Integrated Clutter Estimation and Target Tracking ...

Integrated Clutter Estimation and Target Tracking Using Poisson Point Process Xin Chena , R. Tharmarasaa , T. Kirubarajana and Michel Pelletierb a Electrical

and Computer Engineering, McMaster University, Hamilton, ON, Canada b ICx Radar Systems, Laval, QC, Canada ABSTRACT

In this paper, methods of tracking multiple targets in non-homogeneous clutter background is studied. In many scenarios, after detection process, measurement points provided by the sensor (e.g., sonar, infrared sensor, radar) are not distributed uniformly in the surveillance region. On the other hand, in order to obtain accurate results, the target tracking filter requires information about clutter’s spatial density. Thus, non-homogeneous clutter point spatial density has to be estimated based on the measurement point set and tracking filter’s outputs. Also, due to the requirement of compatibility, it is desirable for this estimation method to be integrated into current tracking filters. In this paper, a recursive maximum likelihood method and an approximated Bayesian method are proposed to estimate the clutter point spatial density in non-homogeneous clutter background and both will in turn be integrated into Probability Hypothesis Density (PHD) filter. Here, non-homogeneous Poisson point processes, whose intensity function are assumed to be mixtures of Gaussian functions, are used to model clutter points. The mean and covariance of each Gaussian function is estimated and used in the update equation of the PHD filter. Simulation results show that the proposed methods are able to estimate the clutter point spatial density and improve the performance of PHD filter over non-homogeneous clutter background. Keywords: clutter estimation, non-homogeneous Poisson point process, PHD filter, target tracking, normalWishart distribution

1. INTRODUCTION Although widely used, the drawback of homogeneous clutter model is that the measurement point provided by the sensor after detection is not usually distributed uniformly in surveillance region. The mismatch between the true spatial distribution of clutter points and the spatial distribution model used in the tracking filter may result in poor performance. Thus, spatial density of clutter has to be estimated from measurement points. However, due to the fact that target caused measurement points and clutter caused measurement points are not distinguishable before tracking filtering, the output of tracking filter should also be used if the goal is to get an unbiased estimate of clutter spatial density. One way of estimating clutter spatial density is by assuming that clutter points are uniformly distributed in the validation gate and by using sample spatial density as the estimate of clutter’s spatial density.1 However, this method is based on current measurement set, rather than whole available measurement sets. Also, the volume of validation gate has to be carefully chosen: on one hand, if the validation gate is too small that there are only a few measurements falling in, the estimate of clutter’s spatial density may suffer from large variance; on the other, if the validation gate is large, the uniform distribution assumption of clutter points may not hold anymore. Furthermore, this estimation method is biased, because it does not take into account target caused measurements in validation gate. To obtain unbiased estimate of clutter’s spatial density, “track quality”, the probability that the target exists at current time given previous measurements, has been used to handle target caused measurements in current measurement set.2 However, there it has still assumed that clutter points are uniformly distributed in validation gate. In paper3, 4 the surveillance region has been divided into cells and assumed that clutter points follow homogeneous Poisson point process in each cell. Based on Poisson assumption, three clutter spatial density estimators have been discussed: first is based on the number of measurements falling in each cell, second is based on the Signal and Data Processing of Small Targets 2009, edited by Oliver E. Drummond, Richard D. Teichgraeber, Proc. of SPIE Vol. 7445, 74450X · © 2009 SPIE · CCC code: 0277-786X/09/$18 · doi: 10.1117/12.826140

Proc. of SPIE Vol. 7445 74450X-1

distance between center of cell and its nearest Poisson measurement point, and third is based on the inter-arrival time between two Poisson measurements in the same cell. However, the proper size for each cell has to be determined before estimation procedure. In this paper, the spatial density of clutter points over the whole surveillance region is estimated and integrated into tracking filter, i.e., not only use the estimate of clutter’s spatial density to help tracking, but also use the output of tracking filter to improve estimates of clutter’s spatial density. By modeling clutter points as a non-homogeneous Poisson point process5 and using mixture Gaussian function as its intensity function, it is shown that it is possible to build estimators for clutter’s spatial density under recursive maximum likelihood estimation scheme or approximated Bayesian scheme. Simulation experiments have confirmed that methods proposed here are capable of estimating the clutter’s spatial density and improve the performance of PHD filter over non-homogeneous clutter background. The rest of this paper is organized as follows. In section 2, a maximum likelihood(ML) estimator of clutter point spatial distribution is proposed, then a sequential Monte Carlo(SMC) implementation of proposed estimator and corresponding simulation are given. In section 3, enlighten by paper6, 7 a PHD filter is built for nuisance variables, which describes unknown clutter background. Then, this filter is implemented by using mixture normal-Wishart distribution function and its simulation results are presented. Conclusions are provided in section 4.

2. PHD FILTER AND INTEGRATED CLUTTER PARAMETER ESTIMATOR WITHIN ML FRAMEWORK In this section, an ML estimator of clutter parameter is proposed based on the predictive set likelihood function f (Zk+1 |Z (k) ). Then, a gradient-based maximization method, which could be integrated into SMC implementation of PHD filter, has been used to maximize f (Zk+1 |Z (k) ). Simulation results show that this estimator could improve PHD filter over non-homogeneous clutter background.

2.1 Methodology of ML Estimator for Clutter Parameter Based on the theory of Random Finite Set (RFS), PHD filter has been proposed.8 A recursive form, similar to Bayes-update equation, is defined for PHD filter: • Prediction equation

Dk+1|k (x)

pS (x )fk+1|k (x|x )Dk|k (x )dx

= bk+1|k (x) +

(1)

• Update equation Dk+1|k+1 (x)

∼ =

LZk+1 (x)Dk+1|k (x)

(2)

where LZk+1 (x)

1 − pD (x) +

z∈Z

pD (x)Lz (x) λk+1 (zk+1 ) + Dk+1|k [pD Lz (x)]

(3)

Here, to simplify the problem, it has been assumed that there is no new target spawned from already existing targets. fk+1|k (x|x ) is the Markov transition density for single target; pS (x ) is the surviving probability for already existing target whose state is x at time step k; bk+1|k (x) is the PHD for completely new targets who appear at time step k + 1 with state x; Lz (x) is abbreviation of fk+1 (z|x), the sensor likelihood function; pD (x) is abbreviation of pD,k+1 (x), the target detection probability at time step k + 1; λk+1 (zk+1 ) represents the intensity function of non-homogeneous Poisson clutter points. To avoid the calculation of multiple integrals in (1) and (2), SMC implementation of PHD filter has been presented.9 There, target PHD has been approximated by a large number of weighted particles and a modified importance sampling-resampling method has been proposed.


Using PHD filter, predictive measurement set likelihood function fk+1|k (Zk+1 |Z (k) ) can be calculated as8 fk+1|k (Zk+1 |Z (k) ) = exp −λ − pD Dk+1|k (x)dx λk+1 (z) + pD f (z|xk+1 )Dk+1|k (xk+1 )dxk+1 z∈Zk+1

in which λ =

(4) λk+1 (z)dz.

If there are totally K sample frame available and f (Z0 ) is assumed to be known, then f (Z (K) ) = f (ZK , ZK−1 , · · · , Z1 , Z0 ) = f (Z 0 )

K

f (Zk+1 |Z (k) )

(5)

k=0

Thus, if further assumed λk (z) = λ(z), i.e., the intensity function of clutter does not vary with time, it is possible to build a ML estimator for clutter intensity by maximizing f (Z (K) ) with respect to λ(z). However, without further constrained, the non-negative function λ(z) has infinite dimension and is difficult to estimate. Therefore it is necessary to parameterize λ(z). In this paper, for tracking problems whose measurement has 2 dimension, λ(z) is modeled as a mixture of 2 − D Gaussian components

N (x − μi1 )2 ci (y − μi2 )2 exp − + λ(z) = +m (6) √ 2 2 2π σi1 σi2 2σi1 2σi2 i=1 In above equation, it is assumed that there is homogeneous clutter distributed over the whole surveillance region with constant value intensity function m. It is also assumed that in the surveillance region, there are N subregions whose clutter intensity function has higher value than m, consequently each Gaussian function in (6) corresponds to one of those subregions. For ith Gaussian function, [μi1 , μi2 ] represents the center of ith subregion, σi1 , σi2 represent the width of ith subregion along x and y coordinate respectively and ci (x − μi1 )2 (y − μi2 )2 exp −[ + ] dx dy (7) √ 2 2 2σi1 2σi2 S 2π σi1 σi2 should be equal to the expected number of clutter in ith subregion. For each Gaussian function, there are 5 unknown parameters and there are totally 5 × N + 1 parameters need to be estimated in (6). In order to obtain ML estimate of clutter parameter θ = [c1 , μ11 , μ12 , σ11 , σ12 , · · · , m], a maximization problem has to be solved. In order to solve this maximization problem efficiently, not only f (Z (K) ) is needed, but also its partial difference with respect to clutter parameter, ∂θ f (Z (K) ). In order to get ∂θ f (Z (K) ), a method similar to the one used in paper10 is adopted: First, the partial difference of likelihood function is constructed based on (4) and (5). By (5): K

∂θ log f (Z (K) ) = ∂θ log f (Z 0 ) ∂θ log f (Zk+1 |Z (k) )

(8)

k=0

And by (4), there is

∂θ log f (Zk+1 |Z (k) )

=

−

∂θ λ(z)dz − pD

∂θ Dk+1|k (x)dx ∂θ λ(z) + pD f (z|xk+1 )∂θ Dk+1|k (xk+1 )dxk+1 + λ(z) + pD f (z|xk+1 )Dk+1|k (xk+1 )dxk+1 z∈Zk+1

(9)

Thus, once ∂θ Dk+1|k (x) is obtained, ∂θ log f (Zk+1 |Z (k) ) and ∂θ log f (Z (K) ) could be derived. Then, it becomes possible to recursively update ∂θ Dk|k (x) through equations similar to (1) and (2). There is ∂θ Dk|k−1 (x) = ps f (xk |xk−1 )∂θ Dk−1|k−1 (xk−1 )dxk−1 (10)

pD f (z|x)Dk|k−1 (x) ∂θ A(z) pD f (z|x) + (1 − pD ) − ∂θ Dk|k (x) = ∂θ Dk|k−1 (x) A(z) A(z) A(z) z∈Zk


in which A(z) = λ(z) + pD

Dk|k−1 (x)f (z|x)dx.

Based on above two equations, it is possible to obtain a SMC method to derive particle approximation of Dk|k (x) and ∂θ Dk|k (x), in which same set of particles are used for Dk|k (x) and ∂θ Dk|k (x); but are assigned multiple weight, one is for Dk|k (x) and the other is for ∂θ Dk|k (x): • Assume at the beginning of time step k, the estimate of clutter’s parameter θa ∈ Rm and following particle approximation of target PHD are known N (k−1) (i) N (k−1) (i) (i) (i) – Dk−1|k−1 (x) = i wk−1 δ(x − xk−1 ) ∂θa Dk−1|k−1 (x) = i βθa ,k−1 δ(x − xk−1 ) (i)

• For i = 1, · · · , N (k−1) , sample x ˆk (i)

w ˆk|k−1 = pS ·

∼

ˆ k−1|k−1 N N (k−1)

N (k−1) j=1

(j)

(j)

wk−1 f (·|xk−1 ) and give each of them predicted weight

(i) ¯nb /J. Here, there • For i = N (k−1) + 1, · · · , N (k−1) + J, sample x ˆk ∼ fnb (·) and give each of them weight N ¯ is Nnb fnb (x) = bk+1|k (x), in which bk+1|k (x) represents the PHD of completely new target who appears at sampling time k + 1

• For i = 1, · · · , N (k−1) , calculate (i) βˆθa ,k|k−1

Nk−1

=

(i)

(j)

(j)

xk |xk−1 )βθa ,k−1 j=1 f (ˆ pS N (k−1) (j) (i) (j) wk−1 f (ˆ xk |xk−1 ) j=1

• After receiving measurement set Zk , for each z ∈ Zk , calculate Nk−1 +J

A(z) = λc(z)θa + pD

i=1

(i)

Nk−1

(i)

w ˆk|k−1 f (z|ˆ xk ) ∂θa A(z) = ∂θa [λc(z)] + pD

i=1

(i) (i) xk ) βˆθa ,k|k−1 f (z|ˆ

• For i = 1, · · · , N (k−1) , update (i) w ˆk

=

(i) βˆθa ,k

=

(i) pD f (z|ˆ xk ) (i) w ˆk|k−1 1 − pD + A(z) z∈Zk (i) (i) (i) (i) xk )w ˆk|k−1 ∂θa A(z) pD f (z|ˆ xk ) pD f (z|ˆ + (1 − pD ) − βθa ,k|k−1 A(z) A(z) A(z)

z∈Zk

• For i = N (k−1) + 1, · · · , N (k−1) + J, update (i) pD f (z|ˆ xk ) (i) (i) w ˆk|k−1 w ˆ k = 1 − pD + A(z) z∈Zk

(i) βˆθa ,k = −

(i) (i) pD f (z|ˆ xk )w ˆk|k−1 ∂θa A(z) A(z) A(z)

z∈Zk

• Calculate approximation of ∂θa log f (Zk |Z (k−1) ) ∂θa log f (Zk |Z

(k−1)

) ≈ −∂θa λ − pD

N (k−1) +J i=1

(i) βˆθa ,k +

∂θ A(z) a A(z)

z∈Zk

• Update estimate of clutter’s parameters by maximization method, e.g. using a proper gradient based Newton-Raphson method, to maximize log f (Zk , Zk−1 , · · · , Zk−L+1 |Z k−L ) with respect θ. Here, a strategy, similar to Recursive Maximum Likelihood (RML) estimator,11 is adopted. Instead of maximizing f (Z (k+1) ) with respect θ, joint predictive likelihood function f (Zk , Zk−1 , · · · , Zk−L+1 |Z (k−L) ) is used. Unlike ordinary ML estimator, RML has potential to work on-line. For simulation in next subsection, L is equal to 3.


2.2 Simulation of ML estimator for Clutter Parameter A simulation experiment is run based on the algorithm given in previous subsection. All measurements and states in this simulation experiment are in SI units. Consider three targets with unknown clutter over a two-dimensional surveillance region [−150, 150]×[−150, 150]. The first target appears at sampling interval 1 and disappears at sampling interval 33. The second target appears at sampling interval 12 and disappears at sampling interval 42. The third target appears at sampling interval 22 and disappears at sampling interval 50. Each target’s state independently evolves according to the linear constant velocity Gaussian dynamics model given as xk = Fk|k−1 xk−1 + Γν

(11)

where xk = [x1,k , x2,k , x3,k , x4,k ]T , in which [x1,k , x3,k ]T is the position and [x2,k , x4,k ]T is the velocity at time k. Sampling interval T is 1. ν1,k , ν2,k are mutually independent Gaussian white noise whose standard deviation σν1 = 0.3, σν2 = 0.1. The survival probability ek|k−1 for each existing target is 0.975 and independent of state. The number of newly appearing targets in each sampling interval follows Poisson distribution with parameter 0.05. The initial states of three targets are [−7, 5, −2.5, 3]T , [6.5, −3.5, −7.2, 2]T and [20, 4, 100, −2.25]T respectively. The true trajectories of three targets are given in Fig.1. 100 Target1 Target2 Target3

80

Y

60

40

20

0

−20 −100 −80

−60

−40

−20

0

20

40

60

80

100

120

X

Figure 1: Three targets’ true trajectories over 50 sampling steps Only position measurement is available and measurement noise is mutually independent Gaussian white noise, with standard deviation σω = 1. Here assume the target detection probability, pD , is 1. The spatial distribution of clutters is non-homogeneous: 80% of clutter points is distributed in the region [40, 100] × [45, 75], others are uniformly distributed over whole surveillance region. The total expected number of clutter per sample interval is 10. In this simulation, it is assumed that the number of high clutter density area in surveillance region is known. One thousand particles are used for each target. Also, 100 particles are placed around each measurement in previous sampling step to cover possible new-born targets. In order to reduce the variance of particle approximation of Dk|k (x) and ∂θ Dk|k (x), same seed is used for random number generator in each PHD predict-update recursion. After deriving log f (Zk , Zk−1 , Zk−2 |Z (k−3) ) and its partial derivation with respect to θ, these two R are fed as inputs into MATLAB “fmincon” function. “large-scale” algorithm has been chosen in “fmincon” to perform the maximization. Because the method uesd here is similar to RML, it is possible to obtain estimates of clutter’s parameters at each sampling interval, except the first two and the last two sample steps. To initialize clutter’s parameter, following strategies are adopted: (1) the center point of high clutter density area is [0, 0]; (2) its width is 300 in both x-axis and y-axis; (3) the expected number of clutter in high density area is equal to the expected number of uniformly distributed clutter, where both of them are equal to one half of the number of measurements in first sampling interval. The center and the width of high clutter density area is bounded by parameters of surveillance region. c and m are constrained to be non-negative.


9

4

Estimate True

8

3

Number of Targets

Number of Targets

7 6 5 4 3

2.5 2 1.5 1

2

0.5

1 0 3

Estimate True

3.5

6

9

0 3

12 15 18 21 24 27 30 33 36 39 42 45 48

6

9

12 15 18 21 24 27 30 33 36 39 42 45 48

Sample Step Index

Sample Step Index

(a) Without clutter parameter estimation

(b) With clutter parameter estimation

Figure 2: Target number estimate without vs. with clutter parameter estimation In order to demonstrate the improvement brought by the clutter parameter estimator proposed in previous subsection, here outputs of PHD filter, with and without this estimator, are compared. In the first simulation, it is assumed that clutter point’s expected number per sample step is known, but its true spatial distribution is unknown. Thus, in the first simulation, uniform distribution has been used in PHD filter, i.e., λ(z) in PHD update equation is equal to 10/S (S is the area of surveillance region). In the second simulation, ML estimator proposed in previous subsection is used to estimate parameters of clutter point’s spatial density and the resulting estimates are inserted into PHD update equation. Fig.2 shows the estimate of target number in two simulations and Fig.3 shows the estimate of target position in two simulations. Both figures clearly show that, with proposed clutter parameter estimator, the performance of PHD filter has been improved over this non-homogeneous clutter background. Without clutter parameter estimator, the estimate of number of target is outside of ±0.5 region of true value in most sample steps and a lot of false estimates have occurred in high clutter area. In contrast, with clutter parameter estimator, the estimate of number of target is within ±0.5 region of true value for 80% sample steps and fewer false estimates have occurred in high clutter area. 100

100

Target1 Target2 Target3 Estimate

80

60

Y

Y

60

40

40

20

20

0

0

−20 −100 −80

−60

−40

−20

Target1 Target2 Target3 Estimate

80

0

20

40

60

80

100

120

−20 −100 −80

−60

X

−40

−20

0

20

40

60

80

100

120

X

(a) Without clutter parameter estimation

(b) With clutter parameter estimation

Figure 3: Filter’s position estimate (◦) without vs. with clutter parameter estimation

3. PHD FILTER AND INTEGRATED CLUTTER PARAMETER ESTIMATOR WITHIN APPROXIMATED BAYESIAN FRAMEWORK In this section, mixture Gaussian model is still used for intensity function of non-homogeneous clutter. The idea presented here is emulated from paper6, 7 by first adding some nuisance variables to describe the unknown


dynamic clutter background and then by deriving a PHD filter for these nuisance variables. However, in order to get a tractable filter, here one-to-one assumption between nuisance variable and clutter point is made, then pruning and merging technique is used to reduce computation load by combining closely spaced nuisance variable. Furthermore, by using mixture normal-Wishart function to approximate the PHD of nuisance variable for clutter, an implementation method that is similar to Gaussian mixture probability hypothesis density filter (GM-PHD)12 is obtained. This method can be considered an approximated Bayesian estimator for clutter parameter. Simulation has proved the effectiveness of this method.

3.1 Methodology of Approximated Bayesian Estimator for Clutter Background For measurement on 2-D plane, let z = [x y]T , from mixture Gaussian model of clutter intensity function, there is λ(z) =

N˙ i=1

N˙ mi |ρi | 1 1 mi T exp − (z − μi )T Σ−1 (z − μ ) = ) ρ (z − μ ) exp − (z − μ i i i i i 2 2π 2 2π |Σi | i=1

(12)

Here μi ∈ R2 , Σi ∈ R2 , ρi = Σ−1 i is mean, covariance matrix, precision matrix for each Gaussian term respectively and mi ∈ R+ .Each pair ci = [mi , μi , Σi ] is taken as a clutter generator. (12) is more flexible than (6), since (12) does not implicitly assume that the covariance matrix of each Gaussian term is diagonal. N˙ , mi , μi , Σi are all random here, therefore ci = [mi , μi , Σi ](i = 1, 2, · · · , N˙ ) could be thought as a random point process in a high ˙ In paper,6, 7 CPHD/PHD filter for random point process ci = [mi , μi , Σi ](i = 1, 2, · · · , N˙ ) dimension space X. has been derived. However, that CPHD/PHD filter is not tractable, since all possible partitions of measurement set Zk are needed for CPHD/PHD update equation. To avoid this problem, here it is further assumed that there is one-to-one relationship between ci and clutter point zi , i.e., each ci = [μi , Σi ] only corresponds to one clutter point zi and given ci the likelihood function for zi is 1 1 exp − (zi − μi )T Σ−1 (z − μ ) (13) θ(zi |ci ) = i i i 2 2π |Σi | Assuming measurement set at sampling time step k is Zk , Zk could be decomposed into two parts: one part, (X) (C) Zk , is composed of target caused measurements, while the other part, Zk , is composed of clutter points. (X) (C) (X) (C) (C) (X) Obviously, there should be Zk ∪ Zk = Zk , Zk ∩ Zk = ∅ and Zk statically independent of Zk . Using (X) (C) proposition 4 in [8], the probability generating functional(p.g.fl.) of Zk and Zk are (X) Gk [g|X] (C)

Gk [g|C]

g

(X)

Zk

(C)

g Zk

X · f (Z|X)δZ = 1 − pD + pD g(z)fk (z|x)dz · θ(Z|C)δZ =

(14)

C g(z)θ(z|c)dz

(15)

where X, C are finite set of target and clutter generator respectively. It is assumed that there are two possible ˙ for clutter generators. The joint problem kinds of state-vector for “targets”: x ∈ X for real targets and c ∈ X ¨ which is the disjoint union of X and X, ˙ i.e. X ¨ =X ˙ X and the element in X ¨ is x has state space X, ¨. The joint (C) (X) ¨ state-sets of multiple targets and multiple clutter generators is X = X ∪ C. Because Zk , Zk are statistically independent, p.g.fl. for Zk is ¨ = G(X) [g|X] · G(C) [g|C] = Gk [g|X] k k

1 − pD + pD

X C · g(z)θ(z|c)dz g(z)fk (z|x)dz

¨ define it in space X˙ and X separately For any function on X, h1 (¨ x) if x ¨=x ¨ x) = h(¨ h2 (¨ x) if x¨ = c


(16)

(17)

¨ could be defined as, Using above formula, following functions on X pD if x f (z|x) if x ¨=x ¨=x ¨ z (¨ p¨D (¨ x) = x) = , L θ(z|c) if x ¨=c 1 if x¨ = c

(18)

Using (18), p.g.fl. for joint measurement set Zk becomes

X¨

x) + p¨D (¨ x) 1 − p¨D (¨

¨ = G[g|X]

¨ z (¨ g(z)L x)dz

(19)

¨ k , using above equation, there is Thus, at sample time k, for Zk and X ¨ X¨ k g Zk fk (Zk |X ¨ k )f¨k|k−1 (X ¨k |Z (k−1) )δZδ X ¨ F [g, ¨ h] h ¨ X¨k G[g|X] ¨ f¨k|k−1 (X ¨k |Z (k−1) )δ X ¨ = h

¨ x) 1 − p¨D (¨ ¨ z (¨ = Gk|k−1 h(¨ x) + p¨D (¨ x) g(z)L x)dz

(20)

¨ k , assuming f¨k|k−1 (X ¨ k |Z (k−1) ) is an Poisson point process, i.e., To get a PHD filter for X ¨ = exp N ¨ ¨h(¨ ¨ x)¨ s(¨ x)d¨ x−N Gk|k−1 (h)

(21)

in which

¨ k|k−1 (¨ ¨ s¨(¨ D x) = N x),

¨ = N

¨ k|k−1 (¨ D x)d¨ x,

¨ k|k−1 (¨ D x) =

¨ = N + N˙ , N

N=

Dk|k−1 (x)dx,

N˙ =

Dk|k−1 (x) (C) Dk|k−1 (c)

if x ¨=x if x ¨=c

(C)

Dk|k−1 (c)dc

Insert (21) into (20), it could be found out that

¨ x) 1 − p¨D (¨ ¨ s¨(¨ ¨ z (¨ ¨ x)h(¨ x) + p¨D (¨ x) g(z)L x)dz d¨ x−N F [g, ¨ h] = exp N By rule for set derivative, from above equation it is possible to obtain δF ¨ x)¨ ¨ ¨ ¨ s¨(¨ ¨ z (¨ N x)h(¨ pD (¨ [g, h] = F [g, h] x)L x)d¨ x δZk

(22)

(23)

(24)

z∈Zk

Thus, there is ¨ ¨ k|k [h] G

δF ¨ δZk [0, h] δF δZk [0, 1]

¨ ¨ x)¨ ¨ z (¨ F [0, h] s¨(¨ x)h(¨ pD (¨ x)L x)d¨ x z∈Zk = ¨ x)¨ pD (¨ F [0, 1] z∈Zk s¨(¨ x)Lz (¨ x)d¨ x

(25)

From the above equation, following result could be derived ¨ k|k δG ¨ = [h] δ¨ x

1 ¨ z (¨ x)¨ pD (¨ F [0, 1] z∈Zk s¨(¨ x)L x)d¨ x ¨ x)¨ ¨ s¨(¨ ¨ z (¨ ×{F [0, ¨h]N x) (1 − p¨D (¨ x)) x)L x)d¨ x+ s¨(¨ x)h(¨ pD (¨

F [0, ¨ h]

z1 ∈Zk

z2 ∈Zk

z∈Zk

¨ x)¨ s¨(¨ x)h(¨ pD (¨ x)Lz2 (¨ x)d¨ x s¨(¨ x)¨ pD (¨ x)Lz1 (¨ x) } ¨ x)Lz1 (¨ x)¨ pD (¨ x)d¨ x s¨(¨ x)h(¨ (26)


¨ is By definition, the posterior PHD for X ¨ k|k δG ¨ k|k (¨ [1] D x) = δ¨ x

(27)

Insert (26) into above equation to get ¨ k|k (¨ D x) =

¨ s¨(¨ N x)(1 − p¨D (¨ x)) +

z∈Zk

Insert (22) into above equation, there is ⎛ Dk|k (x) (C)

=

Dk|k (c) =

Dk|k−1 (x) ⎝1 − pD + (C)

Dk|k−1 (c)

z∈Zk

z∈Zk

pD

s¨(¨ x)¨ pD (¨ x)Lz (¨ x) ¨ x)Lz (¨ s¨(¨ x)h(¨ x)¨ pD (¨ x)d¨ x

(28)

⎞ pD f (z|x) ⎠ (C) pD Dk|k−1 (x)f (z|x)dx + Dk|k−1 (c)θ(z|c)dc θ(z|c)

Dk|k−1 (x)f (z|x)dx +

(29)

(C)

Dk|k−1 (c)θ(z|c)dc

To implement (29), proper dynamic/measurement models are needed to describe target and clutter generator separately. By (13), the unknown clutter parameter estimation problem considered here could be taken as an estimation problem for Gaussian random variable with unknown mean and covariance. Similar to GM-PHD , (C) here “conjugate prior” is used to get a partially close form recursion method to calculate Dk|k (c). Also, “pruning and truncating” from GM-PHD is used to reduce the number of components propagated to the next time step and to make sure the computation load will be bounded. (C)

To get a partially close form recursion method to calculate Dk|k (c), for state transition of clutter generator,

here it is further assumed that: (1) the covariance component of clutter generator c(i) does not change from sample time k to sample time k + 1; (2) the state transition p.d.f of its position component μ(i) follows

(C) (i) (i) (i) (i) (30) fk+1|k (μk+1 |μk ) = N μk+1 ; μk+1 , aΣ(i) , a > 0 In other words, from sample time k to sample time k+1, it is assumed that the movement for position component, μi , of ci is a zero-mean Gaussian white noise with covariance proportional to its own covariance component Σi . These assumptions are reasonable, because widely used in cluster analysis.

(i)

(i)

(i)

(i)

(μk+1 − μk )T (Σ(i) )−1 (μk+1 − μk ) is the Mahalanobis distance

For p dimension normal distribution with unknown mean μ and unknown covariance matrix Σ (or precision matrix ρ = Σ−1 ), its conjugate prior is normal-Wishart distribution,13 and there is N (z; μ, ρ)N (μ; m, tρ)K(α)|τ |α/2 |ρ|(α−p−1)/2 exp {−tr(τ ρ)/2} ∗ ∗ = q(z, m, t, α, τ )N (μ; m∗ , t∗ ρ)K(α∗ )|τ ∗ |α /2 |ρ|(α −p−1)/2 exp {−tr(τ ∗ ρ)/2}

(31)

Which K(α)−1

=

2αp/2 π p(p−1)/4

p

Γ(

j=1

m∗ t

∗

τ∗ ∗

α

α+1−j ), 2

(32)

=

(tm + z)/(t + 1)

(33)

=

t+1

(34)

= =

t (m − z)(m − z) τ+ t+1 α+1

(35) (36)

∗

q(z, m, t, α, τ ) =

tp/2 Γ( α2 )|τ |α/2 p

∗ α∗ /2 2 (t + 1)p/2 Γ( α−1 2 )π |τ |


(37)

In above equations, Γ(·) is Gamma function, α > p−1, t > 0 is degree of freedom for normal-Wishart distribution, m ∈ Rp , τ ∈ Rp×p and τ is symmetric and positive definite. Assuming that the posterior PHD of clutter generator at time k − 1 is a normal-Wishart mixture of the form (C)

Jk−1

(C)

Dk−1|k−1 (c) =

i=1

(i) (i) (i) (i) (i) κ(k−1) N μ(k−1) ; m(k−1) , t(k−1) ρ(k−1) (i)

(i)

(i)

(i)

(38)

(i)

(i)

(i)

×K(αk−1 )|τk−1 |αk−1 /2 |ρk−1 |(αk−1 −p−1)/2 exp{−tr(τk−1 ρk−1 )/2} Further assuming that the PHD for new-born clutter generator at time k is also a normal-Wishart mixture of the form (Cb )

(C) bk (c)

Jk

=

i=1

(i) (i) (i) (i) (i) (i) (i) (i) (i) (i) (i) η(k) N μk ; mk , tk ρk K(αk )|τk |αk /2 |ρ(i) |(αk −p−1)/2 exp −tr(τk ρk )/2

(39)

then the predicted PHD is also a normal-Wishart mixture, if the state transition follows (30). This results from

N (μk ; m, tρk )K|τ |α/2 |ρk |(α−p−1)/2 exp{−tr(τ ρk )/2} N (μk+1 ; μk , aρk )δ(ρk+1 − ρk )dμk dρk ta ρk K|τ |α/2 |ρk |(α−p−1)/2 exp{−tr(τ ρk )/2}δ(ρk+1 − ρk )dρk = N μk+1 ; m, a+t ta ρk+1 K|τ |α/2 |ρk+1 |(α−p−1)/2 exp{−tr(τ ρk+1 )/2} = N μk+1 ; m, (40) a+t (C)

Suppose predicted PHD for clutter generator Dk|k−1 (c) is a normal-Wishart mixture of the form (C)

Jk|k−1

(C)

Dk|k−1 (c)

=

i=1

(i) (i) (i) (i) (i) (i) κk|k−1 N μk|k−1 ; mk|k−1 , tk|k−1 ρk|k−1 Kk|k−1

(i)

(i)

αk|k−1 /2

×|τk|k−1 |

(i)

(i)

(αk|k−1 −p−1)/2

|ρk|k−1 |

(i)

(41) (i)

exp{−tr(τk|k−1 ρk|k−1 )/2}

Then, the posterior intensity at time k is also a normal-Wishart mixture and it has form (C) (C) Dk|k (c) = Dz, k|k (c)

(42)

z∈Zk

where (C)

(C) Dz, k|k (c)

Jk|k−1

=

j=1

(j) (j) (j) (j) (j) (j) (j) (j) (j) (j) (j) κk (z)N μk ; mk , tk ρk Kk × |τk |αk /2 |ρk |(αk −p−1)/2 exp{−tr(τk ρk )/2}

(j) κk (z)

(j)

(j)

=

(j)

κk|k−1 qk (z) Dk|k−1 (x)f (z|x)dx + (j)

(j)

(43)

(C) Jk|k−1

(l)

l=1

(j)

(44)

(l)

κk|k−1 qk (z) (j)

mk = mk|k−1 + (tk|k−1 mk|k−1 + z)/(tk|k−1 + 1)

(45)

(j)

(j) tk

=

(j) tk|k−1

+1

(j) αk

=

(j) αk|k−1

+1

(j) τk

=

(j) τk|k−1

+

tk|k−1 (j) tk|k−1


+1

(j)

(j)

(mk|k−1 − z)(mk|k−1 − z)

(46)

(j)

α +1 (j) (j) tk|k−1 Γ( k|k−1 )|τk|k−1 | 2

(j)

qk (z) =

α

(j) k|k−1 2

(47) (j) +1 αk|k−1 2 (j) (j) (j) (tk|k−1 + 1)Γ( (mk|k−1 − z)(mk|k−1 − z) 2 Above equations could be derived by insert (31) and (13) into (29). To calculate Dk|k−1 (x)f (z|x)dx, SMC PHD method or GM-PHD method can be used, because if Dk−1|k−1 (x) is known already, then the calculation (C) of Dk|k−1 (x) and Dk|k−1 (x)f (z|x)dx is uncoupled with Dk−1|k−1 (c). (j) αk|k−1 −1

p (j) )π 2 τk|k−1 +

(j) tk|k−1 (j) tk|k−1 +1

Similar to GM-PHD, the algorithm proposed above suffer from computation problem, since the normalWishart components will increase without bounds as time progresses (the number of normal-Wishart component (C) at sampling time k is (Jk−1 + JkCb )|Zk |). Thus, a truncating and merging method is necessary. By truncating, those components, whose weights are below some predefined threshold, will be discarded. Then, those closely spaced components, i.e. U1−1 ≤ |αi τi |/|αj τj | ≤ U1 and (mi − mj )T (αi τi )(mi − mj ) ≤ U2 (U1 , U2 is some predefined threshold), will be merged into one normal-Wishart component. Based on these methods, following pruning algorithm can be derived: (i)

(i)

(i)

(i)

J

(C)

k , threshold T for truncating, threshold U1 , U2 for merging • Given {κk , mk , tk , αk , τ (i) }i=1

(C)

(i)

• Truncating: set I = {i = 1, 2, · · · , Jtrunc, k |κk ≥ T } • Merging: – Initialize l as 0 – repeat (i)

∗ l := l + 1, j := argmaxi∈I κk ∗ L := {i ∈ I|(mi − mj )T (αi τi )(mi − mj ) ≤ U2 , U1−1 ≤ |αi τi |/|αj τj | ≤ U1 } (l) (i) (l) (i) (i) (l) (l) (i) (i) (l) (l) (i) (i) (l) ∗ κ ¯ k = i∈L κk ; m ¯ k = i∈I κk mk /¯ κk ; t¯k = i∈I κk tk /¯ κk ; α ¯ k = i∈I κk αk /¯ κk

−1 (l) (i) (i) −1 (i) (l) (i) (l) ∗ τ¯k = + (mk − m ¯ k )(mk − m ¯ k )T i∈I κk (τk ) ∗ I := I\L – until I = ∅

3.2 Simulation of Approximated Bayesian Estimator for Clutter Background To prove the effectiveness of algorithm proposed in section 3.1, a simulation is performed. Its setup parameters, target true state and measurement sets are identical to those in the simulation of section 2.2. Here, measurements in sample frame k −2 are used to generate PHD of possible new born clutter generator in sample frame k. Following two assumptions are adopted: (1) for each clutter generator’s covariance matrix term, its biggest eigenvalue does not exceed the width of surveillance region; (2) the orientation for each covariance matrix corresponding ellipse is uniformly distributed within [0, π]. Totally 1000 normal-Wishart components are generated in each sample steps to cover possible new born clutter generator. However, after pruning and merging procedure, only 100 to 200 normal-Wishart components are need to be propagated to the next sample step. Fig. 4 shows filter’s estimate of number of target and position of target. When Fig. 2 and Fig. 3 are compared, it can be found that, except first 2 to 3 iterations, the algorithm proposed in sec. 3.1 has almost the same performance of algorithm given in Sec. 2. In simulation here, the estimate of number of target stays within ±0.5 region of true value for more than 80% sample steps and only a few false estimates have occurred in high clutter area. One advantage of the algorithm in Sec. 3.1 is that its computation load is much smaller than the algorithm in Sec. 2, since algorithm in Sec. 2 needs to use gradient based method to recursively maximize likelihood function. Furthermore, in each recursion, algorithm in Sec. 2 uses SMC method to calculate ∂θ log f (Zk |Z k−1 ), whose computation load is O(N 2 ) (N is the number of particles in SMC method).


100

4

Estimate True

3.5

Target1 Target2 Targe3 Estimate

80

60 2.5

Y

Number of Targets

3

2

40

1.5

20 1

0 0.5 0 3

6

9

12

15

18

21

24

27

30

33

36

39

42

45

48

−20 −100 −80

−60

−40

−20

Sample Step Index

0

20

40

60

80

100

120

X

(a) Estimate of Number of Target

(b) Estimate of Target’s Position

Figure 4: Output for Filter Proposed in Sec. 3.1

4. CONCLUSION In this paper, two estimator for non-homogeneous clutter’s spatial intensity function are presented. The first one is an ML estimator based on the predictive measurement set likelihood function f (Zk |Z (k−1) ), while the second is based on using the PHD filter, combined with normal-Wishart mixture function, for clutter generators c = [μ, Σ]. Both estimators could be integrated into PHD filter. Simulations have proved that these two clutter density estimators can improve the performance of PHD filter over unknown non-homogeneous clutter background.

REFERENCES [1] Bar-Shalom,Y., Li,X.Y., [Multitarget-Multisensor Tracking: Principles and Techniques], YBS Publishing (1995). [2] Li,X.Y., Li,N., “Integrated real-time estimation of clutter density for tracking,” IEEE Transaction on Signal Processing 48(10), 2797-2805 (2000). [3] Musicki,D., Suvorova,S., Morelande,M., Mora,B., “Clutter Map and Target Tracking,” 8th International Conference on Information Fusion, 69-76 (2005). [4] Hanselmann,T., Musicki,D., Palaniswami,M., “Adaptive target tracking in slowly changing clutter,” 9th International Conference on Information Fusion, 1-8 (2006). [5] Daley,D.J., Vere-Jones,D., [An Introduction to the Theory of Point Processes Volume I: Elementary Theory and Methods], Springer (2002 (second edition)). [6] Mahler,R., “CPHD and PHD filters for unknown backgrounds, I: Dynamic data clustering,” Proc. SPIE 7330, (2009). [7] Mahler,R., “CPHD and PHD filters for unknown backgrounds, II: Multitarget filtering in dynamic clutter,” Proc. SPIE 7330, (2009). [8] Mahler,R., “Multitarget bayes filtering via first-order multitarget moments,” IEEE Transaction on Aerospace and Electronic System 39(4), 1152-1178 (2003). [9] Vo,B.N., Singh,S., Doucet,A., “Sequential Monte Carlo methods for multitarget filtering with random finite sets,” IEEE Transaction on Aerospace and Electronic System 41(4), 1224-1244 (2005). [10] Poyiadjis,G., Doucet,A., and Singh,S., “Maximum likelihood parameter estimation in general state-space models using particle methods,” Proc. American Statistical Association (2005). [11] Doucet,A., and Tadic,V.B., “Parameter estimation in general state-space models using particle methods,” Annals of the Institute of Statistical Mathematical 55(2), 409-422 (2003). [12] Vo,B.N., Ma,W.K., “The Gaussian mixture probability hypothesis density filter,” IEEE Transaction on Signal Processing 54(11), 4091-4104 (2006). [13] DeGroot,M.H., [Optimal Statistical Decisions], McGraw-Hill Book Company (1970 (second edition)).
