, E c , K, n, N, N P , p B , p S , λ FA , c FA initialize: { s p coll (0) , w p coll (0) } N coll (0) p =1 = { s p new (0) , w p new (0) } N new (0) p =1 = ∅ . while i ≤ n do Merge the sets of persistent and newborn particles with corresponding weights: { s p tot ( i ) , w p tot ( i ) } N tot ( i ) p =1 = (cid:8) s p pers ( i − , w p pers ( i − (cid:9) N pers ( i − p =1 ∪ { s p new ( i − , w p new ( i − } N new ( i − p =1 . for p = 1 , . . . , N tot ( i ) do Predict the new state of each particle and update the weight with the probability of survival p S : s p pers ( i ) = F s p tot ( i ) w p pers ( i | i −
1) = p S w p tot ( i ) . Cluster the measurements using single-linkage clustering and assign each measurement z a label C ( z ) reflectingthe cardinality of its cluster. Update the weights using the measurements of the entire network: w p pers ( i ) = " − p D + X z j ∈ Σ i w pj, update ( i ) w p pers ( i | i − w pj, update ( i ) = p D f i ( z j | x p ( i ))( λ FA c FA ( z j ) + L ( z j )) C ( z j ) , L ( z j ) = N pers ( i ) X q =1 p D f i ( z j | x q ( i )) w q pers ( i | i − . end for Form the set of candidate measurements for ATB: Σ i, cand = Σ i \ n z m p (cid:12)(cid:12)(cid:12) m p = arg max j w pj, update ( i ) , p = 1 , ..., N pers ( i ) , j = 1 , ..., | Σ i | o . Calculate the estimated number of targets: ˆ N tgt ( i ) = N tot ( i ) X p =1 w p pers ( i ) . Resample ˆ N pers ( i ) = ˆ N tgt ( i ) N P particles and reset the weights: w p pers ( i ) = ˆ N tgt ( i ) N pers ( i ) , p = 1 , . . . , N pers ( i ) . Find the set of estimated target states n ˆ s l ( i ) o ˆ N tgt ( i ) l =1 by using k -means clustering and calculating the centroid ofeach cluster. Add an independent jitter to each particle using a component-wise standard deviation of: σ jc ( i ) = KE c N pers ( i ) − /d . Place N P new particles randomly around each candidate measurement z j ∈ Σ i, cand . Set the weights as: w p new ( i ) = p B N new ( i ) , p = 1 , . . . , N new ( i ) . i ← i + 1 end while return TABLE II: The Multi-Sensor Particle PHD Filter.
UBMITTED FOR PUBLICATION 16 • Weighting : Each particle is updated in the weighting step given by Equations (32)-(34). The weight update as well as thedesignation of candidate measurements for ATB depends on the number of measurements N meas = | Σ | . ⇒ O ( N tot N meas ) • Resampling : The estimation of the number of targets and the resampling step in Equations (36)-(38) are linear in thenumber of particles used for the calculation [38]. ⇒ O ( N tot + N tgt N P ) • Clustering : In contrast to the D-PPHDF we can use k -means clustering. The complexity of Lloyd’s implementation isgiven by [44] ⇒ O (( N tgt N P ) dN tgt +1 log( N tgt N P )) . • Roughening : Roughening is linear in the dimensionality of the particles and their number. ⇒ O ( N tgt N P d ) • Adaptive Target Birth : The birth process depends on the number of particles per target N P as well as the number ofcandidate measurements N cand . ⇒ O ( N P N cand ) In summary, the computational complexity of the MS-PPHDF is largely comparable to that of the D-PPHDF. The onlyexception is the pre-custering step, which scales cubicly with the total number of measurements and adds additional complexityto the algorithm. As a tradeoff the communication load of the MS-PPHDF clearly is lower compared to the D-PPHDF becausethere is only the initial transmission of measurements from the nodes to the fusion center. However, considering a setupwith relatively small communication radii, this initial communication step requires a lot of relaying and leads to high trafficdensity in the vicinity of the fusion center. Furthermore, this communication structure exhibits a single point of failure whilea distributed sensor network is inherently redundant. V. S
IMULATIONS
In this section, we evaluate the performance of the D-PPHDF as well as the MS-PPHDF for tracking multiple targetsin a sensor network with 1-coverage. To this end, we consider Gaussian measurement noise of different variance as wellas ε -contaminated noise with different contamination ratios to investigate the robustness of the algorithms. In addition, theperformance for different amounts of clutter is analyzed. We compare the proposed algorithms to the alternative distributedPPHDF from [20], which will be referred to as Distributed Data Fusion Particle PHD Filter (DDF-PPHDF). Here, each noderuns its own PPHDF using only its own measurements. In a subsequent step, the particles are distributed over the neighborhoodand reweighted by fusing their corresponding Exponential Mixture Densities.Furthermore, we formulate the DPCRLB as a lower bound for evaluating the performance of the three algorithms in termsof the OSPA [15] metric. In our simulations, we compute the OSPA metric with respect to the joint set of target state estimatesof the entire active network. The latter is found by clustering the target state estimates of all active nodes. Furthermore, weconsider the squared OSPA metric scaled by the number of targets, i.e., N tgt · (cid:16) ¯ d ( c ) p (cid:17) , as in [45]. That way, we can use theDPCRLB, which will be introduced in the following, as a benchmark. UBMITTED FOR PUBLICATION 17
A. The Distributed Posterior Cram´er-Rao Lower Bound (DPCRLB)
Rather than evaluating the performance of the different MTT algorithms based on an error metric, it makes more sense toderive a minimum variance bound on the estimation error, which enables an absolute performance evaluation. For time-invariantstatistical models, the most commonly used bound is the Cram´er-Rao Lower Bound (CRLB), which is given by the inverseof Fisher’s information matrix [46]. In [45] and [47], the CRLB is used in the context of multi-sensor MTT of an unknownnumber of unlabeled targets in order to evaluate the performance, as well as prove the asymptotic efficiency of the PHD asthe number of nodes goes to infinity. Since we are more interested in the tracking behavior of a fixed network over time, weresort to the Posterior Cram´er-Rao Lower Bound (PCRLB), which is an extension of the CRLB for the time-variant case [16].This bound can be calculated sequentially with the help of a Riccati-like recursion derived in [48]. Furthermore, in [17] and[49], the PCRLB is adapted for an MTT scenario in which the tracker can obtain more than one measurement per target.Let π mi , m = 1 , ..., M denote the probability that any measurement is associated with target m at time instant i as defined in[18]. With the corresponding stochastic process Π mi , the new stochastic process of association probabilities and target states tobe estimated becomes Φ i = (cid:16) Π Mi , Ξ Mi (cid:17) . Fisher’s information matrix J Φ i = J Π i J Ξ i Π i J Π i Ξ i J Ξ i can now be formed as describedin [49] and [17]. However, as the number of targets varies over time, i.e., targets might enter or exit the ROI, J Φ i has to beexpanded or shrunk in the inverse matrix domain as described in [50]. The PCRLB B i at time instant i can be obtained asthe trace of the inverted submatrix J Ξ i according to [49] B i = trace (cid:26)h J Ξ i − J Π i Ξ i J − i J Ξ i Π i i − (cid:27) . (40)Note that, in a distributed MTT scenario, B i corresponds to a lower bound on the estimation error of a central processingunit with access to all measurements. Since we are interested in completely distributed MTT with in-network processing, weextend the PCRLB to the Distributed Posterior Cram´er-Rao Lower Bound (DPCRLB). To this end, each node k computes itsown PCRLB B ki considering only the measurements of its two-hop neighborhood, which is given by N (2) k = [ l ∈N k N l , (41)i.e., the neighbors of node k and their neighbors. Furthermore, only the targets within the sensing range of N k are taken intoaccount. Clearly, only nodes with a neighborhood in the vicinity of at least one target will be able to calculate a PCRLB. TheDPCRLB B i, dist at time instant i is then obtained by averaging over these values according to B i, dist = 1 |M| X k ∈M B ki , (42)where M is the set of all nodes that are able to compute a PCRLB. UBMITTED FOR PUBLICATION 18 V ARIABLE V ALUE D ESCRIPTION ∆ i N
30 number of nodes σ r componentwise power of meas. noise σ q componentwise power of state noise ε R com R sen communication radius R sen E c K N P
500 number of particles per target p B p D p S λ FA c FA ( z ) πR sen PDF of false alarms / clutter (uniform) c p B. Simulation Setup
In the following simulations, a static sensor network as depicted in Figure 1a) is used to perform MTT. The network iscentered around the point of origin [0 , ⊤ and distributed such that 1-coverage of the ROI is guaranteed. It covers an area ofapproximately m . Clutter is assumed Poisson and uniformly distributed over the sensing range of each node with anaverage rate of λ FA = 0 . and . . Moreover, we consider Gaussian measurement noise with variance σ r = 0 . and 0.3 aswell as ε -contamination noise with a ten-times higher variance and a contamination rate of ε = 0 . and 0.3. For the sake ofsimplicity, collisions between targets and sensor nodes are neglected.An overview of all simulation parameters is given in Table III. Since the purpose of this work is to introduce the MS-PPHDFas well as the D-PPHDF, verify their functionality, and compare them to alternative approaches, we consider a rather simplescenario with a high probability of detection and relatively low clutter levels. In our future work, we will study more sophisticatedscenarios to define possible breakdown points of our algorithms.We use the MS-PPHDF, the D-PPHDF, as well as the DDF-PPHDF to track three targets for i = 0 , ..., . The targets enterthe ROI at time steps i = 0 , , from the north, south, and west, respectively. A Monte Carlo simulation with N MC = 1000 runs is performed to evaluate the performance of the tracking algorithms in terms of the Optimal Subpattern Assignment(OSPA) metric. Note that the target trajectories as shown in Figure 1a) are deterministic, as is often the case in target trackingsimulations [50] in order to guarantee the comparability of the different Monte Carlo runs regarding, for instance, the numberof targets present. UBMITTED FOR PUBLICATION 19
PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDFDPCRLB true number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (seconds)
005 5 15 2510 20 30 (a) Squared and scaled OSPA compared to DPCRLB, σ r = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
00 5 15 2510 20 30 N e s t (b) Estimated number of targets, σ r = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDFDPCRLB true number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (seconds)
005 5 15 2510 20 30 (c) Squared and scaled OSPA compared to DPCRLB, σ r = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
00 5 15 2510 20 30 N e s t (d) Estimated number of targets, σ r = 0 . Fig. 2: Simulation I: Results for Gaussian noise and clutter rate λ FA = 0 . . The left part of the figure shows the squared andscaled Optimal Subpattern Assignment (OSPA) metric for each algorithm compared to the Distributed Posterior Cram´er-RaoLower Bound (DPCRLB), while the right part compares the estimated to the true number of targets. C. Simulation I: Results
In the first simulation, we compare the performance of the MS-PPHDF and the D-PPHDF to the alternative DDF-PPHDFand the DPCRLB, which serves as a benchmark. Measurement noise is zero-mean Gaussian with variance σ r = 0 . , . andthe average number of clutter is . .The simulation results are depicted in Figure 2. While the top part considers zero-mean Gaussian measurement noise with aper-component variance of σ r = 0 . , the bottom part shows the results for σ r = 0 . . In addition to evaluating the performanceof the MS-PPHDF, the D-PPHDF, and the DDF-PPHDF in terms of the squared and scaled OSPA metric over time andcomparing it to the DPCRLB as can be seen in the left part of the figure, we also look at the estimated number of targets,which is depicted in the right column. Since the OSPA metric contains a penalty for an erroneous estimate of the number oftargets, this side-by-side comparison facilitates the interpretation of the tracking results.Let us start by considering Figures 2a) and 2b), i.e., the case of σ r = 0 . . First of all, we observe that neither tracking UBMITTED FOR PUBLICATION 20
PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDFDPCRLB true number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (s)
00 5 15 2510 20 30 (a) Squared and scaled OSPA compared to DPCRLB, σ r = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDFDPCRLB true number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (s)
00 5 15 2510 20 30 (b) Squared and scaled OSPA compared to DPCRLB , σ r = 0 . Fig. 3: Simulation I: Results for Gaussian noise and clutter rate λ FA = 0 . (zoomed in). The squared and scaled OptimalSubpattern Assignment (OSPA) metrics of using the Multi-Sensor Particle PHD Filter (MS-PPHDF), the Diffusion ParticlePHD Filter (D-PPHDF), and the Distributed Data Fusion Particle PHD Filter (DDF-PPHDF), are compared to the DistributedPosterior Cram´er-Rao Lower Bound (DPCRLB).algorithm provides an OSPA value or an estimate of the number of targets for i = 0 . This is expected and due to ATB, whichinitializes new particle clouds based on the measurements from the previous time step. Thus, target birth is delayed by onetime step and tracking can only be performed for i > . The same effect can be witnessed at i = 9 and i = 14 , respectively,which mark the time instants at which targets 2 and 3 enter the ROI. Here, the OSPA curves of all trackers exhibit a spike,which is due to the fact that the newborn particles are not yet considered in the tracker and, hence, the number of estimatedtargets is too low, as can be seen in Figure 2b).Another sudden rise of all the OSPA curves can be observed in the time interval ≤ i ≤ with a valley at i = 22 .Looking at the estimated number of targets, we can attribute this phenomenon to the fact that only two of the three targetsare recognized by the tracking algorithms. Since the target trajectories are deterministic, we know that in the given timeinterval targets 2 and 3 cross paths. Due to the inability of the clustering algorithm to separate strongly overlapping sets ofmeasurements, the two targets merge into one as long as they are close to each other. When the two targets occupy almostexactly the same position, i.e., at i = 22 , the OSPA metric decreases due to the decrease in measurement variance. As thetargets drift apart, the variance and with it the OSPA metric increases up to the point where the two targets can be recognizedas separate again and the corresponding penalty is switched off.Looking at the overall picture in Figure 2a), which shows the case of σ r = 0 . , it is evident that the centralized MS-PPHDFand the distributed D-PPHDF achieve approximately the same performance with OSPA values closely approaching the DPCRLBwhen the number of targets stays constant. Furthermore, both algorithms deliver very accurate estimates of the number of targets,given they are separable by clustering, as can be seen in Figure 2b). The DDF-PPHDF, however, continuously exhibits a worseperformance than the D-PPHDF, both in terms of the OSPA metric as well as the estimated number of targets. This is wherethe additional communication in the proposed D-PPHDF shows its strength in reducing uncertainty due to measurement noiseand clutter. Apart from achieving worse tracking results, the DDF-PPHDF also has more difficulty in separating targets 1 and UBMITTED FOR PUBLICATION 21 σ r = 0 . , the overall performance of the different tracking algorithms is very similar to the case of σ r = 0 . .In order to make a statement on how the different tracking algorithms compare, let us neglect the penalty due to an erroneousestimate of the number of targets and take a look at Figures 3a) and 3b), which are zoomed-in versions of Figures 2a) and2c), respectively.In Figures 3a) and 3b) the DPCRLB is given as a benchmark for tracking performance. One can observe that its valueis always smaller or equal to the respective measurement variance. As stated before, the centralized MS-PPHDF and thedistributed D-PPHDF exhibit very similar performance and deliver better tracking results than the DDF-PPHDF. While theMS-PPHDF achieves lower OSPA values than the D-PPHDF when the number of targets stays constant, i.e., for ≤ i ≤ and ≤ i ≤ , the D-PPHDF performs better directly after a new target appears, i.e., for ≤ i ≤ , ≤ i ≤ , and ≤ i ≤ . This is likely due to the fact that the two-step communication scheme employed in the D-PPHDF is able toreduce the impact of measurement noise and clutter faster than the centralized MS-PPHDF can.Looking at the case of σ r = 0 . in Figure 3b), we observe that the higher measurement noise affects the performanceof all algorithms, resulting in higher OSPA curves. While the OSPA curves of the MS-PPHDF and the DDF-PPHDF areproportionally shifted upward by approximately the same value, i.e., they are equally impacted by the higher noise level, theD-PPHDF seems to be slightly more affected by the change. However it still outperforms the DDF-PPHDF at all time instants.All in all, the proposed D-PPHDF yields better performance than the existing DDF-PPHDF in estimating the number oftargets and tracking them, irrespective of the amount of measurement noise. In addition, it is also a bit faster in deliveringcorrect state estimates of new targets than the centralized MS-PPHDF and performs only slightly worse once the number oftargets stays constant. In our future work, we will look at ways to further improve the performance of the MS-PPHDF andthe D-PPHDF in order to approach the DPCRLB even more closely. D. Simulation II: Results
In the second simulation, we evaluate the performance of the MS-PPHDF, the D-PPHDF and the DDF-PPHDF under ahigher clutter rate of . . The remaining parameters are chosen as in the previous simulation. The simulation results are shownin Figure 4. The top part considers zero-mean Gaussian measurement noise with a per-component variance of σ r = 0 . whilethe bottom part shows the results for σ r = 0 . .While the higher clutter rate causes an increase in the OSPA value of all algorithms, the MS-PPHDF is still able to correctlyestimate the number of targets (except for the crossing period ≤ i ≤ ) in both cases. When taking the next lower integerof the estimate, the D-PPHDF also yields acceptable results for σ r = 0 . . For σ r = 0 . the number of targets is overestimatedby 1 for ≤ i ≤ , causing a stronger degradation of the scaled and squared OPSA value in this interval.Apparently, the DDF-PPHDF is not able to cope with a clutter rate of . as the number of targets is largely overestimated.Hence, no accurate target tracking is possible. UBMITTED FOR PUBLICATION 22
PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (seconds)
00 5 15 2510 10 20 30 PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N e s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
00 5 15 2510 10 20 30 N e s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N est N e s t Fig. 4: Simulation II: Results for Gaussian noise and clutter rate λ FA = 0 . . The left part of the figure shows the squared andscaled Optimal Subpattern Assignment (OSPA) metric for each algorithm compared to the Distributed Posterior Cram´er-RaoLower Bound (DPCRLB), while the right part compares the estimated to the true number of targets. E. Simulation III: Results
In the third simulation, we evaluate the robustness of the MS-PPHDF, the D-PPHDF and the DDF-PPHDF in the face of ε -contaminated noise and different clutter rates. We consider a per-component variance of the measurement noise of σ r = 0 . and . , an average number of clutter of λ FA = 0 . and . , as well as a contamination of and . The remainingparameters are chosen as before. The simulation results for clutter rates λ FA = 0 . and . are given in Figures ?? and 6,respectively. The top half of each figure considers σ r = 0 . while the bottom half pertains to σ r = 0 . . Rows 1 and 3 dealwith a noise contamination of
10 % , rows 2 and 4 show the results for
30 % .Let us look at the case of λ FA = 0 . first. We observe that the centralized MS-PPHDF is still the best performing algorithm,being largely unaffected by higher noise variance and outliers. The D-PPHDF is a close second, being primarily affected by thehigher clutter rate and the higher noise variance. It shows only a slight additional performance degradation when increasing thenoise contamination to
30 % . Hence, it can be said that both algorithms are robust against outliers and can handle a fractionof at least
10 % in the given scenario. The MS-PPHDF can also cope with λ FA = 0 . while the target position estimates ofthe D-PPHDF might be too imprecise in this case, depending on the problem at hand. The DDF-PPHDF, in contrast, is moreseverely affected by outliers. Both the OSPA value and the estimated number of targets increase with the introduction of noise UBMITTED FOR PUBLICATION 23 contamination. When the number of targets remains constant and no target crossing takes place, i.e. for i < and i > , thenumber of targets is only slightly overestimated. However, when targets two and three enter the scene, i.e. for ≤ i < ,the estimate is inaccurate, which imposes a penalty on the scaled and squared OPSA metric. Hence, the DDF-PPHDF is nota robust algorithm for the considered tracking scenario.In the case of λ FA = 0 . , the DDF-PPHDF, again, breaks down completely. The MS-PPHDF, however, is still able togive accurate results with only slight deviations from the true number of targets and a small OSPA value. Unfortunately, thecombination of noise contamination and more clutter is too much for the D-PPHDF to handle. It overestimates the number oftargets by one to two, causing the OSPA value to rise as well.In summary, the proposed MS-PPHDF and D-PPHDF are—to a certain extent—robust against outliers of the ε -contaminationkind. This property is due to the employed two-way communication scheme, which vets measurements as well as intermediatetarget position estimates against the entire network or the neighborhood of each node. The alternative DDF-PPHDF, however,breaks down in the face of outliers.In [51]–[53], we successfully proposed to use robust estimators to robustify sequential detectors for distributed sensornetworks. We applied the same concept to the distributed D-PPHDF. However, no further performance improvement could begained here since the twofold neighborhood averaging already exhausted the power of neighborhood communication.VI. C ONCLUSION
In this work, we developed a distributed as well as a centralized PPHDF for MTT in sensor networks. We, furthermore,came up with a distributed version of the PCRLB that served as a benchmark in the performance evaluation. Our simulationresults showed that the distributed D-PPHDF is faster in correctly tracking new targets than the centralized MS-PPHDF andperforms only slightly worse when the number of targets stays constant. In addition, it delivers accurate tracking results as longas the targets are far enough apart so that their corresponding measurement clouds are separable. Our approach outperformsthe existing DDF-PPHDF at the cost of additional communication between sensor nodes. Moreover, the proposed trackersare inherently robust against outliers and the centralized MS-PPHDF is even able to handle higher clutter rates. The existingDDF-PPHDF, in contrast, is neither robust nor able to cope with more than
10 % clutter.A
CKNOWLEDGMENT
The authors would like to thank Dr. Paolo Braca from the NATO Science & Technology Organization, Centre for MaritimeResearch and Experimentation in La Spezia, Italy, for his valuable comments.R
EFERENCES[1] S. Challa, R. Evans, M. Morelande, and D. Musicki,
Fundamentals of object tracking . Cambridge University Press, 2011.[2] S. Maresca, P. Braca, J. Horstmann, and R. Grasso, “Maritime surveillance using multiple high-frequency surface-wave radars,”
Geoscience and RemoteSensing, IEEE Transactions on , vol. 52, no. 8, pp. 5056–5071, Aug 2014.[3] J. Rambach, M. F. Huber, M. R. Balthasar, and A. M. Zoubir, “Collaborative multi-camera face recognition and tracking,” in
Proceedings of the 12thIEEE International Conference on Advanced Video- and Signal-based Surveillance (AVSS2015) , August 2015.
UBMITTED FOR PUBLICATION 24 [4] R. Olfati-Saber, J. A. Fax, and R. M. Murray, “Consensus and cooperation in networked multi-agent systems,”
Proceedings of the IEEE , vol. 95, no. 1,pp. 215–233, Jan 2007.[5] O. Hlinka, F. Hlawatsch, and P. M. Djuric, “Distributed particle filtering in agent networks: A survey, classification, and comparison,”
IEEE SignalProcessing Magazine , vol. 30, no. 1, pp. 61–81, 2013.[6] F. S. Cattivelli, C. G. Lopes, and A. H. Sayed, “Diffusion strategies for distributed Kalman filtering: Formulation and performance analysis,”
Proceedingsof the 1st IAPR Workshop on Cognitive Information Processing (CIP) , pp. 36–41, 2008.[7] M. Arulampalam, S. Maskell, N. Gordon, and T. Clapp, “A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking,”
IEEETransactions on Signal Processing , vol. 50, no. 2, pp. 174–188, 2002.[8] Y. Bar-Shalom, P. K. Willett, and X. Tian,
Tracking and Data Fusion: A Handbook of Algorithms . Storrs, CT: YBS Publishing, 2011.[9] D. Reid, “An algorithm for tracking multiple targets,”
IEEE Transactions on Automatic Control , vol. 24, no. 6, pp. 843–854, 1979.[10] S. Oh, L. Schenato, P. Chen, and S. Sastry, “Tracking and coordination of multiple agents using sensor networks: System design, algorithms andexperiments,”
Proceedings of the IEEE , vol. 95, no. 1, pp. 234–254, 2007.[11] D. E. Clark, “Multiple target tracking with the probability hypothesis density filter,” Ph.D. dissertation, Department of Electrical, Electronic and ComputerEngineering, Heriot-Watt University, UK, October 2006.[12] R. Mahler, “Multitarget Bayes filtering via first-order multitarget moments,”
IEEE Transactions on Aerospace and Electronic Systems , vol. 39, no. 4,pp. 1152–1178, 2003.[13] X. Wang, G. Xing, Y. Zhang, C. Lu, R. Pless, and C. Gill, “Integrated coverage and connectivity configuration in wireless sensor networks,” in
Proceedingsof the 1st international conference on Embedded networked sensor systems . ACM, 2003, pp. 28–39.[14] M. R. Balthasar, S. Al-Sayed, S. Leier, and A. M. Zoubir, “Optimal area coverage in autonomous sensor networks,” in
Proceedings of the 2nd InternationalConference and Exhibition on Underwater Acoustics (UA2014) , June 2014.[15] D. Schuhmacher, B.-T. Vo, and B.-N. Vo, “A consistent metric for performance evaluation of multi-object filters,”
Signal Processing, IEEE Transactionson , vol. 56, no. 8, pp. 3447–3457, Aug 2008.[16] H. L. Van Trees,
Detection, estimation, and modulation theory . John Wiley & Sons, 2004.[17] C. Hue, J.-P. Le Cadre, and P. Perez, “Performance analysis of two sequential Monte Carlo methods and posterior Cramer-Rao bounds for multi-targettracking,” in
Proceedings of the Fifth International Conference on Information Fusion , vol. 1, 2002, pp. 464–473.[18] ——, “Sequential Monte Carlo methods for multiple target tracking and data fusion,”
IEEE Transactions on Signal Processing , vol. 50, no. 2, pp.309–325, 2002.[19] M. Uney, D. Clark, and S. Julier, “Distributed fusion of PHD filters via exponential mixture densities,”
IEEE Journal of Selected Topics in SignalProcessing , vol. 7, no. 3, pp. 521–531, June 2013.[20] M. Uney, S. Julier, D. Clark, and B. Ristic, “Monte Carlo realisation of a distributed multi-object fusion algorithm,” in
Sensor Signal Processing forDefence (SSPD 2010) , Sept 2010, pp. 1–5.[21] G. Battistelli, L. Chisci, C. Fantacci, A. Farina, and A. Graziano, “Consensus CPHD filter for distributed multitarget tracking,”
Selected Topics in SignalProcessing, IEEE Journal of , vol. 7, no. 3, pp. 508–520, 2013.[22] R. Mahler, “The multisensor PHD filter: I. General solution via multitarget calculus,” in
Proc. SPIE Signal Processing, Sensor Fusion, and TargetRecognition XVIII , vol. 7336, May 2009.[23] ——, “Approximate multisensor CPHD and PHD filters,” in
Proceesings of the 13th Conference on Information Fusion (FUSION) , July 2010, pp. 1–8.[24] F. Gustafsson, F. Gunnarsson, N. Bergman, U. Forssell, J. Jansson, R. Karlsson, and P.-J. Nordlund, “Particle filters for positioning, navigation, andtracking,”
IEEE Transactions on Signal Processing , vol. 50, no. 2, pp. 425–437, 2002.[25] B. T. Vo, “Random finite sets in multi-object filtering,” Ph.D. dissertation, University of Western Australia, 2008.[26] R. P. Mahler,
Statistical multisource-multitarget information fusion . Artech House, Inc., 2007.[27] ——,
Advances in statistical multisource-multitarget information fusion . Artech House, 2014.[28] B.-N. Vo, S. Singh, and A. Doucet, “Sequential Monte Carlo implementation of the PHD filter for multi-target tracking,” in
Proceedings of the 6thInternational Conference on Information Fusion , 2003, pp. 792–799.[29] ——, “Sequential Monte Carlo methods for multitarget filtering with random finite sets,”
IEEE Transactions on Aerospace and Electronic Systems ,vol. 41, no. 4, pp. 1224–1245, 2005.[30] R. Mahler, “Multitarget moments and their application to multitarget tracking,”
Technical Report, DTIC Document , 2001.
UBMITTED FOR PUBLICATION 25 [31] ——, “’Statistics 102’ for multisource-multitarget detection and tracking,”
IEEE Journal of Selected Topics in Signal Processing , vol. 7, no. 3, pp.376–389, 2013.[32] B. Ristic, D. Clark, B.-N. Vo, and B.-T. Vo, “Adaptive target birth intensity for PHD and CPHD filters,”
IEEE Transactions on Aerospace and ElectronicSystems , vol. 48, no. 2, pp. 1656–1668, April 2012.[33] B. Ristic, D. Clark, and B.-N. Vo, “Improved SMC implementation of the PHD filter,” in
Proceesings of the 13th Conference on Information Fusion(FUSION) , July 2010, pp. 1–8.[34] S. Hong, L. Wang, Z.-G. Shi, and K. S. Chen, “Simplified particle PHD filter for multiple-target tracking: Algorithm and architecture,”
Progress InElectromagnetics Research , vol. 120, pp. 481–498, 2011.[35] A. H. Sayed, “Diffusion adaptation over networks,” in
E-Reference Signal Processing , R. Chellappa and S. Theodoridis, Eds. New York: Elsevier,2013.[36] B. Everitt, S. Landau, M. Leese, and D. Stahl,
Cluster Analysis . John Wiley & Sons, 2011.[37] N. J. Gordon, D. J. Salmond, and A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” in
IEE Proceedings F – Radarand Signal Processing , vol. 140, no. 2, 1993, pp. 107–113.[38] F. Gustafsson, “Particle filter theory and practice with positioning applications,”
IEEE Aerospace and Electronic Systems Magazine , vol. 25, no. 7, pp.53–82, 2010.[39] F. Murtagh, “A survey of recent advances in hierarchical clustering algorithms,”
The Computer Journal , vol. 26, no. 4, pp. 354–359, 1983.[40] B.-N. Vo and W.-K. Ma, “The Gaussian mixture probability hypothesis density filter,”
IEEE Transactions on signal processing , vol. 54, no. 11, pp.4091–4104, 2006.[41] K. Granstrom and U. Orguner, “A PHD filter for tracking multiple extended targets using random matrices,”
IEEE Transactions on Signal Processing ,vol. 60, no. 11, pp. 5657–5671, Nov 2012.[42] K. Granstrom, A. Natale, P. Braca, G. Ludeno, and F. Serafino, “PHD extended target tracking using an incoherent X-band radar: Preliminary real-worldexperimental results,” in , July 2014, pp. 1–8.[43] J. MacQueen et al. , “Some methods for classification and analysis of multivariate observations,” in
Proceedings of the 5th Berkeley Symposium onMathematical Statistics and Probability , vol. 1, no. 281-297. California, USA, 1967, p. 14.[44] M. Inaba, N. Katoh, and H. Imai, “Applications of weighted voronoi diagrams and randomization to variance-based k-clustering: (extended abstract),”in
Proceedings of the 10th Annual Symposium on Computational Geometry , ser. SCG ’94. New York, NY, USA: ACM, 1994, pp. 332–339. [Online].Available: http://doi.acm.org/10.1145/177424.178042[45] P. Braca, S. Marano, V. Matta, and P. Willett, “Asymptotic efficiency of the PHD in multitarget/multisensor estimation,”
IEEE Journal of Selected Topicsin Signal Processing , vol. 7, no. 3, pp. 553–564, June 2013.[46] S. M. Kay,
Fundamentals of statistical signal processing: Estimation theory . Prentice-Hall, Inc., 1993.[47] P. Braca, S. Marano, V. Matta, and P. Willett, “Multitarget-multisensor ML and PHD: Some asymptotics,” in
Proceedings of the 15th InternationalConference on Information Fusion (FUSION) , 2012, pp. 2347–2353.[48] P. Tichavsky, C. Muravchik, and A. Nehorai, “Posterior Cramer-Rao bounds for discrete-time nonlinear filtering,”
IEEE Transactions on Signal Processing ,vol. 46, no. 5, pp. 1386–1396, 1998.[49] C. Hue, J.-P. Le Cadre, and P. P´erez, “Performance analysis of two sequential Monte Carlo methods and posterior Cram´er-Rao bounds for multi-targettracking,” IRISA, Tech. Rep., 2002.[50] A. Bessell, B. Ristic, A. Farina, X. Wang, and M. Arulampalam, “Error performance bounds for tracking a manoeuvring target,” in
Proceedings of the6th International Conference of Information Fusion (FUSION) , vol. 2, 2003, pp. 903–910.[51] W. Hou, M. R. Leonard, and A. M. Zoubir, “Robust distributed sequential detection via robust estimation,”
Proceedings of the 25th European SignalProcessing Conference (EUSIPCO) , Aug 2017.[52] M. R. Leonard and A. M. Zoubir, “Robust sequential detection in distributed sensor networks,”
IEEE Transactions on Signal Processing , Feb 2018,submitted. [Online]. Available: https://arxiv.org/abs/1802.00263[53] M. R. Leonard, M. Stiefel, M. Fauß, and A. M. Zoubir, “Robust sequential testing of multiple hypotheses in distributed sensor networks,” in
Proceedingsof the 43rd IEEE International Conference on Accoustics, Speech and Signal Processing (ICASSP) , April 2018.
UBMITTED FOR PUBLICATION 26 σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (seconds)
005 5 15 2510 20 30 PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds) N e s t σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
005 5 15 2510 20 30 N est N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds) N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds) N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t Fig. 5: Simulation III: Results for ε -contaminated noise and clutter rate λ FA = 0 . . The two upper rows consider σ r = 0 . ,the lower ones show the results for σ r = 0 . . Rows 1 and 3 consider ε = 0 . , rows 2 and 4 show the results for ε = 0 .3
05 5 15 2510 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t Fig. 5: Simulation III: Results for ε -contaminated noise and clutter rate λ FA = 0 . . The two upper rows consider σ r = 0 . ,the lower ones show the results for σ r = 0 . . Rows 1 and 3 consider ε = 0 . , rows 2 and 4 show the results for ε = 0 .3 . UBMITTED FOR PUBLICATION 27 σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t Time steps i (seconds)
005 515 1525 2510 1020 20 30
PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N e s t σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
005 515 1525 2510 1020 2030 30 N est N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
005 515 1525 2510 1020 2030 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t σ r = 0 . ε = 0 . PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLBtrue number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
005 515 1525 2510 1020 2030 3035 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N t g t · (cid:16) ¯ d ( ) (cid:17) a nd B i , d i s t PSfrag replacements
MS-PPHDFD-PPHDFDDF-PPHDF
DPCRLB true number N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist Time steps i (seconds)
05 5 15 2510 10 20 30 N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N est N tgt · (cid:16) ¯ d (2)2 (cid:17) and B i, dist N e s t Fig. 6: Simulation III: Results for ε -contaminated noise and clutter rate λ FA = 0 . . The two upper rows consider σ r = 0 . ,the lower ones show the results for σ r = 0 . . Rows 1 and 3 consider ε = 0 . , rows 2 and 4 show the results for ε = 0 .3