Maneuvering target track-before-detect via multiple ... - Springer Link

J. Cent. South Univ. (2015) 22: 3935−3945 DOI: 10.1007/s11771-015-2938-3

Maneuvering target track-before-detect via multiple-model Bernoulli particle filter ZHAN Rong-hui(占荣辉), LIU Sheng-qi(刘盛启), HU Jie-min(胡杰民), ZHANG Jun(张军) Science and Technology on Automatic Target Recognition Laboratory, National University of Defense Technology, Changsha 410073, China © Central South University Press and Springer-Verlag Berlin Heidelberg 2015 Abstract: Target tracking using non-threshold raw data with low signal-to-noise ratio is a very difficult task, and the model uncertainty introduced by target’s maneuver makes it even more challenging. In this work, a multiple-model based method was proposed to tackle such issues. The method was developed in the framework of Bernoulli filter by integrating the model probability parameter and implemented via sequential Monte Carlo (particle) technique. Target detection was accomplished through the estimation of target’s existence probability, and the estimate of target state was obtained by combining the outputs of modeldependent filtering. The simulation results show that the proposed method performs better than the TBD method implemented by the conventional multiple-model particle filter. Key words: Bernoulli filter; multiple model; target maneuver; track-before-detect (TBD); sequential Monte Carlo (SMC) technique

1 Introduction In the traditional detect-before-track (DBT), target measurements are first extracted from the threshold observations (detection), and target state is then estimated by incorporating the motion model with the measurement information. The latent defect of threshold processing is that it may incur the loss of target information or even the whole target, especially when the signal-to-noise ratio (SNR) is low. In the track-before-detect (TBD), however, target detection and tracking are implemented simultaneously by processing the raw observation data. Because the non-threshold raw data are used directly to declare the presence/absence of the target and to estimate the target state, the performance of target detection and tracking can be improved greatly. Various TBD methods have been considered in the literature, and these methods can be largely classified into two categories, i.e., batch methods and recursive methods, according to their implementation manner. Maximum likelihood estimation [1], Hough transform [2], dynamic programming [3−4], etc, are common examples for the batch methods, and they usually process several consecutive frames of observations as a whole. One drawback of these methods is the demanding computational load, and the other drawback is that batch methods have difficulty in

modeling complicated target motion [5], which makes the problem of maneuvering target tracking intractable. As the most typical example of recursive methods, Bayesian filter imposes no constraint on the target motion and can flexibly tackle the TBD issue with model uncertainty. Moreover, because there is no need to store and process multiple frames of observation data, Bayesian filter can also reduce the computational burden significantly. To implement Bayesian recursion, the sequential Monte Carlo (SMC) method or particle filter [6−8] has served as an effective way and is extensively applied to TBD context [9−12]. Bernoulli filter is a class of exact Bayesian filter for the general non-linear/non-Gaussian dynamic systems, and has recently emerged from the random set theoretical framework. The Bernoulli filter is designed for stochastic dynamic systems which can switch on and off randomly. For this special and important feature, Bernoulli filter is quite suitable for describing the phenomena of target presence or absence, and for this reason it has been successfully used to TBD implementation [13−14]. Up to now, the existing TBD literatures mainly focus on the performance improvement for the general target motion without dynamics uncertainty. The TBD problem with motion model transition, however, is seldom investigated. In this work, a multiple-model Bernoulli particle filter is proposed to tackle the issue of maneuvering target TBD. The developed method is an

Foundation item: Projects(61002022, 61471370) supported by the National Natural Science Foundation of China Received date: 2014−09−20; Accepted date: 2015−01−31 Corresponding author: ZHAN Rong-hui, PhD; Tel: +86−731−84574484; E-mail: [email protected]

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effective integration of the multiple model approach and the Bernoulli particle filter, which not only can keep the state-of-the-art processing mechanism of TBD but also can flexibly accommodate the uncertainty of target motion models. Simulations are carried out under various conditions to validate the effectiveness of the presented method.

2 System model

 i =diag [Qi , qw2 ] , i=1, 2, 3

(3) where qw2 is the power of the target’s intensity noise. At a certain time step, the model variable only takes one discrete value out of M={1, 2, 3}, and its transition process from time k−1 to time k can be modeled by first order Markov train. The transitional probability matrix (TPM) is [ ]i ,j =P (rk =j|rk -1 =i ), i, j  M

Without loss of generality, the problem of maneuvering target TBD is considered in the 2D plane. The target motions are characterized by different dynamics, which can be formulated as (1)

xk =Fk -1 (rk -1 )xk -1 +wk (rk -1 )

where xk =[xk xk yk y k ]T denotes the target state, (xk , yk ) and (xk , y k ) are the position and velocity of the target, wk (rk -1 ) models the process noise, rk -1  M={1, 2,  , M } is the model variable representing different target motion. Suppose that the target motion consists of three models (M=3), which can be described in the unified form:

sin iT  1 i  0 cos iT Fk (rk =i )=  1 cos iT 0  i  sin iT 0

Fi =diag [Fk (rk =i ), 1] ,

1 - cos iT   i  0 - sin iT   sin iT  1  i  0 cos iT  0

(2)

where i (i =1, 2, 3) is the turn rate for the constant turn (CT) model, and T is the sampling interval. For the special case with ωi=0, the model reduces 1 T 0 0  0 1 0 0   , which is equivalent to to lim Fk (rk =1)=  0 0 1 T  1 0   0 0 0 1  the constant velocity (CV) model. For the Gaussian process noise wk (rk -1 =i ), it is characterized by the covariance matrix Qi  i Q =  T 3 /3 T 2 /2 0 0   2  T 0 0  T /2 i   , where ηi (i=1, 2, 3) is 0 T 3 /3 T 2 /2   0   0 T 2 /2 T   0

the power spectra density of the process noise. The target intensity Ik is usually unknown and can be modeled by a random walk model. The augmented state xk is now denoted as xk =[xk xk yk y k I k ]T , and the corresponding state transition equation and the associated process noise covariance matrices are

(4)

where the initial probability distributions (r0 =i ) are assumed known and

 P0 (i)=1 .

P0 (i )=P

i

In the TBD application, the target can appear/ disappear from the surveillance region at arbitrary time step, which can be modeled by a binary random variable ek  {0, 1} referred to as existence variable [15]. Conventionally, ek=1 denotes the event that a target is present and ek=0 means the absence of a target. Dynamics of ek is modeled by first order two-state Markov chain with TPM [ ]i ,j =P{ek =j - 1|ek -1 =i - 1}, where i, j  {1, 2}. The TPM can be further written as 1 - pb 1 - ps

 =

pb  ps 

(5)

where pb =P{ek =1|ek -1 =0} is the probability of target birth, and ps =P{ek =1|ek -1 =1} is the target’s survival probability during the sampling interval. Suppose that the sensor provides 2D images of the surveillance region, and each frame of the image observation consists of Mx×Ny cells with resolution Δx×Δy, and then the observed intensity zk( m, n ) at the cell (m, n) can be given as

hk( m, n ) ( xk , rk )  vk( m, n ) , H1: if target is present zk( m, n )   ( m, n ) vk , H 0 : if there is no target (6) where vk( m,n ) ~N(; 0,  2 ) denotes the observation noise with known statistics, and hk( m, n) ( xk ,rk ) 

x y I k 2π h2

Gaussian

 (mx  xk ) 2  (ny  yk ) 2   exp   2 h2   (7)

where σh represents the blurring amount of the sensor. The cell (m, n) can be denoted in more compact form as ordered index array i=(m, n), and accordingly Eq. (6) is rewritten as

 (zk(i ) |xk ,rk )=p(zk(i ) |xk ,rk )  N(zk(i ) ;hki (xk ,rk ), 2 ), i  V (xk ) = (i ) 2  N(zk ;0, ), i  V (xk )

(8)

where  (zk(i ) |xk , rk ) denotes the likelihood function, V(xk)

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represents the index set for the cells which are significantly affected by the target. The likelihood function Eq. (8) can be further given in the form of likelihood ratio as

where p(xk ,ek =1|z1:k 1 ) is the prediction density and p(zk |xk ,ek =1) is the likelihood function given by Eq. (8).

 N(zk(i ) ;hki (xk ,rk ), 2 ) , i  V (xk )   l (x )= iV (x ) N(zk(i ) ;0, 2 )  1, i  V (xk )

3.2 Particle implementation based on multiple model approximation The above recursive propagation of the posterior density is only a conceptual solution, and it generally cannot be solved analytically. As an effective numerical method, the particle filter approximately represents the required posterior PDF by using a set of random samples with associated weights. Consider that for the maneuvering target, since the dynamics may vary when the target exhibits different motions, the posterior PDF at instant k can then be represented as

(9)

3 Multiple-model particle filter 3.1 Recursive propagation of state density According to the problem formulation in the previous section, target TBD can be treated as the estimation of hybrid state yk =[xkT ek ]T , which can be recursively estimated in the Bayesian framework as follows. Given the posterior PDF (probability density function) of the hybrid state p(yk 1 ,ek 1|z1:k 1 ) at time k−1, the goal is to construct the posterior PDF of time k. Since the target state is not defined when the target does not exist (ek−1=0) in the observation data, then for ek−1=1, the recursion can be presented as a two-step procedure. Prediction: The predicted target state can be written in terms of the target state and existence at the previous time:

p(xk ,ek =1|z1:k 1 )=   p(xk ,ek =1|xk 1 ,ek 1 ,z1:k 1 ) p (xk 1 , ek 1|z1:k 1 )dxk 1dek 1 =  p (xk ,ek =1|xk 1 ,ek 1 =1, z1:k 1 )p(xk 1 ,ek 1 =1|z1:k 1 )dxk 1 +  p (xk ,ek =1|xk 1 , ek 1 =0,z1:k 1 )p (xk 1 ,ek 1 =0|z1:k 1 )dxk 1

(10)

Because p(xk ,ek =1|xk 1 ,ek 1 =1,z1:k 1 )=p (xk |xk 1 ,ek 1 =1,ek =1, z1:k 1 )p(ek =1|xk 1 ,ek 1 =1,z1:k 1 )

(11)

p(xk ,ek =1|xk 1 ,ek 1 =0,z1:k 1 )=p (xk |xk 1 ,ek 1 =0,ek =1, z1:k 1 )p(ek =1|xk 1 ,ek 1 =0,z1:k 1 )

(12)

Therefore,

 p(xk |xk 1 ,ek 1 =0,ek =1,z1:k 1 )p(ek =1|xk 1 ,ek 1 =0, z1:k 1 )p(xk 1 ,ek 1 =0|z1:k 1 )dxk 1 =p(xk )  pb

(13)

 p(xk |xk 1 ,ek 1 =1,ek =1,z1:k 1 )p(ek =1|xk 1 ,ek 1 =1, z1:k 1 )p(xk 1 ,ek 1 =1|z1:k 1 )dxk 1 =ps  p (xk |xk 1 ,ek 1 =1,

ek =1,z1:k 1 )p(xk 1 ,ek 1 =1|z1:k 1 )dxk 1

p(xk ,ek =1|z1:k 1 )p (zk |xk ,ek =1)  p (zk |z1:k 1 )

p(xk ,ek =1|z1:k 1 )p (zk |xk ,ek =1)

n =1 N





  wkn x n ,r n (xk ,rk ) n =1

k

(16)

k

where δ(·) denotes the Dirac delta function, N is the number of particle and wkn is the normalized weight of the nth particle. Given the particle set {xkn-1 ,ekn-1 ,rkn-1}nN=1 at time k−1, the multiple model particle propagation [10, 16] for a complete recursion can be summarized as Algorithm 1. Algorithm 1: Recursion procedure for multiplemodel particle filter (MMPF) {xkn , ekn , rkn }nN=1  =MMPF-TBD {xkn-1 , ekn-1 , rkn-1}nN=1 , zk     

Prediction: For n=1:N Predict the existence variable ekn according to ekn-1 and the TPM in Eq. (5) If ekn-1 =0 and ekn =1 Draw xkn from the new born proposal density qb (|zk ) and generate the model variable rkn according to the prior model probability Pi =P{rk =i} (i =1, 2, 3) Else if ekn-1 =1 and ekn =1 Predict the model variable rkn according to rkn-1 and the TPM in Eq. (4) Generate xkn from the proposal density q(|xkn-1 , ekn-1 zk ) End if End for

(14)

Update: The updated density of the target state can be obtained by using Bayes’ rule: p(xk ,ek =1|z1:k )=

N

p(xk ,ek =1|z1:k )=  wkn (xk ,rk ) - (xkn ,rkn )

Update for n=1:N Evaluate the important weight  kn according to Eq. (9) for each particle End for N

(15)

Normalize the weights to get wkn =1/  w kn n =1

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Resampling {xkn , ekn , rkn }nN=1  =Resample {xkn , ekn , rkn , wkn }nN=1     

The posterior probability of target existence (also termed as detection probability) at time k is Pk= 1 N P{ek =1|z1:k } , and it can be estimated by Pˆk =  ekn . If N n =1 Pˆ exceeds a given threshold Th, the target’s presence is k

Y1:k 1 denotes the cumulative observations up to time k−1. Then, the stochastic filtering problem in the random finite set framework can be expressed as follows [13]:

f k |k 1 (X k |Y1:k 1 )=  k |k 1 (X k |X k 1 ) f k 1|k 1 (X k 1|Y1:k 1 ) 

k (Yk |X k )f k |k 1 (X k |Y1:k 1 )

f k |k (X k |Y1:k )=

 k (Yk |X ) fk |k 1 (X |Y1:k 1 )δX

declared and the target state estimation is obtained as N

xˆ k =  xkn ekn n =1

N

 ekn

(17)

n =1

Accordingly, the estimated model probability is given by N

Pˆrk =i =   (rkn - i ) n =1

N

 ekn

(18)

4 Multiple-model Bernoulli particle filter 4.1 Bernoulli filter In the TBD context, the target may be present or absent in the surveillance region at a particular time. The discrete-time target state can therefore be modeled by a random finite set (RFS) which can be either empty or a singleton. The finite set statistics (FISST) [17] provides a practical tool for statistical description and mathematical manipulations of finite-set random variables, including the notion of FISST PDF and its integral. A convenient model of target state is the Bernoulli RFS on x  R nx . A Bernoulli RFS has a probability q of being a singleton (with PDF p(x) defined on x) and a probability 1−q of being empty. The FISST probability density of a Bernoulli RFS X is defined as 1 - q, if X = f (X )=  q  p(x ), if X ={x}

(19)

Suppose that at time k, there is nk  {0, 1} target with state  or xk, both the number of target and the state in x are random and time-varying. The target state, represented by a finite set Xk, can be conveniently be modeled as a RFS on x. Also assume that the target state is a Markov process with transitional density k |k 1 (X k |X k 1 ) , and the posterior FISST PDF of the target state is known as f k 1|k 1 (X k 1|Y1:k 1 ), where

(21)

If the target appears during the sampling interval Tk =tk  tk -1 , the PDF bk |k 1 (x ) denotes its birth density, and the dynamics of the Bernoulli Markov process Xk is characterized by the transitional FISST PDF k |k 1 (X |X ) as follows: 1  pb , if X =  pb  bk |k 1 (x ), if X ={x}

k |k 1 (X |)= 

n =1

The MM-PF is generally initialized by drawing the samples e0n (n=1, 2, , N ) according to the assumed target existence probability Pe. For the particles with e0n =0 , the target states are set as x0n =; while for those particles with e0n =1, they are sampled from the initial proposal density p0(x) and the associated model variables are generated from the initial model probability P0(i) (i=1, 2, 3).

(20)

δX k 1

1  ps , if X =  ps   k |k 1 (x|x ), if X ={x}

k |k 1 (X |{x })= 

(22) (23)

where  k |k 1 () is the transition function of state distribution and ps denotes target’s survival probability from time k−1 to k. 4.2 Bernoulli filter in multiple model framework When the dynamics uncertainty is taken into consideration, the target state can be augmented with a model variable r  M in the hybrid state space x=R nx  M , and the target posterior densities have the form [18]: 1 - q, if X = f (X )=  T T q  (r )  sr (x ), if X ={[x r ] }

(24)

where η(r) is the model probability when there is target present and

M

 (r )=1, and

pr (x ) denotes the density

r =1

function of the augmented state vector conditioned on the model variable r. By the analog to Eq. (22), it is easy to know that a target will appear with a probability pb if there is no target in the surveillance region. When the event of target’s appearance happens, the density function of the augmented state conditioned on the dynamic model r of the birth target can be denoted as br ,k |k 1 (x ) and the associated model probability for the birth target is characterized by γ(r)>0,

M

  (r )=1. Therefore, the target r =1

transition density of FISST must have the form: 1  pb , if X = T T  pb   k (r )  br ,k |k 1 (x ), if X ={[x r ] }

k |k 1 (X |)= 

(25) Accordingly, if the target continues to survive from

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time k−1 to k, the target transition density has the form: 1  ps , if X = T T  ps   k |k 1 (x ,r|x ,r ), if X ={[x r ] }

k |k 1 (X |x , r )= 

(26) Suppose that at time k−1, the posterior FISST density f k -1|k -1 (X ) is known. That is, if qk -1|k -1 , k -1|k -1 (r ), and sr (xk -1 ) are given, then when X k =, we have f k |k 1 (|Y1:k 1 )=  k |k 1 (|X ) f k 1|k 1 (X |Y1:k 1 )δX 



  (1  ps )qk 1|k 1k 1 (r )sr ,k 1|k 1 (x )dx r =1

=(1  pb )(1  qk 1|k 1 )+(1  ps )qk 1|k 1

(27)

Since the predicted FISST PDF is in the form Eq. (24), the left-hand side of Eq. (27) equals 1  qk |k 1 and we can therefore obtain 1  qk |k 1 =(1  pb )(1  qk 1|k 1 )+(1  ps )qk 1|k 1

(28)

which leads to the prediction equation for the probability of existence: qk |k 1 =pb (1  qk 1|k 1 )+ps qk 1|k 1

(29)

t =1

[pb (1  qk 1|k 1 )+ps qk 1|k 1 ]

predicted

density f k |k -1 (X ) is

1 - qk |k -1 , if X = f (X )=  T T qk |k -1 k |k -1 (r )  sr , k |k -1 (x ), if X ={[x r ] }

(34) where qk |k -1 , k |k -1 (r ) and sr , k |k -1 (x ) are represented by Eqs. (29), (33) and (31), respectively. For the image observation model, Yk =zk , the updated FISST PDF given by Eq. (21) follows from the Bayes rule as f k |k (X k |z1:k )=

 (zk |X k )f k |k 1 (X k |z1:k 1 ) pk (zk |z1:k 1 )

(35)

= (zk |)f k |k 1 (|z1:k 1 )+    (zk |{x, r})  rM

M

f k |k 1 ({x , r}|z1:k 1 )dx = (zk |)(1  qk |k 1 )+

t =1

k |k 1 (r )qk |k 1   (zk |{x, r})sr ,k |k 1 (x)dx

=pb k (r )br ,k |k 1 (x )(1  qk 1|k 1 )+  (ps π k |k 1 

M



({xk ,r}|{x,t})) qk 1|k 1k 1 (t )st ,k 1|k 1 (x ) dx M

t =1

= (zk |)f k |k 1 (|z1:k 1 )/[ (zk |)(1  qk |k -1 )+

=pb k (r )br ,k |k 1 (x )(1  qk 1|k 1 )+ps qk 1|k 1  M

For the case X k =, the update step Eq. (35) can be expressed as f k |k (|z1:k )= (zk |)f k |k 1 (|z1:k 1 )/pk (zk |z1:k 1 )

 ps π k |k 1 ({xk ,r}|{x,t})st ,k 1|k 1 (x)dx

k 1 (t )p(r|t )   k |k 1 (xk |x,t )st ,k 1|k 1 (x )dx

(36)

r =1

=pb k (r )br ,k |k 1 (x )(1  qk 1|k 1 )+ qk 1|k 1k 1 (t ) 

(30)

t =1

According to Eq. (24), the left-hand side of Eq. (30) equals qk |k 1k |k 1 (r )sr ,k |k 1 (x ), which leads to the predicted spatial PDF in the form: =[pb k (r )br ,k |k 1 (x )(1 

qk 1|k 1 )+ps qk 1|k 1 k 1 (t )p (r|t )   k |k 1 (xk |x ,t )  t =1

pb k (r )(1  qk 1|k 1 )+ps qk 1|k 1  k 1 (t ) p(r|t )

pk (zk |z1:k 1 )=   (zk |X )f k |k 1 (X |z1:k 1 )δX

tM

M

(32)

where

({xk ,r}|{x,t})f k 1|k 1 ({x , t}|Y1:k 1 )dx

f k |k 1 (X k |Y1:k 1 )

=1

or equivalently

r ] , we have

=k |k 1 (X k |)f k 1|k 1 (|Y1:k 1 )+   k |k 1 

qk |k 1k |k 1 (r )

t =1

[pb (1  qk 1|k 1 )+ps qk 1|k 1 ] k |k 1 (r )

T

f k |k 1 (X k |Y1:k 1 )=  k |k 1 (X |X ) f k 1|k 1 (X |Y1:k 1 )δX 

sr ,k |k 1 (x )=

M

pb k (r )(1  qk 1|k 1 )+ps qk 1|k 1  k 1 (t ) p (r|t )

Therefore, the characterized as

M



 sr ,k |k 1 (x)dx =1 , and we have

(33)

=(1  pb )f k 1|k 1 (|Y1:k 1 )+

Similarly, if

Note that

k |k 1 (r )=

 k |k 1 (|{x，r}) f k 1|k 1 ({x，r}|Y1:k 1 )dx

X k =[xkT

(31)

M

=k |k 1 (|Y1:k 1 )f k 1|k 1 (|Y1:k 1 )+ rM

st ,k 1|k 1 (x )dx ]/[pb (1  qk 1|k 1 )+ps qk 1|k 1 ]k |k 1 (r )

 k |k 1 (r )qk |k 1   (zk |{xk , r})sr ,k |k 1 (x )dx ]

rM

M

=f k |k 1 (|z1:k 1 )/[(1  qk |k 1 )+k |k 1 (r )qk |k 1  r =1

 lr (zk |x )sr ,k |k 1 (x )dx ]=(1  qk |k 1 )/[(1  qk |k 1 )+ M

qk |k 1  k |k 1 (r )  lr (zk |x )sr ,k |k 1 (x )dx ] r =1

where lr (zk |x )=

 (zk |{xk , r}) .  (zk |)

(37)

J. Cent. South Univ. (2015) 22: 3935−3945

3940

Since f k |k (|z1:k )=1  qk |k , then probability can be updated by

the

existence

M

qk |k =

qk |k 1 k |k 1 (r )  lr (zk |x )sr ,k |k 1 (x )dx

N r ,k

are normalized, that is  wrn,k =1 .

r =1

M

(1  qk |k 1 )+qk |k 1 k |k 1  lr (zk |x )sr ,k |k 1 (x )dx

n =1

r =1

(38) Using

the

fact

sr ,k |k (x ) for the case

that X k =[xkT

f k |k (X k |z1:k )=qk |k k |k (r ) 

M

(1  qk |k 1 )+qk |k 1 k |k 1 (r )  lr (zk |x )sr ,k |k 1 (x )dx

Suppose that at time k−1, the probability of existence is qk 1|k 1 and the model-dependent spatial PDF is approximated by N r ,k 1

r ]T , we have

qk |k 1k |k 1 (r )sr ,k |k 1 (x )lr (zk |x )

N

r ,k , where N r ,k is the particle set {rn,k ,(xkn ,rkn )}n =1 number of particle for the rth model at time k. Since sr ,k |k (x ) is a conventional PDF, the weights

sr ,k 1|k 1 (x )= =

(39)

which is equivalent to sr ,k |k (x )=1/qk |kk |k (r )  [qk |k 1k |k 1 (r )sr ,k |k 1 (x )lr (zk |x )] / M

[(1  qk |k 1 )+qk |k 1  k |k 1 (r )  sr ,k |k 1 (x )lr (zk |x )dx ] r =1

M

 sr ,k |k (x)dx =1 ,

(40) and the updated

model probability is then given as

k |k (r )=

 k |k 1 (r )  lr (zk |x )sr ,k |k 1 (x )dx M

k |k 1 (r ) lr (zk |x)sr ,k |k 1 (x)dx

(41)

r =1

By inserting Eq. (41) into Eq. (40), we immediately get sr ,k |k (x )=

sr ,k |k 1 (x )lr (zk |x )

 lr (zk |x)sr ,k |k 1 (x)dx

(42)

Now, the updated posterior density f k |k (X ) is characterized by 1 - qk |k , if X = f (X )=  T T  qk |k  k |k (r )  sr , k |k (x ), if X ={[x r ] }

N r ,k +Br ,k



n =1

r =1

Also note that

wrn,k 1 x n

(43)

where qk |k , k |k (r ) and sr , k |k (x ) are in the forms of Eqs. (38), (41) and (42), respectively. 4.3 Particle implementation of multiple-model Bernoulli filter The sequential Monte Carlo method provides an effective tool for the implementation of Bernoulli filter. In this work, the general particle implementation of Bernoulli filter is extended to the multiple-model framework in the TBD context. The resulting multiplemodel Bernoulli particle filter (MM-BPF) approximates the model-dependent spatial PDF sr ,k |k (xk ) with a

n k 1 ,rk 1



(xk 1 ,rk 1 )

(44)

The predicted probability of existence qk |k 1 is straightforward, and can be calculated from Eq. (29). The prediction spatial PDF sr ,k |k 1 (x ) in Eq. (31), however, involves the sum of two terms, and the particle approximation can be written as sr ,k |k 1 (x )=

=1/k |k (r )  [k |k 1 (r )sr ,k |k 1 (x )lr (zk |x )]/

[ k |k 1 (r )  sr ,k |k 1 (x )lr (zk |x )dx ]



n =1

r =1

qk |kk |k (rk )sr ,k |k (x )

wrn,k 1 (xk 1 ,rk 1 ) - (xkn1 ,rkn1 )

n =1 N r ,k 1

=





wrn,k |k 1 x n

n k |k 1 ,rk |k 1

(xk |k 1 ,rk |k 1 )

(45)

where the samples are drawn from two proposal distributions: qk (xk |xkn1 ,rkn1 ,z1:k ), n =1, 2,  , N r ,k xkn|k 1   qb (xk |z1:k ), n =N r ,k +1,  , N r ,k +Br ,k

(46)

with the weights M  k 1 (t )p(r|t ) k |k 1 (xk |xkn1 ,t ) p q   s k 1|k 1 t =1 wrn,k 1 ,  qk |k 1 k |k 1 (r )  qk (xk |xkn1 ,z1:k )   wrn,k |k 1 =  n =1, 2, , N r ,k  n  pb  k (r )(1  qk 1|k 1 ) br ,k |k 1 (xk ) ,  qk |k 1 k |k 1 (r ) qb (xk |z1:k )  Br ,k   n =N r ,k +1,  , N r ,k +Br ,k (47)

where Br ,k is the number of target-birth particles drawn from the proposal qb (xk |z1:k ). The update spatial PDF can be obtained by inserting Eq. (45) into Eqs. (38), (41) and (42), respectively. Firstly, the update weighs are evaluated as w rn,k =lr (zk |xkn|k 1 )wrn,k |k 1

(48)

and the integral term in Eq. (42) is approximated by I r ,k =  lr (zk |x )sr ,k |k 1 (x )dx =

N r ,k +Br ,k



lr (zk |xkn|k 1 )wrn,k |k 1

n =1

(49) The update probability of existence qk |k and the model probability k |k (r ) at time k have the forms

J. Cent. South Univ. (2015) 22: 3935−3945

k |k (r )=

3941

I r ,k k |k 1 (r )

(50)

M

 I r ,k k |k 1 (r ) M

qk |k 1  I r ,k k |k 1 (r ) r =1

(51)

M

(1  qk |k 1 )+qk |k 1  I r ,k k |k 1 (r )

Therefore, the model-dependent posterior spatial PDF at time k is N r ,k +Br ,k

 n =1

wrn,k  x n

n k |k 1 ,rk

(52)

(xk |k 1 ,rk )

where wrn,k is the normalized weight and N r ,k +Br ,k



wrn,k =1/

w rn,k .

M

(i=1, 2, …, M), where B =  Br ,k , and Br ,k is the particle number for each dynamic model. Predict weights wkn|k 1 for n=1, 2, … , N+B according to Eq. (47) Update for r=1:M Evaluate the important weight w rn,k = wkn (xkn|k -1 ,rkn ) according to Eq. (48) Normalize the weights to get wrn,k =1/

n =1

For the purpose of avoiding particle degeneracy and quick growth of particle number, a resampling procedure has to be involved after each time recursion. Any conventional resampling technique such as residual sampling and systemic sampling [7] can be adopted. To prevent the dynamic model with temporarily low likelihoods from being permanently lost, resampling is recommended to be separately performed on each modeldependent particle subset with normalized weights [19], i.e., the particles with the same model variable should be resampled separately and the associated weights are normalized. From the above derivation, it is easy to infer that the multiple-model mechanism can be implemented by the use of M parallel Bernoulli filters (each corresponds to a separate, model-dependent particle set). If the target state is treated as a mixed approximation of the particles with different model variables, the multiple-model method can also be implemented sequentially by using only one Bernoulli filter. In this work, for the convenience of performance comparison, the serial implementation fashion is adopted by concatenating the model-dependent particle subset into a complete set. The procedure of the multiple-model Bernoulli particle filter (MM-BPF) is summarized as Algorithm 2. Algorithm 2: Recursive procedure for multiplemodel Bernoulli particle filter (MM-BPF)  qk |k ,{xkn , wkn , rkn }nN=1  =MMPF-TBD  qk -1|k -1 ,{xkn-1 ,    n n N wk -1 , rk -1}n =1 , zk  Prediction: Predict the existence probability using Eq. (29) Predict the model probability using Eq. (33) M

Draw N =  N r ,k -1 particles {xkn|k -1}nN=1 from the r =1

proposal

model variables according to the new born proposal qb (xk |z1:k ) and the prior model probability Pi= P{rk =i} r =1

r =1

sr ,k |k (x )=

generated according to rkn-1 and the TPM in Eq. (4). Draw B particles {xkn|k -1}nN=+NB+1 with associated

r =1

qk |k =

particles, and the model variable for each particle ( rkn ) is

qk (xk |xkn1 , rkn1 , z1:k ) for

the

continuing

N r ,k -1 +Br ,k



w rn,k

n =1

Calculate the model probability k |k (r ) using Eq. (50) End for Update the existence probability qk |k using Eq. (51) Model-dependent resampling {x n ,r n ,1/N }N r ,k  =Resample {x n ,r n , wn }N r ,k +Br ,k  n =1   k k  k |k -1 k r ,k n =1 

If the existence probability qk |k >Th , it means that there is target present and the associated state is estimated as M

xˆ k =  k |k (r )  xˆ k (r ) r =1

N

N

n =1

n =1

where xˆ k (r )=   (r  rkn )xkn /   (r  rkn ) is the model dependent estimation of the target.

5 Simulation study Maneuvering target TBD is considered in this section. The target motions consist of three typical dynamics, which are characterized as the turn rates ω1=0, ω2=3° and ω3=7.5°. Suppose that the target appears at time k=6 and disappears at time k=55. The trajectory of the target is divided into 5 segments according to different motion model: 1) k=[6, 15], CV motion(Mode 1); 2) k=[16, 25], CT motion with turn rate 7.5° (Model 2); 3) k=[26, 35], CV motion (Model 1); 4) k=[36, 45], CT motion with turn rate 3° (Model 3); 5) k=[46, 55], CV motion (Model 1). The corresponding model variables are listed in Table 1.

J. Cent. South Univ. (2015) 22: 3935−3945

3942 Table 1 Segment parameters for maneuvering trajectory Time k k k k k interval [6, 15] [16, 25] [26, 35] [36, 45] [46, 55] Model 1 2 1 3 1 variable

The process noise is characterized by η1=η2=η3= 1×10−5 and qw=0.1. The intensity of the target Ik is given by the peak signal-noise-ratio (SNR, Rsn), where the SNR is defined as Rsn=20lg(ΔxΔyIk/2π  h2 ) . A typical target trajectory is shown in Fig. 1, and the corresponding observations at time k=20 are shown in Fig. 2, where the true target position is denoted by a circle. It is seen that the target is fairly weak even when Rsn=12, which means that the detection and tracking of such target is very difficult if the traditional DBT method is used.

Fig. 1 Typical trajectory of maneuvering target

Both the multiple-model particle filter (MMPF) and multiple-model Bernoulli particle filter (MMBPF) are used to examine the TBD performance of the maneuvering target. The effective region V(xk) is defined a s V (xk )={(m, n), m  [m0 -  x , m0 + x ]; n  [n0 -  y , n0 +

 y ]}, where (m0, n0) represents the resolution cell most affected by the target, i.e., (m0 ,n0 )= max hk( m, n ) ( xk ) . In m, n

the simulations, T=2 s and a total of 60 frame observations are generated. The related parameters to implement the filters are listed in Table 2. For both of the methods, the model probabilities for the new born particles are P (r =1)=P (r =2)=P (r =3)=1/3 and the TPM 0.8, i =j of target motion is set as  i ,j =  .  0.1, i  j The miss-distance, or error between the estimated and true state is evaluated by the metric optimal sub-pattern assignment (OSPA) [20] which is defined as 0, if X = and Y =  (c) d p (X ,Y )= d (c) p (x , y ), if X ={x} and Y ={y}  c, others

(53)

where X={x} and Y={y} denote the true and estimated

Fig. 2 Image observations of target at time k=20: (a) Rsn=12; (b) Rsn =9; (c) Rsn =6 Table 2 Simulation parameters for TBD methods Variable

σ

σh

ps

pb

Mx, Ny

Value

3

0.7

0.95

0.05

36

Variable

Δx, Δy

δx, δy

N

B

Th

Value

1

3

4000

1000

0.6



target states, respectively; d p (x,y )=min c, x  y where



p

p

,

represents the operator of p-norm and c is a

cut-off parameter. Under the condition of Rsn=12 dB, the target existence probability (also equivalent to the detection probability) estimated over 200 Monte Carlo trials is plotted in Fig. 3, and the model probability estimation during the period of target’s existence is shown in Fig. 4.

J. Cent. South Univ. (2015) 22: 3935−3945

Fig. 3 Estimated target existence probability

3943

recursion of the filter. In comparison, the detection probability of MMPF is much lower, especially at the initial stage of target’s appearance. Additionally, an obvious one-step delay of target detection is observed in MMPF. The results shown in Fig. 4 also indicate that the MMBPF has more accurate estimation of model probabilities for different target motion dynamics. The performance difference between MMPF and MMBPF mainly results from the implementation mechanism of the filters. In the conventional MMPF, the particles representing new born target at time k are generated according to the particles at time k−1 and the predefined TPM. Generally, the birth probability is very low (for example pb=0.05 in the simulation), and the number of particle to represent the new target is very small at the beginning time steps of true target’s appearance. Accordingly, the occurrence of target cannot be detected in time with a high detection probability. In the proposed MMBPF, however, a fixed part of the particle set with much larger number (B) is adopted to detect the target birth and to explore the state space of the target. With the more effective approximation of the spatial PDF, the appearance of the target can be detected instantly and more accurate state estimation is obtained in the framework of multiple-model Bernoulli filter. To elaborate on this point, the average OSPA distance (for c=5 and p=2) is shown in Fig. 5. As can be clearly seen from the figure, the miss-distance of the MMBPF is consistently lower than that of MMPF, and it reduces quickly before reaching a stable level.

Fig. 5 Performance comparison in terms of OSPA distance

Fig. 4 Estimated model probabilities of different dynamic models: (a) Model 1; (b) Model 2; (c) Model 3

As can be seen from Fig. 3, the presence and absence of target can be correctly detected by MMBPF, and the detection probability increases rapidly with the

To evaluate the effect of target intensity on TBD performance, the simulation results under different SNR conditions are plotted in Figs. 6 and 7. It is observed from the figures that for both of the methods, the detection probabilities reduce consistently with the decrease of SNR, and the resulting OSPA distances increase at the same time. However, the MMBPF significantly outperforms the MMPF and has more slight performance deterioration. Moreover, the TBD

J. Cent. South Univ. (2015) 22: 3935−3945

3944

Fig. 6 Estimated existence probability at different SNR

Fig. 7 Average OSPA distance at different SNR

performance of MMBPF for SNR=6 dB is even better than that of MMPF with 9 dB SNR, meaning that the proposed method is more applicable to the low SNR condition. To gain an insight into the effect of particle number on the performance of MMPF, the detection probabilities and OSPA distances obtained with different particle number are plotted in Figs. 8 and 9. As a benchmark for performance comparison, the results obtained by MMBPF using 5000 particles are also plotted.

Fig. 9 Average OSPA distance under different particle number

As indicated by Figs. 8 and 9, with the increase of particle number, an improved TBD performance is observed for MMPF, especially when the number of particle is smaller than 20000. Despite the improvement of the detection probability, the detection delay of the target’s presence and absence is still not eliminated even the particle number is increased up to 40000. Apparently, when 5000 particles are used, the MMBPF has approximately equivalent estimation error with MMPF using 40000 particles except at the time steps when the target appears and disappears, and it consistently achieves better detection performance than MMPF in the whole simulation scenario. The complexity of the filters is measured by the computational time for a single Monte Carlo run conducted in the PC environment with 3 GHz CPU and 3 GB RAM. The simulation software used to implement the filters is Matlab 7.12 and the results are listed in Table 3. Table 3 Comparison in computational time Method Particle number Mean time/s Standard devitation MMBPF

MMPF

Fig. 8 Estimated existence probability under different particle numbers

5000

40.150

0.607

5000

9.282

1.009

10000

19.477

1.171

20000

40.297

1.447

40000

81.152

1.776

It is seen from Table 3 that the time consumption of the MMPF is approximately proportional to the particle number. On the premise that no optimization strategy is considered for the implementation procedure, the MMBPF consumes more execution time than MMPF when the same number of particles is used. Specifically, the average computational time of MMBPF with 5000 particles is almost equivalent to that of MMPF with 20000 particles except that the former has a smaller standard deviation. The extra time consumption of

J. Cent. South Univ. (2015) 22: 3935−3945

MMBPF is introduced by more complicated implementation procedure. It is also seen from Table 3 that even 40000 particles are used by the MMPF, the computational time for each time step is still less than T=2 s, which means that both of the methods can meet the requirement of real-time processing.

6 Conclusions Weak target detection and tracking with uncertain target dynamics in low-SNR environment are concerned. By utilizing the key feature of the Bernoulli filter that it can jointly estimate the posterior PDF of the system state and the probability of its existence, the problem of maneuvering target TBD is exactly formulated as the recursions of the state PDF, existence probability, and model probability with multiple motion models, and it is implemented via sequential Monte Carlo method. The effectiveness of the resultant multiple-model Bernoulli particle filter (MM-BPF) is verified by simulations under different conditions and the superior performance is also demonstrated by comparison with the MM-PF based TBD method.

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