Identical Maxiu Lielihood State Estiation Based on Increental Finite Mixture Model in PHD Filter Gang Wu Eail: xjtuwugang@gail.co Jing Liu Eail: elelj20080730@ail.xjtu.edu.cn Chongzhao Han Eail: czhan@ail.xjtu.edu.cn Xiaoxi Yan School of Electrical and Inforation Engineering Jiangsu University Zhenjiang, China Eail: yanxiaoxi1981@gail.co Xueen Wang Eail: wangxueen@gail.co Abstract The increental finite ixture odel is proposed for the ultiple target state estiation in sequential Monte Carlo ipleentation of probability hypothesis density filter. The odel is constructed in increental way. It consists of two steps, the step of inserting new coponent into odel and the step of estiating ixing paraeters. The axiu lielihood criterion is adopted for both steps. At the inserting step, the inserted coponent is selected fro the candidate set of new ixture coponents by axiu lielihood, while the ixing paraeters of existing coponents reain unchanged. Expectation axiization algorith is adopted at the step of ixing paraeters estiation by axiu lielihood. The step of inserting new coponent into ixture odel, and the step of estiating ixing paraeters by expectation axiization algorith, are alternately applied until coponent nuber is equal to the estiate of target nuber. The candidate set of new ixture coponents for inserting into ixture odel is generated by -diensional tree. The increental finite ixture odel unifies the increasing tendency of coponent nuber and that of lielihood function so that it contributes to search axiu lielihood solution of ixture paraeters step by step. Siulation results show that the proposed state estiation algorith based on increental finite ixture odel is slight superior to the existing two algoriths in sequential Monte Carlo ipleentation of probability hypothesis density filter. Index Ters probability hypothesis density filter, sequential Monte Carlo ipleentation, increental finite ixture odel, axiu lielihood, expectation axiization. I. INTRODUCTION Multiple target tracing is to estiate the target nuber and target states siultaneously fro a sequence of easureent sets. At one given tie, the obtained easureents are given in the for of set, and the achieved estiates of ultiple target states are also presented in the for of set. Therefore, ultiple target tracing is a proble of set-valued estiation in nature, in which both target nuber and target states are unnown. It is ore reasonable to solve the proble of ultiple target tracing directly in the integral view of set-valued estiation. Most of existing ultiple target tracing algoriths are based on data association [1] - [3]. The correspondence of targets and easureents in these algoriths should be set up first. The basic character of these algoriths is to brea the proble of ultiple target tracing into several probles of single target tracing. They solve the proble of ultiple target tracing in the individual view, and try to update the individual target state with the corresponding easureent. Point process theory is recognized as the atheatical fundaent for solving ultiple target tracing in the view of integration. The intensity of point process, which is an integral paraeter on the whole space, ay be ade use for ultiple target tracing in further. There have not been systeatic and rigorous algoriths in the 181
view of integration until probability hypothesis density PHD filter is proposed by Mahler by the tools of rando finite sets statistics [4], [5]. The integral of PHD in the whole state space is the expectation of target nuber. PHD collapses the ultiple target posterior density defined in ultiple target state space into single target state space. PHD filter estiates the intensity function fro which the ultiple target states can be extracted with additional algorithic support. Recently, there are two new insights of PHD filter. One is in the view of bin-occupancy in [6], and the other is in the view of Poisson point process in [7]. There have been two ipleentations of PHD filter, Gaussian ixture ipleentation [8] - [10] and sequential Monte Carlo ipleentation [11], [12], which are suitable for linear-gaussian dynaics and nonlinear non-gaussian dynaics, respectively. And their convergence is presented in [13], [14]. Analogous to the superiority of Kalan filter to alphabeta filter, the perforance of PHD filter can be iproved by introducing ore inforation. The cardinalized PHD CPHD filter, which also propagates the distribution of target nuber besides PHD, is proposed to iprove the perforance of PHD filter [15]. In general, CPHD filter is intractable in coputation for applications. Under the siplified ultiple target linear Gaussian assuptions, Gaussian ixture ipleentation of CPHD filter is developed [16]. PHD and CPHD filters are the proising and unified ethodologies. They have been widely applied in any fields, such as aneuvering target tracing [17], sonar target tracing[18], video tracing [19] - [21] and so on. In the sequential Monte Carlo ipleentation of PHD filter, SMC ethod is adopted in approxiating the integral of PHD in single target state space [11]. This approxiation is done according to the relation of probability density function and rando finite set statistics. The weighted particles are used to represent the PHD. These particles are propagated over tie by iportance sapling and resapling operations. As the integral value of PHD in single target state space is the expectation of target nuber, the suation of particle weights is the estiate of target nuber. The output of this ipleentation is a weighted particle set approxiating the PHD. However, the goal of ultiple target tracing algoriths is to estiate the ultiple target states rather than PHD. Therefore, the algorith of estiating the ultiple target states fro the weighted particle set is very iportant in SMC ipleentation PHD filter. The reaining parts of this paper are organized as follows. The proble description of state estiation in sequential Monte Carlo ipleentation of probability hypothesis density filter is presented in part 2. The identical axiu algorith for ultiple target state estiation based on increental finite ixture odel is given in part 3. Siulation study of proposed algorith is shown in part 4. The conclusion is presented in the last part. II. PROBLEM DESCRIPTION The PHD filter and its sequential Monte Carlo ipleentation are briefly suarized as follows according to [4] and [11]. The proble of ultiple target state estiation is described at the final subsection. A. Probability Hypothesis Density Filter The predicting forulation of PHD filter is D 1 x Z 1 = γ x + [β 1 x x 1 + e 1 x 1 f 1 x x 1 ] D 1 1 x 1 Z 1 dx 1 1 where D the intensity, γ x the birth intensity, β 1 x 1 the spawning probability density, e 1 x 1 the survival probability, and f 1 x x 1 the state transition density. The updating forulation of PHD filter is D x Z 1 = 1 p D x D 1 x Z 1: 1 + ϕ,z x D z D 1x Z 1 z Z where p D x the detection probability, g z x the lielihood function, D z = κ z + C z, κ the clutter intensity, ϕ,z = p D x g z x, and C z = pd x g z x D 1 x Z 1 dx. B. Prediction The prediction in SMC ipleentation of PHD filter consists of existing targets and birth targets. the prediction { operation of existing } targets is done on the particle set x 1 1,, xl 1 1 at tie 1 x i q x i 1, Z 3 where q x i 1, Z is the iportance function. The weight of predicted particle is coputed by φ w i 1 x i 1 =, xi 1 w i q x i xi 1, Z 1 4 where i = 1,..., L 1, L 1 the particle nuber at tie 1, φ 1 x, x 1 the transition density φ 1 x, x 1 = e 1 x 1 f 1 x x 1 +β x x 1 For the birth targets at tie, the particle is sapled by 2 5 x i p Z 6 182
where p Z is probability density of birth targets. The corresponding weight is coputed by w i 1 = 1 J γ p x i x i Z where J the particle nuber for birth targets. C. Update 7 For the easureent z Z, the easureent intensity is coputed by C z = L 1 +J j=1 ϕ,z Then, the particle weight is updated by w i = 1 p D x i D. Resapling Operation + z Z x j w j 1 8 ϕ,z x i κ z + C z The suation of all the particle weights is ˆN = L 1 +J j=1 w i 1 9 w j 10 { The particle set w i / ˆN }, x i is resapled to get { w i / ˆN }, x i L { }. Then, w i, L xi is the particle representation of intensity D x Z at tie. E. State Estiation The tie index is neglected, and L is replaced by n for the siple expression of particle set. Then, the resapled particle set is represented as X = { x 1,, x n}. The goal of this paper is to estiate the ultiple target states ˆX = {ˆx { 1,, ˆx ˆN} fro the resapled particle set X = x 1,, x n}, given the estiate of target nuber ˆN. III. ALGORITHM The increental finite ixture odel is proposed for estiating the ultiple target states ˆX = {ˆx 1,, ˆx ˆN} fro the resapled particle set X = { x 1,, x n}. The iportant character of the proposed odel is that it unifies the increasing tendency of coponent nuber and the axiizing tendency of lielihood function of finite ixture odel. A. Idea The resapled particle set X = { x 1,, x n} is fitted by the proposed finite ixture odel for ultiple target state estiation. The increental finite ixture odel is constructed by the insertion of ixture coponent one by one. The axiu lielihood criterion is adopted for the construction of the increental finite ixture odel. The whole construction of increental finite ixture odel consists of two steps, the step of the insertion of ixture coponent and the step of the estiation of ixture paraeters. These two steps are alternately applied until the ixture coponent nuber is equal to the estiate of target nuber ˆN. The ˆN eans of ultiate finite ixture odel are the estiates of ultiple target state ˆX = {ˆx 1,, ˆx ˆN}. B. The Suary of Algorith The particle x is assued to be fro one of the estiated target states. Then, the distribution of x can be odeled by the finite ixture odel. p M x θ = π p x θ 11 where M the coponent nuber, p x θ the Gaussian ixture coponent, µ the ean, C the covariance, θ = {µ, C } the paraeters of ixture coponent, θ = {θ 1,, θ M, π 1,, π M } the paraeters of finite ixture odel. The weights of all the ixture coponents π 1,, π M are subject to π = 1 12 The log-lielihood of particle set X = { x 1,, x n} under the finite ixture odel p M x θ is log p M X θ = n log π p x i θ 13 The ain steps of ultiple target state estiation algorith based on increental finite ixture odel is briefly suarized as follows. 1 Initialization: When the coponent nuber M is equal to 1, the axiu lielihood estiates of ixture paraeters are achieved by averaging operation of all the particles. 2 Maxiu Lielihood Selection of Inserted Mixture Coponent: The inserted coponent is selected fro the candidate set of new ixture coponents by axiizing lielihood { θ M+1, πm+1 } = arg ax θ M+1,π M+1 log pm+1 X θ 14 where log p M+1 X θ is the lielihood of p M+1 x θ after inserting the ixture coponent p M+1 x θ M+1 into finite ixture odel p M x θ 183
log p M+1 X θ = n log p M+1 x i θ 15 The finite ixture odel p M+1 x θ with M + 1 coponents by inserting the ixture coponent p M+1 x θ M+1 into p M x θ is p M+1 x θ = 1 πm+1 M π p x θ +πm+1p M+1 x θ M+1 18 M = M + 1 is set up for the next recursion. 4 Maxiu Lielihood Estiation of Mixture Paraeters: The paraeters of finite ixture odel p M x θ are estiated by expectation axiization algorith under the criterion of axiu lielihood. The axiu lielihood solution of paraeters θ is represented by θ. 5 Judgeent of Coponent Nuber: If the coponent nuber M is saller than the estiate of coponent nuber ˆN, the operation is transferred to step 2. Otherwise, the whole algorith is end. { The inputs of the whole algorith is particle set X = x 1,, x n} and the estiate of target nuber ˆN. The outputs of the whole algorith are the paraeters { π 1,, π ˆN, µ 1,, µ ˆN, C 1,, C ˆN} of finite ixture { odel with } ˆN coponents. The eans of ixture odel µ1,, µ ˆN are the estiates of ultiple target states. The iportant steps of the algorith are presented in detail in the following subsections. C. The Selection of Inserted Mixture Coponent The inserted ixture coponent is selected fro the candidate set of new ixture coponents by axiu lielihood. Every coponent in candidate set is inserted into the existing finite ixture odel p M x θ, and the corresponding lielihood is coputed. The ixture coponent corresponding to the axiu lielihood is selected as the inserted ixture coponent. When the lielihood is coputed, the eans and covariances of ixture coponents in existing finite ixture odel p M x θ reain unchanged. As there is the constraint of ixing weights in finite ixture odel in forulation 12, the ixing weights after inserting the new ixture coponent are subject to π M+1 + π = 1 19 where ixing weight is π = 1 π M+1 π. p M+1 x θ = D. The Maxiu Lielihood Estiation of Paraeters 1 π M+1 p M x θ+π M+1p M+1 x θ M+1 The axiu lielihood solution of paraeters θ of ixture odel p M x θ with M coponents is 16 where the paraeter set of ixture odel p M+1 x θ is { } ˆθ θ = { θ 1,..., θ M, θ M+1, = arg ax log p M X θ 20 1 π M+1 {π1,..., π M }, π M+1 } θ The expectation axiization algorith is adopted here 17 for the axiu lielihood estiation of paraeters θ. The 3 Reconstruction of Finite Mixture Model: The inserted correspondence of particles and ixture coponents is not ixture coponent under the criterion of axiu lielihood is p M+1 x θ included in the particle set X = { x 1,, x n}. Therefore, M+1. The finite ixture odel with M + 1 coponents is constructed by p M+1 x θ M+1 and p M X = { x 1,, x n} can be assued to be the unco- x θ plete observation of finite ixture odel p M x θ. The correspondence Z = { z 1,, z n} is regarded as the issing part, which indexes the sources of n[ particles fro] the ixture coponents. The eleent z i = z i 1,..., zi M is a binary vector indexing the source of particle x in M ixture coponents. Then, the coplete log-lielihood of particle set X = { x 1,, x n} under the finite ixture odel p M x θ is log p M X, Z θ = n z i log π p x i θ 21 The expectation step and axiization step are alternately applied during the axiu lielihood iteration. The iteration is stopped when the relative change of coplete log-lielihood log p M X, Z θ is saller than the given threshold. The log p M X, Z θ is linear to the Z since the eleent z i is binary. Thus, the coputation of conditional expectation of log p M X, Z θ can be reduced to that of conditional expectation of Z [ W E Z X, ˆθ ] t 1 22 where t is the iteration ties. The quality function is Q θ, ˆθ t 1 = log p M X, W θ 23 where the conditional expectation w i is w i = of index eleent z i ˆπ t 1 p x i ˆθ t 1 24 ˆπ j t 1 p x i ˆθ j t 1 j=1 At the axiization step, the paraeters θ is axiized by 184
ˆθ t = arg ax Q θ, ˆθ t 1 θ 25 All the ixture coponents p x θ = 1,, M in ixture odel p M x θ are assued to be Gaussian. Then, the ixing weight is updated at this step by ˆπ t = n w i n The ean of ixture coponent is updated by n 1 n ˆµ t = w i 26 x i w i 27 The covariance of ixture coponent is updated by Ĉ t = n w i 1 n T x i ˆµ t w i E. The Generation of Candidate Coponents x i ˆµ t 28 The candidate set of inserted ixture coponents is generated through the partition of particle set X = { x 1,, x n} by -diensional tree. Each terinal node in -diensional tree defines one subspace of the particle set. The average of all the particles in one subspace is adopted as the ean of one candidate ixture coponent. For the siplicity, the weights and covariances of all the candidate coponents are assued to be identical. Then, the particle set X = { x 1,, x n} is sparely covered by the ixture coponents in the candidate set. IV. SIMULATION STUDY A two-diensional siulation scenario is considered to verify the proposed ultiple target state estiation algorith. The surveillance region is [-1000, 1000] [-1000, 1000] in eter. The target state is ade up of position and velocity, and the easureent is the position. The target oves in the following dynaic odel x = 1 0 T 0 0 1 0 T 0 0 1 0 0 0 0 1 x 1 + T 2 /2 0 0 T 2 /2 T 0 0 T [ v1, v 2, ] where x = [x 1,, x 2,, x 3,, x 4, ] T the whole state, [x 1,, x 2, ] T the position, [x 3,, x 4, ] T the velocity at tie, T = 1 s the period. There are 100 periods in total. v 1, and v 2, are zero-ean Gaussian white process noises with standard deviations σ v1 = 5 /s 2 and σ v2 = 5 /s 2. They are utually independent. There are four targets over 100 scans. Target 1 and target 2 reain existing over 100 scans. Target 3 appears at the first period and disappears at the 96th period. Target 4 appears at 6th period and reains over the following periods. The survival probability e 1 x 1 is 0.95. The birth target obeys to a Poisson point process with intensity function γ = 0.2N ; x, Q, where x = 600 500 0 0, Q = 100 0 0 0 0 100 0 0 0 0 25 0 0 0 0 25 N ; x, Q the Gaussian coponent with ean x and covariance Q. The detected probability p D x is assued to be 1. The observation equation is y = [ 1 0 0 0 0 1 0 0 ] [ ] w1, x + w 2, where w 1, and w 2, are utually independent zero-ean Gaussian white observation noises, and their standard deviations are σ w1 = σ w2 = 10. The clutter are odeled as a Poisson rando finite set with intensity κ z = λ c c z where λ c the averaged clutter nuber, and c z the clutter distribution. λ c is assued 50, and c z is assued unifor in siulation. 1000 particles are given for each expected target, and 500 particles are given for each birth target. The single target state transition density f 1 x x 1 is adopted as the iportance function q x i 1, Z, and the birth target density N ; x, Q is regarded as the iportance density of birth target p Z. The desired nuber of bucets in - diensional tree is 30. The weight of ixture coponent in candidate set is 0.1, and the covariance is σ 2 = 1 10d tr 1 n n T x i x i where d the diensions of x, tr the trace, and the ean of all the particles. The estiates of proposed algorith in one siulation are presented in Fig. 1 and Fig. 2. The proposed algorith is able to detect and trac ultiple targets. 185
copared to the -eans algorith and expectation axiization ipleentation of Gaussian ixture odel EMIGMM in [22]. The log-lielihoods of IFMM and EMIGMM are presented in Fig.3. It is obvious that the IFMM is superior to EMIGMM in log-lielihood. The unifying tendency of coponent nuber and lielihood function in IFMM is responsible for this superiority. Fig. 1 True traces and estiates of X coordinate Fig. 4 The averaged OSPAs of algoriths Fig. 2 True traces and estiates of Y coordinate The Optial Sub-Pattern Assignent OSPA distance between the estiated and true ultitarget state is adopted as the estiation error here, which can jointly capture differences in cardinality and individual eleents between two finite sets in a atheatically consistent yet intuitively eaningful way [23]. For the true ulti-target state set X = {x 1,..., x } and the estiated ulti-target state set X = { x1,..., x n }, the OSPA between X and Y is defined by d c p X, X = 1 n in π Π n d c p x i, x πi + c p n if n, where d c x, x := in c, x x, Π the set of perutations on {1, 2,..., } for any positive integer, p 1 and c > 0. If > n, d c p X, X := d c p X, X. If = n = 0, d c p X, X := d c p X, X = 0. c is 100 here. The coparison of these algoriths in OSPA of 80 Monte Carlo siulations is presented in Fig.4. It is obvious that the proposed algorith is superior to -eans algorith and EMGMM algorith. 1 p Fig. 3 The loglielihoods of algoriths in one Monte Carlo siulation The proposed increental finite ixture odel IFMM is V. CONCLUSION An identical axiu lielihood estiation algorith based on increental finite ixture odel is proposed for ultiple target state estiation in sequential Monte Carlo ipleentation of probability hypothesis density filter. The proposed finite ixture odel is constructed in increental way. The whole algorith consists of the step of inserting the ixture coponent and the step of axiu lielihood estiation of ixing paraeters. The inserted coponent is 186
selected fro the candidate set of new ixture coponents by axiu lielihood. The expectation axiu algorith is adopted to estiate the ixing paraeters by axiu lielihood yet. The proposed algorith unifies the tendency of coponent nuber and the tendency of lielihood function. Siulation results show that the proposed algorith is slight superior to the two algoriths for ultiple target state estiation in sequential Monte Carlo ipleentation of probability hypothesis density filter. ACKNOWLEDGMENT This wor is supported by Foundation for Innovative Research Groups of the National Natural Science Foundation of China No.: 60921003 and the National Science Foundation of ChinaNo.: 61074176. [19] Y. D. Wang, J. K. Wu, A. A. Kassi, and W. M. Huang. Datadriven probability hypothesis density filter for visual tracing, IEEE Transactions on Circuits and Systes for Video Technology, vol. 18, no. 8, pp. 1085 1095, 2008. [20] E. Maggio, M. Taj, and A. Cavallaro. Efficient ultitarget visual tracing using rando finite sets, IEEE Transactions on Circuits and Systes for Video Technology, vol. 18, no. 8, pp. 1016 1027, 2008. [21] E. Maggio, and A. Cavallaro. Learning scene context for ultiple object tracing, IEEE Transactions on Iage Processing, vol. 18, no. 8, pp. 1873 1884, 2009. [22] D. E. Clar, and J. Bell. Multi-target state estiation and trac continuity for the particle PHD filter, IEEE Transactions on Aerospace and Electronic Systes, vol. 43, no. 4, pp. 1441 1453, 2007. [23] D. Schuhacher, B.-T. Vo, and B.-N. Vo. A consistent etric for perforance evaluation of ulti-object filters, IEEE Transactions on Signal Processing, vol. 56, no. 8, pp. 3447 3457, 2008. REFERENCES [1] S. Blacan, and R. Popoli. Design and Analysis of Modern Tracing Systes, Boston: Artech House, 1999. [2] Y. Bar-shalo, and T. E Fortann. Tracing and Data Association, San Diego: Acadeic, 1988. [3] Y. Bar-shalo, and X. R Li. Multitarget-ultisensor Tracing: Principles and Techniques, Storrs: YBS Publishing, 1995. [4] R. Mahler, Multi-target Bayes filtering via first-order ulti-target oents, IEEE Transactions on Aerospace and Electronic Systes, vol. 39, no. 4, pp. 1152 1178, 2003. [5] R. Mahler, Statistical Multisource-Multitarget Inforation Fusion. Norwood: Artech House, 2007. [6] O. Erdinc, P. Willett and Y. Bar-Shalo, The bin-occupancy filter and its connection to the PHD filters, IEEE Transactions on Signal Processing, vol. 57, no. 11, pp. 4232 4246, 2009. [7] R. Streit, Bayes derivation of ultitarget intensity filters, The 11th International Conference on Inforation Fusion, Coloney, vol. 11, pp. 1687-1693, 2008. [8] B.-N. Vo and W.-K. Ma, The Gaussian ixture probability hypothesis density filter, IEEE Transactions on Signal Processing, vol. 54, no. 11, pp. 4091 4104, 2006. [9] S.-A. Pasha, B.-N. Vo and H.-D. Tuan, A Gaussian ixture PHD filter for jup Marov syste odels, IEEE Transactions on Aerospace and Electronic Systes, vol. 45, no. 3, pp. 919 936, 2009. [10] X. X Yan, C. Z Han, and H. Y Zhu. Coponent Pruning Algorith Based on Entropy Distribution In Gaussian Mixture PHD filter, The 14th International Conference on Inforation Fusion, Chicago, vol. 14, pp. 547-552, 2011. [11] B.-N. Vo, S. Singh and A. Doucet, Sequential Monte Carlo ethods for ulti-target filtering with rando finite sets, IEEE Transactions on Aerospace and Electronic Systes, vol. 41, no. 4, pp. 1224 1245, 2005. [12] N. Whiteley, S. Singh and S. Godsill, Auxiliary particle ipleentation of probability hypothesis density filter, IEEE Transactions on Aerospace and Electronic Systes, vol. 46, no. 7, pp. 1437 1454, 2010. [13] D.-E. Clar and B.-N. Vo, Convergence analysis of the Gaussian ixture PHD filter, IEEE Transactions on Signal Processing, vol. 55, no. 4, pp. 1204 1212, 2006. [14] D.-E. Clar and J. Bell, Convergence results for the particle PHD filter, IEEE Transactions on Signal Processing, vol. 54, no. 7, pp. 2252 2261, 2006. [15] R. Mahler, PHD filters of higher order in target nuber, IEEE Transactions on Aerospace and Electronic Systes, vol. 43, no. 4, pp. 1523 1543, 2007. [16] B.-T. Vo, B.-N. Vo and A. Cantoni, Analytic ipleentations of the cardinalized probability hypothesis density filter, IEEE Transactions on Signal Processing, vol. 55, no. 7, pp. 3553 3567, 2007. [17] K. Punithauar, T. Kirubarajan, and A. Sinha. Multiple-odel probability hypothesis density filter for tracing aneuvering targets, IEEE Transactions on Aerospace and Electronic Systes, vol. 44, no. 1, pp. 81 98, 2008. [18] D. Clar, I. T. Ruiz, Y. Petilot, and J. Bell. Particle PHD filter ultiple target tracing in sonar iages, IEEE Transactions on Aerospace and Electronic Systes, vol. 43, no. 1, pp. 409 416, 2007. 187