Particle labeling PHD filter. for multi-target track-valued estimates

Transcription

1 4th International Conference on Information Fusion Chicago Illinois USA July 5-8 Particle labeling PHD filter for multi-target trac-valued estimates Hongyan Zhu Chongzhao Han Yan Lin School of Electronic & Information Engineering Xi an Jiaotong University Xi an Shaanxi 749 P. R. China Abstract--Multi-target tracing is a difficult problem due to the measurement origin uncertainty. Recently the probability hypothesis density (PHD filter provides a promising tool for oint estimation of target number and multi-target states without using data association technique. In particle implementations of the PHD filter clustering is used to extract the target state from the particle population. This technique yields poor performance when the estimated number of targets differs from the number of clusters in the particle population. A particle labeling PHD filter for multi-target trac-valued estimates is developed in this paper. By implementing the efficient sampling and particle labeling technique the proposed method can yield not only better state estimate but also trac-valued estimate. The multi-scan measurement information is also employed to reduce the uncertainty of the estimates. Simulation results demonstrate the efficiency of the proposed method. Keywords: Probability hypothesis density random finite set trac-valued estimate. Introduction For the multi-target tracing (MTT problem in the presence of measurement origin uncertainty data association [-8] can be done to determine the assignments between measurements and targets. The performance of state estimate will be seriously affected by the accuracy of data association. The random finite sets (RFSs theory [9]-[] provides a promising tool to the MTT problem. The probability hypothesis density (PHD filter [3] is a suboptimal but computationally tractable alternative to the RFSs Bayesian multi-target filtering. It is a recursive mechanism which propagates the first-order statistical moment---the posterior expectation instead of the posterior distribution. Multi-target state estimates can be obtained by applying the pea extraction technique over the PHD function. The PHD filter can be especially used in the cases with unnown target number and gives the estimation of target number and multi-target states simultaneously. Reference [4] relaxed the Poisson assumption on the target number and derived a generalization of the PHD recursion nown as the Cardinalized (CPHD recursion by ointly propagating the posterior intensity function and the posterior cardinality distribution. Since the PHD filter equations still include some high-dimension integrals there is no closed-form solution in general. In [5] the Gaussian mixture PHD (GM-PHD filter is developed which is a closed-form solution to the PHD filter under linear Gaussian assumptions. In non-linear non-gaussian cases the Sequential Monte Carlo (SMC techniques [6]-[9] are regarded as an efficient means to approximate the density function by recursively propagating a group of weighted particles. An SMC implementation of the PHD filter is proposed in []-[] and the analogous idea can also be seen in []-[4]. Unfortunately since the original PHD filter [] eeps no record of trac identities the estimation of individual target can t be obtained directly. Some approaches [5]-[8] have been developed in order to acquire the trac-valued estimate. In [5]-[6] multi-target state estimates are firstly obtained by standard clustering ISIF 84

2 techniques and then trac-valued estimates were acquired by performing trac-to-estimate association based on the MHT mechanism. Reference [7] eeps a separate tracer for each target and constructs a two-dimensional assignment problem to perform pea-to-trac association across the adacent two scans. Reference [8] also introduced extra indices to each one in the whole particle set to store the trac identity. But when the estimated number of target differs from the number of clusters in the particle population the original particle PHD filter fails to give good state estimates. In the above methods the clustering is done among the whole particle set to obtain multi-target state estimates. But when the targets are close the particle cloud from different targets can overlap each other. The particles generated from one target can be partitioned wrongly into another cluster from other target which will yield poor performance. In this paper a particle labeling PHD filter for multi-target trac-valued estimates is proposed. Each one in the whole particle set has its own label which indicates the target identity. The particles are evolved with its label and the weights are predicted and updated according to the measurement information. The estimation for individual targets will be obtained by the weighted sum of particles with the same label in order to avoid being influenced by those particles from other targets. When the targets are close high weights of some particles are possibly due to the measurement generated by the other target. So the directly weighted combination of all the particles with the same label may be wrong. For this reason we need to develop an efficient method to extract some particles from the particle set with the same label which can represent the target state with a high quality. Here we do clustering in those particles with the same label firstly and then establish a temporary node for each cluster. Next we should mae the decision to tell which node will be used to update the target state. Hence we define a decision window with a given width by using of the multi-scan measurement information. When the time index is within the decision window each temporary node will be extended into several new nodes according to the clustering result. Each node records all its particles the weights and the node history indicating how it is generated from the beginning. If the time index ust lies on the decision time then the decision is made based on the maximal weight criteria and the whole trac-valued estimate can be obtained by going bac the node history. In addition we draw the samples based on a mixture proposal based on the measurement set instead of using the transition density. In this way the sampling efficiency can be improved greatly. Problem formulation. Multi-target state model and measurement model LetX andz denote the multi-target state space and measurement space N and M be the number of targets and measurements at time respectively. Then the multi-target states are x x... x N X and the measurements are z z... z M Z. The multi-target state set and measurement set at time can be described as the finite set variables: X = { x x... x N } X ( F X Z = { z z... z M } Z ( F Z where F ( X and F ( Z represent the respective collections of all finite subsets generated from X and Z. Let Z be the cumulative measurement set until time and f ( X X and g( Z X be the multi-target transition probability and the lielihood function. The optimal multi-target Bayesian recursive formula are given by p ( X Z ( f ( X X p ( X Z dx = g( Z X f ( X Z ( ( ( p X Z = ( g Z X p X Z dx. Probability hypothesis density filter The optimal multi-target Bayesian recursive formula is computationally intractable. The Finite set statistics (FISST theory [3] provides a powerful tool that allows the extension of the Bayesian inference to multi-target tracing cases directly. The PHD filter is a suboptimal 843

3 alternative to the multi-target Bayesian recursive formula in the FISST framewor. The probability hypothesis density D ( x Z is the first order statistical moment of the multi-target posterior density f ( X Z which is a non-negative function with the following property Nˆ ( S (3 = X S f ( X Z dx= D ( x Z dx S S where N ˆ ( S is the expected number of targets in region S. The PHD filter involves a prediction step and an update step as follows. Prediction step: D ( x Z γ ( x ϕ( x x D ( x dx where = + (4 ϕ x x Ps x f x x b x x ( = ( ( + ( (5 P x The target survives with a probability s( and f ( x x denotes the transition probability density for individual targets. Let b (. x be the intensity of the target RFS spawned from previous state x and γ ( x be the intensity function of the spontaneous birth RFS. Update step: ( D x Z PD( x g( z x = PD ( x + D ( x z Z κ ( ( z + C z where C z = P x g z x D x Z dx (7 ( ( ( ( D where PD ( x and g( z x are the detection probability and the lielihood function for individual targets respectively. κ ( z denotes the intensity of the clutter RFS. It can be also written to be rc (. and r is the average clutter number per scan assuming a Poisson distribution c (. is the probability distribution for each clutter measurement..3 Particle PHD filter Since the PHD filter still involves high dimensional (6 integrals the SMC technique is combined with the PHD filter to deal with the computational intractability []. For simplicity assuming Ps ( x =. Let L - be the total number of particles at time - and set L =. Given a particle representation of D by the particle L x w at time - L D x w δ ( i x x set { } ( = ( (8 Prediction step: For i =... L sample x q(. x Z where qx ( x Z is the proposal distribution. And then the predicted weights can be given by w ϕ( x x = w qx ( x Z To explore newborn targets J new particles are generated. For i = L +... L + J sample x p (. Z and compute the weights w γ ( x J p x Z = ( (9 ( Where p(. Z is the probability density for newborn targets. Update step: For each measurement z Z compute L + J ( ψ z( C z = x w ( where ψ z ( x = PD( x g( z x and then update the weights ψz ( x w = PD( x + w ( z Z κ ( ( z + C z Resampling step: Firstly the sum of weights are given by Nˆ L + J = w (3 and then resample { x ˆ w N } i L { ˆ } x w N where L + J to get = L = L + J. Finally 844

4 rescale the weights by N ˆ to get the updated particle set{ L } x w at time. State estimate: Multi-target state estimates can be obtained by the pea extraction over the PHD function represented by a set of weighted particles and standard clustering techniques [33] can be used to acquire these peas. 3 Particle labeling PHD filter 3. Basic idea In the original particle PHD filter described in Section multi-target state estimates are obtained by standard clustering technique. But if the estimated number of targets is incorrect multi-target state estimates given by standard clustering technique are unreliable. Actually each one in the whole particle set is produced from different origin either from some existing target or newborn target. This information is ignored in the original particle PHD filter. If we add a label to each particle which records the target identity and can be inherited along with the particle evolution procedure then the state estimate for individual target can be obtained by using of the particles with the same label. If the targets are well-separated the weighted sum of particles with the same label can give good state estimate. But the result is unsatisfactory when the targets are close together because the high weight of some particles may be produced by measurements from other targets. determine which cluster will be used to compute the target state the multi-scan measurement information is used to lessen the uncertainty. In the above figure the width of the decision window is set to be. The circles denote the nodes which are produced according to the clustering result among the particles with the same label at each scan. If a node has no a father node then it corresponds to a newborn target. The label in the circle is the trac identity and it will be inherited by all its descendant nodes. At the decision moment for each target the cluster with the maximal sum of weights (blac circles in the figure is ept and the other ones are eliminated. How to draw the samples is a ey step to implement the particle filter. In this paper we sample from a mixture proposal instead of the transition probability done in the original particle PHD filter. The presented algorithm is described as follows. 3. Algorithm L Let { } x w l denote the whole particle ( set at time - x i and w represent the i th particle and the corresponding weight l - means the target identity with l ˆ - Γ = {... Nmax} and ˆN max is the maximal target identity until time -. At time Sample for existing targets. For i =... L sample x q(. x Z where q(./ x Z denotes the proposal distribution. Here a mixture proposal is given by M = p(./ x Z = b p(. x z (4 Fig.. Basic principle of the presented method Here we do clustering among the particles with the same label. Each cluster is a set of particles which is a candidate to be used to compute the target state. To where b p( θ Z( =... M and θ ( > is the association hypothesis which means that the measurement z originates from the target. The hypothesisθ means no measurement comes from the target. The probability can be computed similar to the association probability in probabilistic data association []. The sample procedure is drawn from the mixture proposal as follows. 845

5 Sample from the proposal q (. /. for the measurement index m q(. x Z where the proposal q (. /. is chosen to be discrete distribution {( (...( } b b M b. M Sample from the proposal q (. /. for the target state x q (./ x z m where the proposal q (. /. is set to be the posterior distribution which can be approximated easily by applying the local-linearization technique or the unscented transformation. If m = then the sample can be drawn from the transition probability. The predicted weights can be computed by the following formula w ϕ( x x = w px ( / x Z (5 Sampling for newborn targets. For L +... L + J Sample x p(. Z and compute the weights for newborn targets w = γ ( ( x J p x Z (6 L 3 The whole particle set { } J x w l + can be obtained by combining all the particles generated from the above two steps. For i =... L the label l can be inherited from l directly that is l = l. Fori = L +... L + J a temporary label is assigned until it is finally confirmed in Step 5. Then the weights can be updated as the following formula ψ z ( x w = PD( x + w (7 z Z κ ( ( z + C z L + J ( ψ z( C z = x w (8 Then the resampling procedure is done as the original particle PHD filter does and then the whole particle set { } J x w l + = can be obtained. The estimated L i number for newborn targets is given by L + J ˆ n ( L + N = round w. 4 Clustering In this step clustering is done among those particles with the same label. We denote P as the particle set with label ( { Γ }. For Γ the particle set P is classified into Nˆ e clusters. Where Nˆ min{ ˆ e = m } Na and m is the measurement number in the validation gate of target. Nˆ =+ ˆ a N c where N ˆ c means the target number which are close to target. It can be computed as the number of targets whose validation gate intersects with the one of target. For = the particle set P is partitioned into N ˆ n clusters. 5 Trac Extension and decision We set a time window using the multi-scan measurement information. When the time index is within the time window we establish a node for each cluster l ( l Nˆ e from P which carries all the information about the particles the weight and its generation history. When the time index is ust on the decision time the final decision will be made as follows. a we denotes ω l as the sum of weights for each cluster l ( l Nˆ e from P ( Γ and let * * = ω = ω (9 l arg max{ } w * l Nˆ l l e * If w is larger than some given thresholdg T then the *th l cluster is ept. The multi-scan target states within the time window can be obtained by going bac in the node history of cluster l *th. The other clusters with label will * be deleted along with its ancestor nodes. If w is less than the thresholdg T the trac will be terminated. As a L result the particle set { x w l } at time will be updated by all the left particles. b For each cluster from P if the sum of weights is greater than some given threshold G I then a new target can be initiated and a new trac identity is assigned to all the particles in this cluster. The maximal target identity is also modified by N ˆ ˆ max = Nmax +. Otherwise the 846

6 cluster will be deleted. 4 Simulation results For simplicity we only consider linear Gaussian model in -D plane. The target moves in the surveillance region [- ] m [- ] m. The target state taes the form x( =[x( x( y( y( ] T where x( and y( are the position components x( and y( denote the velocity components of the target state at time. The target dynamics is given by T / T T x( + = x ( + v ( T T / T where the process noise v ( N (.; Q Q σ σ T diag([ vx vy] ( with =. The measurement equation is z ( = x ( + w ( ( where the measurement noise w ( N (.; R T with R = diag([ σ σ ] σ =m and σ =.5m. wx wy The process noise is assumed independent of the measurement noise. We set the sample interval T = s. The clutter measurements are uniformly distributed over the surveillance region with an average rate r points per scan. New-born targets can appear according to a Poisson point process with the intensity function γ =.5Ν (.; x Qn+.5 Ν (.; x Qn with x = [- -] T x = [8-7 5] T T and Qn = diag([4..4].the particle number for each existing target is set to be 5 and for newborn targets. For simplicity we only consider the unity detection probability and survival probability. The Optimal Sub-pattern Assignment (OSPA metric presented in [9] is employed to evaluate the performance. It is a mathematically and intuitively consistent metric which tries to capture the estimate quality both the cardinality and multi-target states. Let X = { x... x m } and Y = { y... y n } represent the real multi-target state set and the estimated one with the cardinality m and n ( mn N = {...} wx wy and be the permutations on the set {... } for any N = {...}. For p< c> The OSPA metric of order p with cut-off c is defined as : if m n ( c d ( X Y p m ( min ( c ( ( p p = d xi yπ i c ( n m n + π n if m > n ( c ( c dp ( X Y = dp ( Y X (3 where ( c d ( x y = min( c d( x y denotes the distance between x y with cut off c >. We define d( x y as the Euclidean distance and set c = and p =. The thresholds for trac initiation and trac termination are set G =.8 and G =.5 respectively. The I T measurements are supported by 5 MC runs performed on the same target traectory but with independently generated measurements for each trial. Example In this scenario the average number of clutter measurement per scan is set r = and the standard derivation of process noise isσ vx = σ vy =. The width of decision window is set to be. The multi-target traectories and state estimates against time are plotted in Fig.. In Fig.3 the OSPA error is shown. ym OSPAm real trac presented method original particle phd xm Fig.. Multi-Target Traectories and State Estimates presented method original particle phd times Fig.3. OSPA Error p 847

7 Example ym ym ym In this example the expected number of clutter measurement per scan is set r = 6 and the standard derivation of process noise isσ σ vx = =. The width of decision window is set to be and 3 respectively. The multi-target traectories and state estimates against time are illustrated by the Fig real trac original particle phd xm Fig.4. State Estimates by the Original Particle PHD Filter real trac presented method with single-scan xm Fig.5. State Estimates by the Presented Method with Single-Scan real trac presented method with multi-scan xm Fig.6. State Estimates by the Presented Method with Multi-Scan The OSPA error is shown Fig. 7. vy OSPAm presented method with single scan original particle phd presented method with multiple scan times Fig.7. OSPA Error In this scenario when we apply the original particle PHD filter and the presented method with single-scan measurements trac loss phenomena happened. But if we set the width of decision window to be 3 good tracing performance can be achieved. It shows that multi-scan measurement information is needed to improve the performance in cases with large process noise and high clutter density 5 Discussions and conclusions A Particle labeling PHD filter for multi-target trac-valued estimates is developed in this paper. There are some possible future research directions. To reduce the uncertainty of the problem the multi-scan measurement is employed in the presented method. As a result the decision delay and high computational burden is inevitable. So a variable width of the decision window may be more economic and can eep the balance between computational burden and algorithm performance according to the current measurement distribution and the target maneuvering level. Acnowledgement The wor is sponsored by the national ey fundamental research & development programs (973 of P.R. China (7CB36 and National Natural Science Foundation of China (6487/F39. References [] Bar-Shalom Y. and Fortmann T. E. Tracing and Data Association. Academic Press 988. [] Y. Bar-Shalom and X. R. Li Multitarget-Multisensor Tracing: Principles and Techniques YBS

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