BAO Zhichao,JIANG Qiuxi,and LIU Fangzheng

College of Electronic and Engineering,National University of Defense and Technology,Hefei 230037,China

Abstract:It is a tough problem to jointly detect and track a weak target,and it becomes even more challenging when the target is maneuvering.The above problem is formulated by using the Bayesian theory and a multiple model(MM)based filter is proposed.The filter presented uses the MM method to accommodate the multiple motions that a maneuvering target may travel under by adding a random variable representing the motion model to the target state.To strengthen the efficiency performance of the filter,the target existence variable is separated from the target state and the existence probability is calculated in a more efficient way.To examine the performance of the MM based approach,a typical track-before-detect(TBD)scenario with a maneuvering target is used for simulations.The simulation results indicate that the MM based filter proposed has a good performance in joint detecting and tracking of a weak and maneuvering target,and it is more efficient than the general MM method.

Key words:particle filter,track-before-detect(TBD),maneuvering target tracking,multiple model(MM).

1.Introduction

Classic target tracking,including the state-of-art random finite sets(RFS)based approach[1–6],is usually performed on the point measurements which originate from the threshold processing of the raw sensor data.For example,the raw image data received by the radar tracking system will be first converted to detections for further processing[7–9].It is efficient to convert the raw data to a finite group of detections,however,this method will inevitably lead to some information loss,which makes it unsuitable for situations with low signal-to-noise ratio(SNR)or weak targets.To resolve this contradiction,a new approach which is referred to as track-before-detect(TBD)[10–13]is proposed.Instead of converting the raw data into detections,this approach uses the raw data directly and enhances target information using observation data at multiple moments.

The TBD approach is proved to be an effective method in joint detecting and tracking of weak targets,and many algorithms can be used to implement the TBD approach.Those algorithms may be divided into two sorts,including the batch algorithms and the recursive algorithms.The batch algorithms mainly include the Hough transform(HT)algorithm[14,15]and the dynamic programming(DP)algorithm[16–20].They enhance and extract target information by integrating and processing raw data from multiple moments simultaneously,while the recursive algorithms predict and update the estimation of the target state recursively.The particle filter is a typical recursive algorithm to implement the TBD approach[21–24].It can be derived from the Bayesian theory[25].As is examined in the survey[26],the particle- filter-based TBD(PF-TBD)approach has drawn wide interest because of its superiority in nonlinear and non-Gaussian circumstances.The TBD scenario with a maneuvering target is an essentially dynamic state estimation problem,which is almost nonlinear and non-Gaussian.As a result,in this paper,we concentrate on applying the PF-TBD to fulfill the joint detecting and tracking of a weak and maneuvering target.

The PF-TBD approach was first studied in[27].The TBD approach is treated as a problem of hybrid estimation.This is accomplished by adding a variable which represents the target existence state into the state vector.The proposed PF-TBD method can process the raw sensor data directly and preserve the target information as much as possible.However,attaching a variable which stands for the target existence to the target state vector degrades the efficiency of the particle group,as the particle group has to not only explore the target space,but also compute the existence probability at the same time.The ratio of the number of particles that are used to describe the posterior probability density function(PDF)of the target state varies greatly depending on if the target is present at a certain moment.Rutten et al.[28]formulated the target existence probability recursively using a more efficient method and re-derived the particle filter based on the Bayesian theory,making the particles only explore the target space.The proposed method can ensure that there is always a fixed number of particles exploring the newborn target state space.Explicit comparison has been made in the conference paper[29]between the two PF-TBD approaches.The results show that the PF-TBD proposed by Rutten is superior than Salmond’s.

To accommodate the maneuvering target in the TBD scenario,the multiple model(MM)approach[30]is incorporated in the basic PF.This is accomplished by extending the target state to include a finite-valued random variable that represents the motion model that the target is traveling under.Similar work has been done in[31,32],and the approach is referred as MM-PF-TBD.However,all these studies are based on the PF-TBD proposed by Salmond as it is easy and intuitive to add the model variable to the target state vector and implement hybrid estimation.As a result,the general MM-PF-TBD approach bears the problem of low particle efficiency like the basic PF-TBD approach.To strengthen the efficiency performance of the general MM-PF-TBD approach,similar to the method proposed by Rutten,the variable representing the existence of the target is separated from the target state,and the target existence probability is formulated recursively in a more efficient way.Essentially,this method can ensure the particles only explore the target space,and there is no need to explore the target’s existence.

This paper can be divided into five sections.In Section 2,we construct the target state transition and observation data model.In Section 3,we derive the proposed MM based approach based on the Bayesian theory and summarize the main implementation steps of the filter.In Section 4,a typical TBD scenario with a maneuvering target is designed to test the effectiveness and efficiency of the proposed approach.Section 5 is the final conclusion.

2.Target state transition and measurement model

2.1Target state transition model

In the tracking system,the performance of the filter can be guaranteed if the transition model matches the actual target motion,if not,the filter is unable to converge.Typically,we just need one state transition model to describe the straightline motion,and one or two models to handle the maneuvers of the target in MM filtering.Consider a single target performing a couple of maneuvers in a two-dimensional plane.The general equation for target state transition is

in whichskrepresents the target state,τkstands for the target motion model,tkrepresents the time stamp,g(sk,τk,tk)is the corresponding process noise input matrix,w(tk,τk)represents the statistically independent Gaussian noise.The process noise describes the model uncertainties in the state transition equation.A large variance of the process noise indicates a large model uncertainty.In target tracking we typically use the maximum acceleration of the target as the variance of the process noise.We start with presenting two models that are commonly used.The first model illustrates the constant velocity(CV)motion without any maneuvers.The second one describes a turn performed by the target with a CV.

(i)CV tracking model

The CV tracking model is one of the simplest models.The discrete-time target transition model is given[33]by

in which the target state issk=[xkvxkykvyk]T.The variablesxk,ykstand for the position of the target,vxk,vykstand for the velocity of the target,Trepresents the sampling period of the sensor.

The model uncertainty is described by the process noisew(tk,τk)=[ax,ay]T.The variablesaxandaydenote the acceleration in the two axes at timek,both of which are assumed to be zero-mean Gaussian noise.Variances ofaxandayrepresent the uncertainty of the model.Typically,the variance values are small as we assume that the maneuverability of the target is low based on this model.The corresponding input matrix is defined as

(ii)Coordinate turn(CT)tracking model

When a target performs a turn,the tracking directions are changing all the time as shown in Fig.1.A turning rateω=d?/dtthat is used to describe the changing directions is typically assumed to be constant.If the turning rate is positive,the target performs left turn.In reverse,if the turning rate is negative,the target performs right turn.Assuming the velocityis constant,we can derive the acceleration components as

Followed by a serial of processing[34],we can finally get

Fig.1The principle of CT tracking model

The corresponding input matrix is the same with the CV model in(3).

Assume the evolution ofτksubjects to the first-order Markov process and usesNτto represent the model number.Transitions between these models are according to a Markov chain.The probabilities of the transitions are constant and form a matrixwhich is referred to as the Markov transition matrix,where each element of the matrix represents the probability of transitioning between the corresponding two models,i.e.,

Using a variableEkto describe if the target is present or not,i.e.,Ek=1 represents the target is alive,whileEk=0 means the target is lost.Assume the evolution ofEksubjects to the first-order Markov process and the corresponding transition matrix is

in whichPb=Pr(Ek=1|Ek?1=0)represents the target“birth”probability,meanwhile,Pd=Pr(Ek=0|Ek?1=1)stands for the target“death”probability.

2.2Observation model

The observation raw data in this paper is assumed to ann×mimage.The information in each image cell is the target intensity defined by

in whichiandjrepresent the index of the axes,respectively.Define

in whichAkis the target complex amplitude,h(sk)stands for the level of target intensity spread on each image cell,and they can be extended as

in whichΣxandΣystand for the standard deviations of target influence in the two axes,xkandykrepresent the true target positions,ΔxandΔyare the lengths of each image cell in the two axes.The noise is assumed independent between cells and defined by

wherevI,kandvQ,kare zero-mean Gaussian noise with varianceσ2.Thus,the measurement in each image cell becomes

Although the observation likelihood function can,in general,be dependent on the motion model variableτ,it is typically independent of it.As is shown in[28],the observation likelihood function is exponentially distributed,i.e.,

in whichstands for the expectation of measurement in each cell,i.e.,

where E[·]is the expectation function.

If the target is not present,i.e.,Ik=0,by(15),we can get that the average power of noise is 2σ2.Thus,it is natural to define the signal noise ratio(SNR)as

When the target stateskis given,the power intensity of the target in each image cell can be calculated by using(16).Besides,the observation noise among different cells in each image is assumed to be independent,and hence the power intensity in different cells is conditionally independent from the target statesk.As a result,the target likelihood densityp(zk|sk)can be formulated as the product of marginal densities

Since when the target is not present,p(zk|Ek=0)has nothing to do with target statesk,define the likelihood ratio in each cell as

then

To simplify the computation of the likelihood ratios,we can ignore the image cells which are far away from the target position and only consider these image cells affected by the target greatly,and thus we can approximate the likelihood ratio in(20)as

in whichCx(sk),Cy(sk)represent the sets of image cells influenced by the target in the two axes respectively.

3.Derivation of the proposed MM based approach using Bayesian theory

The Bayesian method has been applied to various fields in recent years due to its superiority in state estimation of nonlinear systems.The main idea of the Bayesian theory is using the posterior density of previous moment and current measurement information to recursively realize the estimation of the posterior PDF of the system state.Since whenEk=0,the target state has no definition,all we have to do is to focus on estimating the target density withEk=1,i.e.,p(sk,τk,Ek=1|Zk),in whichZkrepresents the sequence observations from beginning to momentk,i.e.,

To derive the posterior PDF of the target state and variables we want to estimate,the normal and intuitive approach is to add the model variableτkand target existence variableEkto the target state and treat this problem as a hybrid estimation problem.Define an augment state vectorOk=[sTkτkEk]T.In this way,we can derive the general MM-PF-TBD approach.The concrete derivation can be found in[31].

We propose to separate the target existence variableEkfrom target stateskand model variableτkfirst,using the Bayesian theory,i.e.,

in whichPk=P(Ek=1|Zk)represents the estimation of existence probability.In case of MM,the model variableτkhas a definition if and only if the target is present.Therefore,we can add model variableτkto the target stateskto form an augmented state vector,let

and thus,the single target state is extended with the variableτkthat represents the motion model.The density of the augmented state is connected to the density of the basic state by parameterizing over the discrete model variable.Thus,

Given the observations and the target existence state,the augmented target state,which is the first factor in(23),may be extended as

As shown in(26),the augmented target state can be expanded overEk?1.The part withEk?1=1 means that the target stays alive from momentk?1 to momentk,and the corresponding density is called the continuing density.In case ofEk?1=1,the target does not show up at timek?1,but presents at timek,and the corresponding density is referred to as the birth density.The two densities can be further expanded based on the Bayes theory[35],achieving

where in MM model circumstances,

and

The predicted density in(28) may be further expanded as

while the density in(29)is usually assumed to be known as

As for the two mixing probabilities in(26),the first one can be further written as

where

As a result,the mixing probability can be calculated recursively.Similarly,the second one may be formulated as

where

Besides,the target existence probability can be derived[28]as

in whichCEis a normalization term,which can be defined by

As stated in the Introduction part,we choose to adopt the PF to realize the proposed MM based approach.First,both of the birth and continuing densities are approximated by two separate particle groups,recursively.Then the mixing probabilities and target existence probability can be formulated and calculated by using these two particle groups.After that a large particle group combined by the two groups of particles is resampled and the particle group after resampling can be used to estimate the posterior density of the target state.Considering at timek?1,the posterior target state can be approximated by a particles groupwhereNcstands for the particle number,and given the estimated target existence probability,we can summarize the implementation steps of the proposed approach as follows.

Step 1Based on the prior density of the augment target state space,generateNbparticles.

One day when she had grown up, her father looked at her, and saw that she was exactly like her mother, so he said to his councillors, I will marry my daughter to one of you, and she shall be queen, for she is exactly like her dead mother, and when I die her husband shall be king

Step 2Calculate the weight of the new born particles and normalize it.

Step 3Predict the state of the continuing particles according to the state transition probability density.

Step 4Calculate the continuing particle weight and normalize it.

Step 5Calculate the mixing probability using nonnormalized weights and normalize it.

Step 6Calculate the target existence probability at timek.

Step 7Weighted combination of the state of the new born and continuing particles is as follows:

Combine the new born particle set and the continuing particle set into a large particle set as

Step 8Resample fromNb+Ncparticles down toNcparticles.

4.Simulations and results

To test the performance of the MM based approach proposed in this paper,we design a scenario of a target performing a couple of maneuvers in the plane for a total of 45.We assume that the target moves in three modes,i.e.,we use three models to describe the target state transition.

(i)τk=1,constant velocity straight-line motion with CV tracking model;

(ii)τk=2,a left turn with the CT model;

(iii)τk=3,a right turn with the CT model.

We assume the model variableτktransition matrix is

The values on the matrix diagonal dominate in each row as the target has the largest probability to remain the same model at the next time.Besides,we set Pr(τk=2|τk?1=3)=0 and Pr(τk=3|τk?1=2)=0 based on the prior information that the target turning left is unlikely to switch to turning right immediately,and vice versa.

Assume that one target shows up at time 4 and“dies”at time 40.The target motion has changed three times during the whole moving process.Firstly,the target performs 8 s of constant velocity straight-line motion.In the second stage,the target performs 10 s of left CT motion,with the turning rateω=0.2 rad/s.In the third stage,the target switches back to perform 8 s of constant velocity straight-line motion.In the last stage,the target performs a 10 s right CT motion with the turning rateω=?0.2 rad/s.The initial target state is set as

whereA0is the target amplitude corresponding to a 10 or 6 dB SNR peak in the image observation defined in(17).The true target trajectory can be generated through target transition models.The reference trajectory is showed in Fig.2.

Fig.2Reference trajectory

4.1Filter configuration

The sensor monitoring area has been divided inton×mcells,withn=30 andm=60,and the length of cells in two directions are set asΔx=Δy=1 m.The sampling period of the sensor isT=1 s and the target influence has a standard deviationΣx=Σy=0.7.Given the true target state over time,the observations can be achieved using equations defined in Section 3.2.

The evolvement of the target existence variableEkis treated as a Markov process,withPb=Pd=0.05.The corresponding transition matrix is

As we consider the uncertainty in the target amplitude throughout the target motion process,we have to include the target amplitudeAkin the target state.Thus,the corresponding transition matrix in(2),(5)and the input matrix in(3)should be expanded as

The noise is expanded asw(tk,τk)=[ax,ay,ae]T.The variableaedenotes the uncertainty in the target amplitude.Set the varianceax=ay=ae=0.001,the noise variance is set toσ2=1.The prior density of the birth particles accords to the following uniform distributions:x0～u(0,30),y0～u(0,60),vx0～u(?2,2),vy0～u(?2,2),A0～u(1,7).

The motion models are given a initial probability[0.60.20.2].The total number of particles used by the two filters is all set to 20 000,i.e.,N=20 000 andNb=Nc=N/2.

4.2Results

To test the effectiveness of the MM based filter proposed in this paper,we choose the CV model,the single model filter,for comparison.The settings of parameters for the proposed filter and the single model filter are all the same.The intensity of the target is set as 6 dB.The performance of the detection is described by the target existence probability,which is shown in Fig.3.Fig.4 shows the position root mean square error(RMSE),which is a good benchmark for the performance of tracking.The position RMSE is defined as

wheremrepresents the Monte Carlo test times and is set as 20,andrepresent the estimations of the target position in two axes respectively at timekin theith Monte Carlo test,xkandykstand for the true target position in two axes at timek.

Fig.3 illustrates the estimated target existence probability of the MM based filter and the single model filter.A better probability of target existence means a better detection performance.When the probability exceeds a certain value,which is called the detection threshold,we believe that the target has been detected at that moment.In this simulation,the threshold value is set as 0.6,which is denoted as a dotted line in this figure.

Fig.3Probability of target existence for MM based filter and single model based filter

The estimated probability of target existence given by the single model filter is unstable and has many values be-low the threshold.This means that the detection performance of the single model filter is not good because the single model filter cannot accommodate the maneuvering target that can switch its motion between different motion models.Meanwhile,the estimated probability of target existence given by the proposed MM based filter is stable and consistent with the true probability of target existence.

Fig.4 shows the RMSE for the MM based filter and the single model filter.As can be expected,poor detection performance will result in large RMSE.The RMSE given by the single model based filter is unstable and remains at a high level,however,the RMSE given by the MM based filter is stable and decreases gradually.

Fig.4Target position RMSE for MM based filter and single model based filter

Analyze the results given in Fig.3 and Fig.4 comprehensively,we can be sure that the proposed MM based filter is effective in detecting and tracking a weak and maneuvering target.

To further study the efficiency of the proposed MM based approach and the influence of the target intensity on the performance of joint detection and tracking,we design a comparison test with the general MM based filter proposed in[31].Four sets of filters are designed for comparison:the general MM-PF-TBD filter with a 6 dB target,the proposed MM based efficient particle filter for TBD(MMEPF-TBD) filter with a 6 dB target,the general MM-PFTBD filter with a 10dB target,and the proposed MM-EPFTBD filter with a 10 dB target.The results can be found in Fig.5 to Fig.8.

Fig.5 shows the estimated target existence probability of the four filters.First,we can see that both the general MM-PF-TBD and the proposed MM-EPF-TBD filter can realize the detection of the 10 dB and 6 dB targets.

Fig.5Probability of target existence

Fig.6Target position RMSE

Fig.7 MM-PF-TBD for 6 dB target

When the target intensity increases,the estimated probability given by each algorithm is closer to the true one as expected.Then,it is obvious that when target maneuvers,the probability given by the general MM-PF-TBD degrades a lot and that given by the proposed MM-EPF-TBD remains stable and close to the true one at each target intensity.Furthermore,from the perspective of the initiation delays of target detection,the proposed MM-EPF-TBD filter performs better and can detect the 6 dB target one second earlier than the general MM-PF-TBD in this simulation.

Fig.6 shows the position RMSE for the four filters.As can be expected,both the detection and tracking performance will become worse when the target intensity gets lower.The position RMSE is a good benchmark for the tracking performance of the filter and has a close correlation with the corresponding target existence probability.For example,the 6 dB target which has a low probability of existence also has a large RMSE initially.As the existence probability does not exceed the threshold value until the 5th or 6th second,after which the filters may begin to track the target,and hence the RMSE is meaningful only after the detection moment.Similar to the detection performance,the RMSE given by the general MM-PF-TBD degrades a lot when the target maneuvers at each target intensity.Meanwhile,the RMSE given by the proposed MMEPF-TBD remains stable and converges gradually.

Fig.8 MM-EPF-TBD for 6 dB target

Fig.7 and Fig.8 are the probability of models for the 6 dB target provided by the general MM-PF-TBD filter and the MM-EPF-TBD filter proposed in this paper,respectively.Typically,the filters treat the mode that has the largest probability as the estimated mode.Both approaches can realize the estimation of modes and the proposed MMEPF-TBD has a better accuracy.

5.Conclusions

The TBD is proved to be an effective method for weak targets and the MM approach is suitable for tracking maneuvering targets.Thus,it is promising to combine the two approaches to solve the problem of jointly detecting and tracking a weak and maneuvering target.In this paper,an MM based approach designed for jointly detecting and tracking a weak and maneuvering target is proposed.The effectiveness and efficiency of the proposed MM based approach are certified by some comparing simulations.