Multi-Target Tracking Based on Multi-Bernoulli Filter with Amplitude for

Article

Multi-Target Tracking Based on Multi-Bernoulli Filter with Amplitude for Unknown Clutter Rate Changshun Yuan, Jun Wang *, Peng Lei, Yanxian Bi and Zhongsheng Sun Received: 27 August 2015; Accepted: 29 November 2015; Published: 4 December 2015 Academic Editor: Fabrizio Lamberti School of Electronic and Information Engineering, Beihang University, Beijing 100191, China; [email protected] (C.Y.); [email protected] (P.L.); [email protected] (Y.B.); [email protected] (Z.S.) * Correspondence: [email protected]; Tel.: +86-10-8233-9203; Fax: +86-10-8231-7240

Abstract: Knowledge of the clutter rate is of critical importance in multi-target Bayesian tracking. However, estimating the clutter rate is a difficult problem in practice. In this paper, an improved multi-Bernoulli filter based on random finite sets for multi-target Bayesian tracking accommodating non-linear dynamic and measurement models, as well as unknown clutter rate, is proposed for radar sensors. The proposed filter incorporates the amplitude information into the state and measurement spaces to improve discrimination between actual targets and clutters, while adaptively generating the new-born object random finite sets using the measurements to eliminate reliance on prior random finite sets. A sequential Monte-Carlo implementation of the proposed filter is presented, and simulations are used to demonstrate the proposed filter’s improvements in estimation accuracy of the target number and corresponding multi-target states, as well as the clutter rate. Keywords: multi-Bernoulli filter; random finite set; multi-target tracking; amplitude information; clutter rate estimation; sequential Monte-Carlo

1. Introduction As a system, the radar has been widely applied in both civil and military areas due to its all-weather, day and night capability compared with optical and infrared sensors [1]. One of its most significant and important applications is target tracking. With the advancement of radar systems, target tracking has focused on multi-target tracking. The objective of multi-target tracking is to jointly estimate the unknown and time-varying number of targets and the corresponding states of multiple targets from measurements. Since measurements produced by radars usually include dense clutters, multi-target tracking becomes a challenging problem [2–4]. In many studies on the topic of multi-target tracking, it is usually assumed that the clutter rate is known and time-invariant as a priori parameter. However, in real-world applications, this parameter is often previously unknown and its value may be time-varying as the environment changes. Therefore, the ability of tracking multiple targets with an unknown clutter rate is very important in practice. To solve this problem, some approaches for traditional multi-target tracking with unknown clutter rate were proposed, such as the non-parametric Joint Probabilistic Data Association (JPDA) [5] and the Joint Integrated Probabilistic Data Association (JIPDA) [6]. However, due to the necessary process of associations between appropriate targets and measurements, the computational load of the aforementioned techniques increases exponentially as the number of targets and measurements increases. This indicates that they may be unsuitable in many situations, i.e., ground moving targets tracking. Mahler recently proposed a random finite set (RFS) approach, which provided an elegant and mathematical Bayesian formulation to the multi-target tracking problem [4] and which could avoid Sensors 2015, 15, 30385–30402; doi:10.3390/s151229804

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Sensors 2015, 15, 30385–30402

associations. In particular, the development of the probability hypothesis density (PHD) [7] and the cardinalized probability hypothesis density (CPHD) [8] filters, including sequential Monte-Carlo (SMC) and Gaussian mixture (GM) implementations [9–11], as well as convergence analysis [12,13], have verified the practicality of the RFS approach. Furthermore, multi-target tracking with unknown clutter rate has been addressed with methods based on PHD and CPHD filters [14,15]. A closed form GM implementation for linear scenarios is proposed in [15]. Although it is possible to extend this work to general non-linear scenarios via SMC implementations, this solution may not achieve acceptable performance due to the necessary clustering process, which has the drawback of being inherently unreliable in state estimations. A significant step towards addressing this problem was the multi-Bernoulli filter (MBerF), which was itself based on RFS; it proposed by Mahler [4] and modified by Vo [16]. Recently, the MBerF has been widely applied to audio and video tracking [17–19], sensor network tracking [20], and cell tracking [21]. Compared with the SMC-PHD and SMC-CPHD filter, the key advantage of the SMC implementation of MBerF is to avoid the additional clustering process, and achieve a more efficient and reliable state estimation of multiple targets. Moreover, Vo proposed a novel MBerF with unknown clutter rate (UCR-MBerF) which was inspired by the method used for the PHD/CPHD filters in [22]. However, this filter produced a large variance in the clutter rate and cardinality estimation and a bias in both of them, which became noticeable as the clutter rate increased. Moreover, it was assumed that the new-born object RFSs were known a priori. However, prior knowledge of new-born object RFSs could not be achieved in practice. In this paper, we propose an improved UCR-MBerF, which incorporates amplitude information in the state and measurement spaces. The proposed filter is abbreviated as UCR-MBerF-AI. Our major innovations are given below: ‚

‚ ‚

We derive novel prediction and update equations which augment the state and measurement spaces with the amplitude information. In radar sensors, measurements not only include the object’s position, but also contain information about the signal’s amplitude. The signal amplitude from an actual target is typically stronger than that of clutter. Therefore, it provides valuable information, which is useful to determine whether the measurement is from an actual target or from clutter. We adaptively generate the new-born object RFSs using the measurements so as to eliminate reliance of the prior new-born object RFSs. We carry out the SMC implementation of the proposed filter for general non-linear multi-target tracking scenarios.

The structure of this paper is as follows: a RFS model with amplitude for radar sensors with unknown clutter rate is introduced in Section 2. Section 3 presents an improved MBerF without the need of the prior clutter rate and new-born object RFSs assumptions, as well as a generic SMC implementation of the proposed filter for non-linear multi-target scenarios. Section 4 shows the numerical simulations with a non-linear multi-target scenario. Conclusions are finally given in Section 5. 2. RFS Model with Amplitude Information for Radar Sensors under Unknown Clutter Rate In this section, we show how amplitude information is incorporated into an RFS model that accommodates unknown clutter rate scenarios. At the same time, we present the likelihood functions for the clutter and actual target by adopting radar amplitude models. 2.1. RFS Model with Amplitude Information for Unknown Clutter Rate Suppose that there are Npkq actual targets with states xk,1 . . . , xk,N pkq and Mpkq observations with measurements zk,1 . . . , zk,Mpkq at time k. The multi-target states and observations are then represented ! ) ! ) by Xk “ xk,1 . . . , xk,N pkq and Zk “ zk,1 . . . , zk,Mpkq , respectively [4]. In this paper, we consider an 30386

represented by X k  {xk ,1

, xk , N ( k ) } and Zk  {zk ,1

, zk , M ( k ) } , respectively [4]. In this paper, we

consider an augmented state which includes kinematic x , parameter A of amplitude information and label u for actual targets and clutters, i.e., x   x, A, u  , and each measurement contains position z and amplitude r of the return signal, i.e., z =  z,r  . SensorsGiven 2015, 15,a30385–30402 multi-target

state X k 1 at time k  1 , state X k is represented as the union of

surviving objects that survive from X k 1 with the survival probability ps , k  x  , and new-born xr, parameter augmented which includes A of amplitude and label u for objects that state appear at time order to distinguish new-born objects information from the surviving objects, k . Inkinematic r r actual targets and clutters, i.e., x “ p x , A, uq, and each measurement contains position z and amplitude X k k 1,  . Given X k , the measurements Z k are written as the label  is used, i.e., X k  r of the return signal, i.e., z “ pr z, rq.  0,1 Given a multi-target state Xk´ 1 at time k ´ 1, state Xk is represented as the union of surviving   Z k   that x   , where  x  is the RFS which has a probability p k  x  gk  z xobjects   of containing   objects survive from X that appeara  xX  k´1 with the survival probability ps,k pxq, andD ,new-born  k  at time k. In order to distinguish new-born objects from the surviving objects, label β is used, i.e., ˜ thatthethe ¸ objects measurement  z or is empty with probability 1  pD, k  x  [4]. Note contain Ť Ť ř ř pxq pxq is X “ X . Given X , the measurements Z are written as Z “ , where k k k |k´1,β targets. k clutters and kactual x P Xk β“0,1 In the following, it is assumed thatg clutters and actual targets are statistically independent, the RFS which has a probability p D,k pxq k pz |x q of containing a measurement tzu or is empty with f  x, A, u   fu  x, A . The while arbitrary onthe theobjects state space be denoted by targets. pxq [4]. defined Note that containwill clutters and actual probability 1 ´ pfunctions D,k

convention that u  1it denotes actual clutters as wellindependent, as   1 denotes  0 denotes In the following, is assumed thattargets cluttersand and uactual targets are statistically while r r arbitrary defined the statenew-born space willobjects be denoted by adopted f px, A, uqthroughout “ f u px, Aq.this Thepaper. convention survivingfunctions objects and will be   on 0 denotes that u “ 1 denotes actual targets and u “ 0 denotes clutters as well as β “ 1 denotes surviving objects and β “ 0 denotes new-born objects will be adopted throughout this paper. 2.2. Amplitude Likelihoods for Radar Sensors Let us assume that the amplitude of the return signal is independent of the object’s kinematic 2.2. Amplitude Likelihoods for Radar Sensors state. Then, the likelihood g k  z x  is given by: Let us assume that the amplitude of the return signal is independent of the object’s kinematic state. Then, the likelihood gk pz |x q is given by: g k  z x   gu , k  z x  gu , k  r A (1) gk pz |x q “ gu,k pr z |xr q gu,k pr |A q (1) In this paper, we adopt the Rice amplitude model, in which the probability densities of the In this paper, we adopt the Rice amplitude model, in which the probability densities of the amplitude of clutter and actual target before threshold detector are represented as in [1]: amplitude of clutter and actual target before threshold detector are represented as in [1]: $  ˆ r22 ˙  r r ’ g r A  exp  r 2 r0    ´ 0, k |A q “ ’ 2 pr g exp rě0 ’ 2 2  & 0,k   2ψ ψ2 (2)  (2) ˆ r 2r2 `  r ˆrA rA˙ A2A 2 ˙ ’  r ’ g r A  I exp  r  0 ’    1, k |A q “ 2I0 0  2  exp ´ 2 % g1,k rě0  pr 22ψ 2   ψ2 ψ2   where I0I 0p¨q is the modified Bessel function, the standard deviation the noise.  is  the where modified Bessel function, ψ istheisstandard deviation of theofnoise. In the radar system, a typical detection process is to find local maxima among all measurements followed by TheThe flowflow diagram of signal processing sequence for a by thresholding thresholdingatataacertain certainlevel levelτ [1]. diagram of signal processing sequence  [1]. radar receiver is shown in Figure 1. for a radar receiver is shown in Figure 1.

Echo Signal

Band-pass Filter

Envelope Detector

Low-pass Filter

Threshold Detector

Noise

Detection Threshold

Figure radar receiver. receiver. Figure 1. 1. Flow Flow diagram diagram of of signal signal processing processing for for aa typical typical radar

If the amplitude likelihood functions for the measurements which exceed the detection If the amplitude likelihood functions for  the measurements which exceed the detection threshold threshold are denoted g0,k  rτ A and g1,k  r A , then we have: τ pr |Aas are denoted as g0,k q and g1,k pr |A q, then we have: $ g0,k pr |A q ’ ’ gτ pr |A q “ ’ & 0,k pτFA,k 3 ’ ’ τ pr |A q “ g1,k pr |A q ’ % g1,k pτD,k

30387

(3)

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where:

$ ˙ ˆ 8 ş τ2 ’ τ ’ p “ g pr |A q dr “ exp ´ ’ 0,k FA,k ’ ’ 2ψ2 τ & g »d ¸fi ˜ f 2 8 ’ f ş 1 ’ ’ pτ “ g pr |A q dr “ Q – A , e2ln fl ’ 1,k ’ τ 2 % D,k p ψ τ FA,k

are the probabilities of false alarm and detection. Q rα, βs “

8 ş

(4)

` ` ˘ ˘ ζ I0 pαζqexp ´ α2 ` ζ 2 {2 dζ is the

β

MarcumQ function and can be computed by numerical integration offline. 3. UCR-MBerF-AI and SMC Implementation In this section, we derive an improved UCR-MBerF by incorporating the amplitude information into a single object state and the measurement and adaptively generate the new-born object RFSs. In Section 3.1, we describe the UCR-MBerF-AI, while further on, in Section 3.2, we present a detailed SMC implementation of the proposed UCR-MBerF-AI. 3.1. UCR-MBerF-AI The UCR-MBerF-AI can be derived from the UCR-MBerF with a particular chosen state and measurement space. This approach is described below. First, analogous to the UCR-MBerF, an unknown clutter rate is accommodated by modelling individual clutters which have their own separate models for transitions, detections and likelihoods, as well as formation and dissolution. Second, we incorporate amplitude information into the state and measurement variables and generate the augmented variables which were introduced in Section 2.1. However, the UCR-MBerF assumes that the new-born object RFSs are known a priori, which cannot be achieved in practice. To solve this problem, we propose a method inspired by [23], namely the formulation of the new-born object RFSs using measurement information. The UCR-MBerF-AI is fundamentally different to the approach proposed in [23], so it solves problems that are not possible to solve with the latter. The prediction and update step of the UCR-MBerF-AI are shown as follows: 3.1.1. UCR-MBerF-AI Prediction The posterior multi-target density (Po-MTD) for MBerF is approximated by a finite and !´ ¯) Mk pi q pi q pi q pi q time-varying number of Bernoulli RFSs where rk and pk denote the existence rk , pk i “1

probability and the state probability density for the ith Bernoulli components, respectively [16]. Then, !´ ¯) Mk´1 pi q pi q at time k ´ 1, the Po-MTD can be written as πk´1 “ rk´1 , pu,k´1 . i “1

As described in Section 2.1, the predicted objects include the new-born objects and surviving objects. Then, the predicted multi-target density (Pr-MTD) is given by the union of surviving and new-born Bernoulli components [22] and is calculated as follows: ¯) Mk|k´1,β ď !´ piq pi q π k | k ´1 “ rk|k´1,β , pu,k|k´1,β (5) i “1

β“0,1

where:

pi q

pi q

rk|k´1,β“1 “ rk´1

E ÿ A pi q pu,k´1 , pS,u,k u“0,1

A pi q

pu,k|k´1,β“1 pxr, Aq “

pi q f u,k|k´1 pxr, A |¨, ¨ q , pu,k´1 pS,u,k A E pi q pu,k´1 , pS,u,k

30388

E

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¯) Mk|k´1,β“0 pi q pi q are the new-born object RFSs, x¨, ¨y denotes an inner product rk|k´1,β“0 , pu,k|k´1,β“0 i “1 ˇ ´ ¯ ´ ¯ ˇ r ς , while pS,u,k pζ, r ς is the single object transition density given a previous state ζ, r ςq f u,k|k´1 xr, A ˇζ, ´ ¯ rς . is an object’s probability of survival given a previous state ζ, !´

3.1.2. UCR-MBerF-AI Update At time k, the new-born objects are driven by measurements and are always detected, so the probability of detection is always 1. According to the analysis in Section 2.2, for surviving clutter and actual targets, the probability of detection only depends on the threshold τ and parameter A of amplitude information, then the probability of detection is given by:

p D,u,k,β“1 pxr, Aq “

$ & pτD,k

u“1

% pτ FA,k

u“0

(6)

If the Pr-MTD is calculated in the predict step as shown in Equation (5), then for a set of receiving measurements Zk , the Po-MTD is written as a union of legacy and updated components for the new-born and surviving objects, respectively [22]. However, since the probability of detection is always 1 for new-born objects, the legacy components for new-born objects can be disregarded. Consequently, the final form can be written as: !´ πk “

pi q

pi q

r L,k , p L,u,k

¯) M L ď ` k i “1

˘( rU,k pzq , pU,u,k p¨; zq

(7)

zPZk

where: MkL “ Mk|k´1,β“1 ÿ pi q pi q r L,k “ r L,u,k u“0,1

A

E pi q pi q r p , 1 ´ p D,u,k,β “ 1 k | k ´ 1,β “ 1 u,k | k ´ 1,β “ 1 pi q A E r L,u,k “ ř pi q pi q 1 ´ rk|k´1,β“1 u1 “0,1 pu,k|k´1,β“1 , 1 ´ p D,u1 ,k,β“1 ´ ¯ iq r, Aq ppu,k pxr, Aq 1 ´ p p x D,u,k,β “ 1 |k´1,β“1 pi q A E r p L,u,k px, Aq “ ř pi q 1 p , 1 ´ p 1 D,u ,k,β“1 u “0,1 u1 ,k|k´1,β“1 ÿ rU,u,k pzq rU,k pzq “ u“0,1

´

¯A E pi q pi q pi q τ pr |A q p pu,k|k´1,β , gu,k pr z |xr q gu,k D,u,k,β ř ř Mk|k´1,β rk|k´1,β 1 ´ rk|k´1,β ´ A E¯2 i “1 ř pi q pi q β“0,1 1 ´ rk|k´1,β u1 “0,1 pu1 ,k|k´1,β , p D,u1 ,k,β E A rU,u,k pzq “ ř pi q pi q z |xr q guτ1 ,k pr |A q p D,u1 ,k,β ř ř Mk|k´1,β rk|k´1,β u1 “0,1 pu1 ,k|k´1,β , gu1 ,k pr A E ř i “1 pi q pi q β“0,1 1 ´ rk|k´1,β u1 “0,1 pu1 ,k|k´1,β , p D,u1 ,k,β pi q

ř ř Mk|k´1,β β“0,1

i“1

pU,u,k pxr, A; zq “

rk|k´1,β iq τ pr |A q p ´ ¯ ppu,k r, Aq pxr, Aq gu,k pr z |xr q gu,k D,u,k,β p x |k´1,β pi q 1 ´ rk|k´1,β pi q

ř Mk|k´1,β

ř ř β“0,1

u1 “0,1

i“1

rk|k´1,β ´ ¯ pi q 1 ´ rk|k´1,β

30389

A

pi q

pu1 ,k|k´1,β , gu1 ,k pr z |xr q guτ1 ,k pr |A q p D,u1 ,k,β

E

Sensors 2015, 15, 30385–30402

τ pr |A q are given by Equation (3). The amplitude likelihoods gu,k Therefore, the UCR-MBerF-AI is different from the original UCR-MBerF in the sense that it not only considers the amplitude information as an augmented variable to improve discrimination between actual targets and clutters, but it also adopts a measurement-driven approach to adaptively generate the new-born object RFSs.

3.2. SMC Implementation In this section, we present a SMC implementation of the proposed UCR-MBerF-AI. This can accommodate non-linear dynamic and measurement models. The convergence results for this SMC implementation still satisfy the convergence results for the conventional MBer filter in [24]. The SMC implementation´is based ˇ on ¯ the sampling importance resampling (SIS) technique, where the transition ˇ r density f u,k|k´1 xr, A ˇζ, ς is used as the importance density. We assume the measurements include range and bearing. The prediction, update, resample, multi-target state and clutter rate estimation steps are given below: 3.2.1. SMC Prediction !´ If at time k ´ 1, the Po-MTD πk´1 “

pi q

pi q

rk´1 , pu,k´1

¯) Mk´1

pi q

is given, and each pu,k´1 ,

i “1

piq

i “ 1, . . . , Mk´1, , is written as

pi q pu,k´1 pxr, Aq

Lu,k´1

ř

“

j “1

pi,jq

wu,k´1 δ

pi,jq

pi,jq

pr xu,k´1 ,Au,k´1 q

pxr, Aq, then the surviving

components of Equation (5) can be calculated as follows: piq

pi q rk|k´1,β“1

“

Lu,k´1

pi q r k ´1

ÿ

ÿ

pi,jq

pi,jq

pi,jq

wu,k´1 pS,u,k pxru,k´1 , Au,k´1 q

(8)

u“0,1 j“1 piq

pi q pu,k|k´1,β“1 pxr, Aq

Lu,k´1

ÿ “

pi,jq

ru,k|k´1,β“1 δ w

where:

pi,jq

pi,jq

pi,jq

pxr, Aq

piq

ř

Lu1 ,k´1

ř

u1 “0,1 j“1

(9)

pi,jq

wu,k´1 pS,u,k pr xu,k´1 , Au,k´1 q

pi,jq

ru,k|k´1,β“1 “ w

´

pi,jq

pr xu,k|k´1,β“1 ,Au,k|k´1,β“1 q

j “1

(10)

pi,jq pi,jq pi,jq wu1 ,k´1 pS,u,k pr xu1 ,k´1 , Au1 ,k´1 q

¯ ´ ˇ ¯ pi,jq pi,jq pi,jq ˇ pi,jq xru,k|k´1,β“1 , Au,k|k´1,β“1 „ f u,k|k´1 ¨ ˇxru,k´1 , Au,k´1

Since the new-born objects given in Equation (5) could appear anywhere with equal probability in the state space, the new-born object RFSs must cover the entire state space. Although it is possible to cover the entire state space with the particles for the SMC-UCR-MBerF-AI, this approach is not possible in a practical scenario because a large number of particles would be necessary for the algorithm to work properly. Instead, we utilize the measurements to adaptively generate new-born object RFSs. We also find that the measurements near the predicted states Xˆ k|k´1 of estimated " * p Nˆ k´1 q p1q p 2q ˆ multi-target states Xk´1 “ xˆ k´1 , xˆ k´1 , ¨ ¨ ¨ , xˆ k´1 are not likely to be obtained from the new-born object, so we remove measurements located near Xˆ k|k´1 , and obtain the new measurement-driven set ! ) p1q p2q pΓ q ZΓ,k “ zk , zk , ¨ ¨ ¨ , zk k . Finally, the new-born components of Equation (5) can be calculated as follows: pi q

rk|k´1,β“0 “

ÿ u“0,1

30390

pi q

ru,k,β“0

(11)

Sensors 2015, 15, 30385–30402

piq

pi q pu,k|k´1,β“0 pxr, Aq

Nb

pi,jq

ÿ

ru,k|k´1,β“0 δ w

“

pi,jq

pi,jq

pr xu,k|k´1,β“0 ,Au,k|k´1,β“0 q

j “1

where pi q

pxr, Aq

(12)

νkb V

pi q

ru,k,β“0 “ ru,β“0 “ 1

pi,jq

ru,k|k´1,β“0 “ w

pi q

2Nb ´ ¯ ` ˇ ˘ pi,jq pi,jq xru,k|k´1,β“0 , Au,k|k´1,β“0 „ bu,k ¨ ˇZΓ,k ` ˇ ˘ pi q The parameter Nb is the number of particles per new-born object, bu,k ¨ ˇZΓ,k is new-born object density given a set of measurements. The parameter νkb is the expected number of new-born objects. The parameter V is the volume of the measurement space. 3.2.2. SMC Update pi q

If at time k, the Pr-MTD Equation (5) is given, and each pu,k|k´1,β , i “ 1, . . . , Mk|k´1,β , is written piq

as

pi q pu,k|k´1,β pxr, Aq

Lu,k|k´1,β

“

pi,jq

ř

wu,k|k´1,β δ

pi,jq

pi,jq

pr xu,k|k´1,β ,Au,k|k´1,β q

j “1

can be calculated as shown below:

pi q

r L,k “

pxr, Aq, then the updated Po-MTD Equation (7) pi q

ÿ

r L,u,k

(13)

u“0,1

pi q

pi q r L,u,k

“

pi q rk|k´1,β“1

1 ´ η L,u,k,β“1 ř pi q pi q 1 ´ rk|k´1,β“1 η L,u1 ,k,β“1

(14)

u1 “0,1

piq

Lu,k|k´1,β“1

pi q p L,u,k pxr, Aq

ÿ “

pi,jq

r L,u,k δ w

pi,jq

pi,jq

pr xu,k|k´1,β“1 ,Au,k|k´1,β“1 q

j “1

rU,k pzq “

ÿ

pxr, Aq

rU,u,k pzq

(15) (16)

u“0,1

´ ¯ pi q pi q pi q ř ř Mk|k´1,β rk|k´1,β 1 ´ rk|k´1,β ηU,u,k,β ´ ¯2 i “1 ř pi q pi q β“0,1 1 ´ rk|k´1,β u1 “0,1 η L,u1 ,k,β rU,u,k pzq “ ř pi q pi q ř ř Mk|k´1,β rk|k´1,β u1 “0,1 ηU,u1 ,k,β ř i “1 pi q pi q β“0,1 1 ´ rk|k´1,β u1 “0,1 η L,u1 ,k,β

(17)

piq

Mk|k´1,β Lu,k|k´1,β

ÿ

ÿ

ÿ

β“0,1

i “1

j “1

pU,u,k pxr, A; zq “ where:

pi,jq

rU,u,k,β δ w

pi,jq

pi,jq

pr xu,k|k´1,β ,Au,k|k´1,β q

pxr, Aq

piq

Lu,k|k´1,β“1

pi q

ÿ

η L,u,k,β“1 “

pi,jq

pi,jq

pi,jq

wu,k|k´1,β“1 p D,u,k,β“1 pxru,k|k´1,β“1 , Au,k|k´1,β“1 q

j “1

30391

(18)

Sensors 2015, 15, 30385–30402

pi,jq

pi,jq r L,u,k w

pi,jq

pi,jq

wu,k|k´1,β“1 p1 ´ p D,u,k,β“1 pxru,k|k´1,β“1 , Au,k|k´1,β“1 qq

“

piq

Lu,k|k´1,β“1

ř

ř

u1 “0,1

j “1

pi,jq

pi,jq

(19)

pi,jq

wu1 ,k|k´1,β“1 p1 ´ p D,u1 ,k,β“1 pxru1 ,k|k´1,β“1 , Au1 ,k|k´1,β“1 qq piq

Lu,k|k´1,β“0

pi q η L,u,k,β“0

pi,jq

ÿ

wu,k|k´1,β“0

“ j “1

pi q

ηU,u,k,β“1 pzq “ piq

Lu,k|k´1,β“1

ř j “1

¯ ´ ˇ ¯ ´ ˇ pi,jq pi,jq pi,jq ˇ pi,jq ˇ pi,jq τ wu,k|k´1,β“1 gu,k r z ˇxru,k|k´1,β“1 gu,k r ˇAu,k|k´1,β“1 p D,u,k,β“1 pxru,k|k´1,β“1 , Au,k|k´1,β“1 q piq

Lu,k|k´1,β“0

pi q ηU,u,k,β“0 pzq

1

pi,jq

ÿ

wu,k|k´1,β“0

“ j “1

2πσu,r σu,θ

´ ˇ ¯ ˇ pi,jq τ gu,k r ˇAu,k|k´1,β“0

piq

pi,jq

wu,k|k´1,β“1 pi,jq rU,u,k,β“1 w “

ru,k|k´1,β“1 piq 1 ´ ru,k|k´1,β“1

´ ˇ ¯ ´ ˇ ¯ pi,jq pi,jq ˇ pi,jq ˇ pi,jq τ gu,k r z ˇxru,k|k´1,β“1 gu,k r ˇAu,k|k´1,β“1 p D,u,k,β“1 pr xu,k|k´1,β“1 , Au,k|k´1,β“1 q ř

ru1 ,k|k´1,β

ř

β“0,1 u1 “0,1

i“1

piq

1 ´ ru1 ,k|k´1,β

pi q

pi,jq wu,k|k´1,β“0 pi,jq rU,u,k,β“0 w

“

ru,k|k´1,β“0

piq ηU,u1 ,k,β pzq

1

pi q

1 ´ ru,k|k´1,β“0 2πσu,r σu,θ ř

β“0,1 u1 “0,1

´ ˇ ¯ ˇ pi,jq τ gu,k r ˇAu,k|k´1,β“1

ru1 ,k|k´1,β

ř

pi q

1 ´ ru1 ,k|k´1,β

i“1

(21)

pi q

Mk|k´1,β

ř

(20)

piq

Mk|k´1,β

ř

pi q ηU,u1 ,k,β pzq

Here, σu,r and σu,θ are the standard deviations of the range and bearing noise, respectively. Note ´ ˇ ¯ 1 ˇ pi,jq gu,k r z ˇxru,k|k´1,β“0 “ due to the measurement-driven new-born objects. 2πσu,r σu,θ 3.2.3. Resampling Similar to the UCR-MBerF [22], the number of particles´ for each Bernoulli component is ¯ pi q pi q re-allocated in scale to its existence probability, i.e., Lk “ max rk Lmax , Lmin . In order to reduce the number of Bernoulli components, components with existence probability below a threshold G are discarded. 3.2.4. Multi-Target State and Clutter Rate Estimation ř ˆ k “ M k r pi q . Analogous to the UCR-MBerF [22], the estimated number of actual targets is N i“1 1,k ! ) p 1q p 2q p Nˆ q The estimated individual actual target states Xˆ k “ xˆ k , xˆ k , ¨ ¨ ¨ , xˆ k k are obtained from the pi q

means of the corresponding posterior density xˆ k ř M piq ř Lpiq pi,jq pi,jq λˆ c,k “ i“k1 r0,k j“0,k1 w0,k p D,0,k .

“

ř Lpiq 1,k j “1

pi,jq pi,jq

w1,k xr1,k . The estimated clutter rate is

The processing steps of the proposed SMC-UCR-MBerF-AI are given in Algorithm 1.

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Algorithm 1. Processing Steps of the SMC-UCR-MBerF-AI Initialization: !´ ¯) M0 ! ´ ¯) Lpiq u,0 pi,jq pi,jq pi,jq pi q pi q pi q Let π0 “ r0 , pu,0 and pu,0 pxr, Aq :“ wu,0 , xru,0 , Au,0 represent the initial state. i “1

j “1

Input: !´ 1. Given πk´1 “

pi q

pi q

rk´1 , pu,k´1

¯) Mk´1 i “1

!

pi q

, pu,k´1 pxr, Aq :“

´ ¯) Lpiq u,k´1 pi,jq pi,jq pi,jq wu,k´1 , xru,k´1 , Au,k´1 , the j “1

estimated multi-target states Xˆ k´ 1 and the current measurement set Zk , Prediction: !´

pi q

pi q

rk|k´1,β“1 , pu,k|k´1,β“1

2. Compute the surviving Bernoulli components

¯) Mk|k´1,β“1 i “1

.

pi q

(a) Compute the existence probability rk|k´1,β“1 using Equation (8), for i “ 1, 2, ¨ ¨ ¨ , Mk´ 1 . pi,jq

ru,k|k´1,β“1 using Equation (10), for j “ 1, 2, ¨ ¨ ¨ , Lpki´q 1 , (b) Compute the weight w i “ 1, 2, ¨ ¨ ¨ , Mk´ 1 and u “ 0, 1. ´ ¯ ´ ˇ ¯ pi,jq pi,jq pi,jq ˇ pi,jq (c) Draw the particle xru,k|k´1,β“1 , Au,k|k´1,β“1 „ f u,k|k´1 ¨ ˇxru,k´1 , Au,k´1 , for pi q

j “ 1, 2, ¨ ¨ ¨ , Lk´1 , for i “ 1, 2, ¨ ¨ ¨ , Mk´ 1 and u “ 0, 1. !´ ¯) Mk|k´1,β“0 pi q pi q 3. Compute the new-born Bernoulli components rk|k´1,β“0 , pu,k|k´1,β“0 . i “1

(a) Remove measurements near the predicted states Xˆ k|k´1 of the estimated multi-target states ! ) p1q p2q pΓ q Xˆ k´ 1 and obtain the rest of the measurements ZΓ,k “ z , z , ¨ ¨ ¨ , z k , Mk|k´1,β“0 “ k

Γk .

pi q

(b) Compute the existence probability rk|k´1,β“0 “

k

k

pi q

ř u“0,1

ru,k,β“0 , for i “ 1, 2, ¨ ¨ ¨ , Γk , and

u “ 0, 1. 1

pi,jq

ru,k|k´1,β“0 “ (c) Compute the weight w

pi q

pi q 2Nb

for j “ 1, 2, ¨ ¨ ¨ , Nb , for i “ 1, 2, ¨ ¨ ¨ , Γk and

u “ 0, 1. ´ ¯ ` ˇ ˘ pi,jq pi,jq (d) Draw the particle xru,k|k´1,β“0 , Au,k|k´1,β“0 „ bu,k ¨ ˇZΓ,k , for j “ 1, 2, ¨ ¨ ¨ , Nb , for i “ 1, 2, ¨ ¨ ¨ , Γk and u “ 0, 1. 4. Using

the union of the Bernoulli ¯) Mk|k´1,β Ť !´ piq pi q π k | k ´1 “ rk|k´1,β , pu,k|k´1,β i “1

components,

obtain

the

Pr-MTD

as

β“0,1

Update: !´ 5. Compute the legacy Bernoulli components pi q

(a) Compute the existence probability r L,k “

pi q

pi q

r L,k , p L,u,k ř u“0,1

¯) M L k

i “1

pi q

r L,u,k via Equation (14), for i “ 1, 2, ¨ ¨ ¨ , MkL .

pi,jq

iq r L,u,k|k via Equation (19), for j “ 1, 2, ¨ ¨ ¨ , Lpu,k (b) Compute the weight w , for |k´1,β“1

i “ 1, 2, ¨ ¨ ¨ , MkL and u “ 0, 1. ´ ¯ pi,jq pi,jq (c) Obtain the particle xrL,u,k|k , A L,u,k|k pi q

´ “

¯ pi,jq pi,jq xru,k|k´1,β“1 , Au,k|k´1,β“1 ,

j “ 1, 2, ¨ ¨ ¨ , Lu,k|k´1,β“1 , for i “ 1, 2, ¨ ¨ ¨ , MkL and u “ 0, 1.

30393

for

Sensors 2015, 15, 30385–30402

Algorithm 1. Processing Steps of the SMC-UCR-MBerF-AI ` ˘( 6. Compute the updated Bernoulli components rU,k pzq , pU,u,k p¨; zq zPZ

k

ř

(a) Compute the existence probability rU,k pzq “

u“0,1

rU,u,k pzq via Equation (17), for

i “ 1, 2, ¨ ¨ ¨ , |Zk |. pi,jq

pi q

rU,u,k|k,β via Equations (20) and (21), for j “ 1, 2, ¨ ¨ ¨ , Lu,k|k´1,β , for (b) Compute the weight w Ť pi,jq pi,jq rU,u,k|k, “ rU,u,k|k,β . i “ 1, 2, ¨ ¨ ¨ , Mk|k´1,β , β “ 0, 1 and u “ 0, 1. Obtain the weight w w β“0,1

´ (c) Obtain

the

particle

pi,jq pi,jq xrU,u,k|k , AU,u,k|k

¯

¯ Ť ´ pi,jq pi,jq xru,k|k´1,β , Au,k|k´1,β ,

“

for

β“0,1

pi q

j “ 1, 2, ¨ ¨ ¨ , Lu,k|k´1,β , for i “ 1, 2, ¨ ¨ ¨ , Mk|k´1,β and u “ 0, 1. 7. Using the union of the Bernoulli components, !´ ¯) M L Ť ` ˘( k pi q pi q rU,k pzq , pU,u,k p¨; zq zPZ πk “ r L,k , p L,u,k

obtain

the

Po-MTD

as

k

i “1

Resample: 8. Discard the Bernoulli components with existence probability below a threshold G and obtain !´ ¯) Mk ! ´ ¯) Lpiq u,k|k pi,jq pi,jq pi,jq pi q pi q pi q ru,k|k , xru,k|k , Au,k|k πk “ rk , pu,k , and pu,k pxr, Aq :“ w . i “1

pi q

9. Resample Lk !

j “1

´

pi q

¯

“ max rk Lmax , Lmin

! times from

pi,jq

´

pi,jq

pi,jq

ru,k|k , xru,k|k , Au,k|k w

¯) Lpiq

u,k|k

j “1

to obtain

´ ¯) Lpiq u,k pi,jq pi,jq pi,jq pi,jq pi q pi q pi q wu,k , xru,k , Au,k with weights wu,k “ 1{Lk and Lu,k “ Lk . j “1

Multi-target state and clutter rate estimation: ř ˆ k “ M k r pi q 10. Estimate number of actual targets N i“1 1,k ! ) ř Lpiq pi,jq pi,jq p 1 q p 2q p Nˆ q pi q 11. Estimate actual targets’ state Xˆ k “ xˆ k , xˆ k , ¨ ¨ ¨ , xˆ k k with xˆ k “ j“1,k1 w1,k xr1,k . 12. Estimate clutter rate λˆ c,k “

ř Mk piq ř Lpiq pi,jq pi,jq 0,k i“1 r0,k j“1 w0,k p D,0,k .

Output: !´ πk “

pi q

pi q

rk , pu,k

¯) Mk i “1

! ´ ¯) Lpiq u,k pi,jq pi,jq pi,jq pi q , pu,k pxr, Aq :“ wu,k , xru,k , Au,k , Xˆ k , λˆ c,k j“1

4. Simulation In this section, we demonstrate the performance of the proposed UCR-MBerF-AI using a non-linear multi-target tracking scenario. We also compare it with the UCR-MBerF and the PHD filter for unknown clutter rate with amplitude information (UCR-PHDF-AI), whose new-born object RFSs are known a priori. The Optimal Subpattern Assignment (OSPA) metric [25] is used to evaluate the filters’ performance. 4.1. Simulation Scenario A non-linear multi-target scenario with the unknown clutter rate was used to demonstrate the performance of the UCR-MBerF-AI. The observation region is a half disc with a radius of 2000 m and has a total of ten targets. It is assumed that all of the targets have the same signal-to-noise ratio (SNR), ˆ 2˙ A defined in dB as SNR “ 10log . Figure 2 shows the targets’ true trajectories, where the start 2ψ2 and stop positions of each trajectory are indicated by the symbols ∇ and ˝, respectively. 30394

performance of the UCR-MBerF-AI. The observation region is a half disc with a radius of 2000 m and has a total of ten targets. It is assumed that all of the targets have the same signal-to-noise ratio

 A2  (SNR), defined in dB as SNR = 10 log  2  . Figure 2 shows the targets’ true trajectories, where the  2ψ  Sensors 2015, 15, 30385–30402 start and stop positions of each trajectory are indicated by the symbols ∇ and  , respectively. angle (deg)

90

radius (m)

Start position Stop position

2000

120

60 1500

1000

150

30

500

180

0

Figure 2. 2. True True target trajectories. Figure

4.1.1. Actual Target Model

10

The actual target motion follows a coordinated turn model. The dynamics of an actual target’s state are written as: # xrk “ F1 pωk´1 qxrk´1 ` G1 wk´1 (22) Ak “ Ak´1 ` Tδk´1 ” ıT . . where xrk “ p x,k , p x,k , py,k , py,k , ωk contains the actual target’s position, velocity and turn » fi 1 ´ cosωT sinωT 0 ´ 0 1 — ffi ω ω — 0 cosωT 0 ´sinωT 0 ffi — ffi — ffi sinωT rate, the transition matrix is F1 pωq “ — 0 1 ´ cosωT 1 , process noise 0 ffi — ffi ω ω — ffi – 0 sinωT 0 cosωT 0 fl 0 0 0 0 1 » 2 fi T 0 0 ffi — 2 — ffi — T 0 0 ffi — ffi ffi, the process noise distributions follow wk´1 „ N p¨; 0, Q1 q T2 matrix is: G1 “ — — 0 ffi 0 — ffi 2 — ffi – 0 T 0 fl 0 0 T ´” ı¯ ´ ¯ 2 2 2 2 Q1 “ diag σ1,w , σ1,w , σ1,ω and δk´1 „ N ¨; 0, σ1,δ with σ1,w “ 5 m{s2 , σ1,ω “ π{180 rad{s and σ1,δ “ 3, while the sampling time is T “ 1. Then, we have f 1,k|k´1 pxrk , Ak |xrk´1 , Ak´1 q “ ´ ¯ ´ ¯ 2 . The survival probability of an actual target is fixed at N xrk ; xrk´1 , G1 Q1 G1 T N Ak ; Ak´1 , σ1,δ pS,1,k “ 0.98. The measurements of actual targets include bearing, range and amplitude with likelihoods given by: ` ˘ g1,k pr zk |r xk q “ N r zk ; mrz,1,k pr xk q , Rrz,1,k ˜ ¸ ˆ ˙ rk rk Ak rk 2 ` Ak 2 I0 exp ´ ψ2 ψ2 2ψ2 τ g »d g1,k prk |Ak q “ ˜ ¸fi f 2 f A 1 k fl Q– , e2ln pτFA,k ψ2

where mrz,1,k pxrq

“

(23)

(24)

ˆ”

ı ” ´ ¯ b arctan p x,k {py,k , p2x,k ` p2y,k and Rrz,1,k

“ pi q

σ1,θ “ π{180 rad, σ1,r “ 5 m and ψ “ 1. For every measurement zk target particle position is given by [23]:

diag

ıT ˙

with

P ZΓ,k , the new-born actual

$ ´ ¯ ´ ¯ pi,jq p jq p jq pi q pi q ’ & p x,k “ p x,s ` zk r2s ` σ1,r ν1 cos zk r1s ` σ1,θ ν2 ´ ¯ ´ ¯ ’ % ppi,jq “ py,s ` zpiq r2s ` σ1,r νp jq sin zpiq r1s ` σ1,θ νp jq 2 1 y,k k k 30395

2 , σ2 σ1,θ 1,r

(25)

Sensors 2015, 15, 30385–30402

pi q

pi q

pi q

where j “ 1, . . . , Nb , zk r1s and zk r2s are the bearing and range measurement respectively, and ` ˘ . pi,jq . pi,jq pnq pnq ν1 , ν2 „ N p¨; 0, 1q. p x,s , py,s is the position of the sensor. The particle velocities follow p x,k , py,k „ ` ˘ ` ˘ pi,jq N ¨; 0, σV2 , where σV “ 50 m/s, the particle turn rates are distributed as ω1,k , „ N ¨; 0, σω2 , with ` ˘ pi,jq σω “ 0.1 rad/s, while the particle amplitudes follow Ak , „ N ¨; m A , σδ2 , where σδ “ 5, m A “ b 2ψ2 ˚ 10SNR{10 . The expected number of new-born actual targets at each scan is νkb “ 0.32. 4.1.2. Clutter Model ” ıT The clutters are only modelled by their positions xrk “ p x,k , py,k , while their turn rates and velocities are ignored and parameter Ak are set to zero. The positions of ´ clutters follow a¯random walk model, with a transition density given by f 0,k|k´1 pxrk |xrk´1 q “ N xrk ; xrk´1 , Px,0,k|k´1 , where ´” ı¯ Px,0,k|k´1 “ diag σx2 , σy2 with σx “ 1000 m and σy “ 500 m. The survival probability of clutters is fixed at pS,0,k “ 0.90. The measurements of clutters are also include bearing, range and amplitude. Their corresponding likelihoods are given by: ` ˘ g0,k pr zk |xrk q “ N r zk ; mrz,0,k pxrk q , Rrz,0,k ˆ 2 ˙ rk τ ´ rk 2 prk |Ak q “ 2 exp ψ 2ψ2 ” ´ ¯ b ı arctan p x,k {py,k , p2x,k ` p2y,k and Rrz,0,k

(26)

τ g0,k

where mrz,0,k pxrq

“

(27) ˆ” “

diag

2 , σ2 σ0,θ 0,r

ıT ˙

with

σ0,θ “ 20π{180 rad, σ0,r “ 400 m. The clutter birth process is similar that of the actual targets, but we only consider their position. 4.1.3. Filter Parameters In the proposed SMC implementation of the UCR-MBerF-AI, the maximum and minimum number of the particles per Bernoulli component are set to be Lmax “ ´ 1000 and Lmin “ 300, ¯ respectively. pi q

pi q

The number of particles for each new-born object is Nb “ max ru,β“0 ˚ Lmax , Lmin . Additionally, G “ 10´3 is the threshold of the probability of existence, while a maximum Tmax “ 100 is used for pruning the number of Bernoulli components. 4.2. Results 4.2.1. Multi-Target Tracking with the Fixed Clutter Rate In this section, we evaluate the performance of the proposed UCR-MBerF-AI with a fixed clutter rate. There were 100,000 clutter measurements before thresholding. We assume that the measurements can be generated from anywhere within the observation region. The probability of a false alarm is set as pτFA,k “ 1 ˆ 10´ 4 , and the corresponding clutter rate after thresholding is λ “ 10. The SNR of all targets was fixed at 13 dB. According to Equation (4), the probability of detecting actual targets in this case is pτD,k “ 0.98. Other scenario parameters are given in Section 4.1. Figure 3 shows the observations immersed in the clutters, while Figure 4 shows the estimated position output of the UCR-MBerF-AI from a single run. Compared with the true trajectories, the results indicate that the UCR-MBerF-AI can correctly determine actual target appearance, motion and disappearance, and achieve accurate multi-target tracking without the need of the new-born objects’ prior RFSs. Moreover, it should be noted that, although false estimates occur occasionally, due to the fact that new-born objects are driven by the measurements, the false estimates die out very quickly.

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Sensors 2015, 15, page–page Sensors 2015, 15, page–page radius(m) 2000 radius(m)

90 90

angle(deg) angle(deg) 120

Measurements Measurements

2000

60 60

120

1500 1500 150 150

30 30

1000 1000

Sensors 2015, 15, page–page

500 500 angle(deg)

180 180

radius(m)

90

Measurements

2000

120

0 0

60 1500

Figure 3. Observations immersed in clutters. Figure 3. 3. Observations Observations immersed immersed in in clutters. clutters. Figure 150

angle(deg) 120 angle(deg)

210

30

1000

90 90

radius(m) 2000 radius(m)

500

Filter Estimates True Filter tracks Estimates 60True tracks 330

2000

120

60

1500 1500

180

150 150

0

Figure 3. Observations immersed in clutters. 240

angle(deg) 120

300

1000 1000

radius(m)

270 90

30 30

Filter Estimates True tracks

2000

60

500 500

1500

180 180

150

0 0

30

1000

Figure 4. Position estimations of the UCR-MBerF-AI. 500 Figure Figure 4. 4. Position Position estimations estimations of of the the UCR-MBerF-AI. UCR-MBerF-AI.

15 15 10 10 5 5

15 True Cardinality Statistics

Cardinality Statistics Cardinality Statistics

To validate the performance of the UCR-MBerF-AI, it was compared to that of the UCR-MBerF 180 0 To validate the performance of the UCR-MBerF-AI, it was compared to that of the UCR-MBerF 210 330 [22] over 100 Monte Carlo (MC) trials. The target trajectories shown Figure 2, with random To validate the performance of the UCR-MBerF-AI, it was compared to that of along the UCR-MBerF [22] Figure 4. Position estimations of the UCR-MBerF-AI. [22] over 100 Monte Carlo (MC) trials. The target trajectories shown Figure 2, along with random clutters were used for each trial. Figures 5 and 6 show the cardinality and clutter rate statistics versus over 100 Monte Carlo (MC) trials. The target trajectories shown Figure 2, along with random clutters clutters were used for each trial. Figures 5 and 6 show the cardinality and clutter rate statistics versus To validate the performance of the UCR-MBerF-AI, it was compared to that of the UCR-MBerF time and the respectively. These results confirm were for usedthe forUCR-MBerF-AI each trial. Figures 5 and 6 UCR-MBerF, show the cardinality and clutter rate statistics versusthat timethe for [22] UCR-MBerF-AI over 100 Monte Carlo (MC)the trials. The target trajectories shown Figure 2, along withconfirm random that the time for the and These results 240 UCR-MBerF, respectively. 300 UCR-MBerF-AI produces more accurate estimations of both the number of actual targets and the the UCR-MBerF-AI and the UCR-MBerF, respectively. These results confirm that the UCR-MBerF-AI clutters produces were used for each trial. Figuresestimations 5 and 6 show the clutter rate statisticstargets versus and the UCR-MBerF-AI more accurate of cardinality both the and number of actual clutter ratetime than the In addition, the270 UCR-MBerF has atargets negative bias thethe clutter rate produces more estimations oftheboth the number of actual theinthat clutter rate than foraccurate theUCR-MBerF. UCR-MBerF-AI and UCR-MBerF, respectively. These resultsand confirm clutter rateUCR-MBerF-AI than the UCR-MBerF. In addition, the UCR-MBerF has a negative bias in the clutter rate producespositive more accurate estimations of both the number of actual the clutters estimate and a corresponding bias in the estimate. This is targets because some the UCR-MBerF. In addition, the UCR-MBerF hascardinality a negative bias in the clutter rateand estimate and a estimate and a corresponding positiveInbias in the estimate. This isin because some clutters rate than the UCR-MBerF. the cardinality UCR-MBerF has a negative bias the clutter rate are treatedclutter as actual targets, while the addition, UCR-MBerF-AI information distinguish corresponding positive bias in the cardinality estimate. uses This amplitude is because some clutterstoare treated as and atargets, corresponding bias in the cardinality estimate. This is because some clutters are treatedestimate as actual whilepositive the UCR-MBerF-AI uses amplitude information to distinguish between actual targets and clutters. actual targets, while UCR-MBerF-AI amplitude information to distinguish between actual are treated asthe actual targets, while theuses UCR-MBerF-AI uses amplitude information to distinguish between actual targets and clutters. targets andbetween clutters. actual targets and clutters. True Mean

Mean StDev 10

StDev 5 0

0 00 0

True Mean StDev

0

10 10

10

20

20 20

30

30 30

40

40 40

5 5

Cardinality Statistics

Cardinality Statistics Cardinality Statistics

10 10

10

True True Mean 5 Mean StDev StDev 0

0

True Mean StDev

10

60

70

60 60

80

70 70

90

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(a) (a)

15

15 15

50

50 t/s 50 (a)t/s t/s

20

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50 t/s

60

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90

100

(b) 0 0 10 20 30 40 50 scenario.60 70 80 90 100 0 Figure 5. Cardinality statistics for the fixed clutter rate (a) UCR-MBerF-AI; (b) UCR-MBerF. 0 10 20 30 the fixed 40 clutter rate 50 scenario. 60 (a) UCR-MBerF-AI; 70 80 (b) UCR-MBerF. 90 100 Figure 5. Cardinality statistics for t/s 13t/s

(b) (b)

Figure 5. Cardinality statistics for the fixed clutter rate scenario. (a) UCR-MBerF-AI; (b) UCR-MBerF. 30397 Figure 5. Cardinality statistics for the fixed clutter rate scenario. (a) UCR-MBerF-AI; (b) UCR-MBerF.

13 13

Sensors 2015, 15, page–page

5 0

Clutter Rate

Clutter Rate

20 True Sensors 2015, 15, 30385–30402 15 Mean15, page–page Sensors 2015, StDev 10 20

0

True Mean StDev

15 10

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0

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30 20

40 30

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(a)50

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8090

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True Mean 15StDev

(a)

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15 Clutter Rate

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60

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t/s

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True Mean StDev

0 10

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4040

50 50 t/s t/s

60

60

70

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(b)

(b) Figure 6. True and estimated clutter rates for the fixed clutter rate scenario. (a) UCR-MBerF-AI; (b)

Trueand andestimated estimatedclutter clutterrates ratesforfor the fixed clutter rate scenario. (a) UCR-MBerF-AI; Figure 6. True the fixed clutter rate scenario. (a) UCR-MBerF-AI; (b) UCR-MBerF. (b) UCR-MBerF. UCR-MBerF. We assessed the performance of the given multi-target filters using the OSPA metric, which can

jointly evaluate the localizationof and cardinality error between filters two finite sets [25]. The OSPA metricwhich can We assessed the the given multi-target multi-target using the OSPA OSPA metric, We assessed the performance performance of the given filters using the metric, which can is given by Equation (28) with p = 1, c = 300 : jointly evaluate metric is jointly evaluate the the localization localization and and cardinality cardinalityerror errorbetween betweentwo twofinite finitesets sets[25]. [25].The TheOSPA OSPA metric 1 given by Equation (28) with p “ 1, c “ 300: m p is given by Equation (28) with ( cp) = 1, c = 300 1 :  (28) d p ( X , Z ) =   min  d ( c ) ( xi , zπ ( i ) ) p + c p (n − m)   1  ˜ ˜ n  π ∈∏ i =1 ¸¸ 1 m ÿm p 1 1  pcq p pcq( cUCR-PHDF-AI p  p We compared UCR-MBerF-AI the Figure 7 p pn the ( c )Zq ) i, z (28) d p pX, min “ `ccand świth d (28) = d the X Z d px ( , ) min ( xi , zππp(iiq) q) p + (n −´mmq )UCR-MBerF.    p   n π P shows the mean estimation errors obtained using the OSPA metric for the 100 MC simulations over n i“ π ∈∏ n 1 n i =1    n

time. It can be seen that the performance of the UCR-MBerF-AI is better than the UCR-PHDF-AI,

while the UCR-MBerF performs significantly among the three filters. main reason Figure is that Figure the with theworst UCR-PHDF-AI and theThe UCR-MBerF. 7 shows We compared compared theUCR-MBerF-AI UCR-MBerF-AI with the UCR-PHDF-AI and the UCR-MBerF. 7 the UCR-MBerF-AI performs more accurate estimations and obtains a smaller variance of the the mean errors obtained using the OSPA metricmetric for thefor 100 MC over time. shows theestimation mean estimation errors obtained using the OSPA the 100simulations MC simulations over cardinality than the UCR-MBerF, as shown in Figure 5, while being free of the reliance on clustering It canItbecan seen that the performance of the of UCR-MBerF-AI is better than the UCR-PHDF-AI, while time. that the the UCR-MBerF-AI is better the UCR-PHDF-AI, thatbeis seen a necessary stepperformance in the implementation of the UCR-PHDF-AI. Due to than the change of the the UCR-MBerF performs significantly worst among the three filters. The main reason the bothperforms the UCR-MBerF-AI and the worst UCR-MBerF produce peaksThe at 10, 20, 30,reason 50,is that while the cardinality, UCR-MBerF significantly among the significant three filters. main is that 70 and 82 s, however the UCR-PHDF-AI only produces significant peaks at 10, 20 and 30 s. This is UCR-MBerF-AI performs more accurate estimations and obtains a smaller variance of the cardinality the UCR-MBerF-AI performs more accurate estimations and obtains a smaller variance of the because there are more targets in the second half of the simulation, and the UCR-PHDF-AI than the UCR-MBerF, as shown in freebeing of thefree reliance clustering that is a cardinality than the UCR-MBerF, asFigure shown5, inwhile Figurebeing 5, while of theon reliance on clustering performs worse due to the clustering process. necessary step in thestep implementation of the UCR-PHDF-AI. Due to the Due change of the cardinality, that is a necessary in the implementation of the UCR-PHDF-AI. to the change of the both the UCR-MBerF-AI and the UCR-MBerF produce significant peaks at 10, 20, 30, 50, 70 and s, 10 cardinality, both the UCR-MBerF-AI and the UCR-MBerF produce significant peaks at 10, 20, 30,8250, UCR-PHDF-AI however UCR-PHDF-AI only produces significant peaks at 10, 20peaks and 30 s. is because there 70 and 82the s, however the UCR-PHDF-AI only produces significant atUCR-MBerF-AI 10,This 20 and 30 s. This is UCR-MBerF are more there targetsare in the second halfin of the simulation, and the UCR-PHDF-AI performs worse due to because more targets second half of the simulation, and the UCR-PHDF-AI 10 the clustering process. performs worse due to the clustering process. OSPA(m) (c=300,p=1)

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Figure 7. OSPA metrics of UCR-PHDF-AI, UCR-MBerF-AI and UCR-MBerF for the fixed clutter rate scenario. Figure 7. OSPA metrics of UCR-PHDF-AI, UCR-MBerF-AI and UCR-MBerF for the fixed clutter rate scenario.

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4.2.2. Multi-Target Tracking with the Time-Varying Clutter Rate 4.2.2. Multi-Target Tracking with the Time-Varying Clutter Rate demonstrate the performance of the UCR-MBerF-AI in clutter rate estimation, we consider 4.2.2.To Multi-Target Tracking with the Time-Varying Clutter Rate To demonstrate the performance of the UCR-MBerF-AI clutter rate estimation, werate. consider the scenario with the target trajectories shown in Figure 2 in and a time-varying clutter The To demonstrate thetarget performance of theshown UCR-MBerF-AI in clutter rate estimation,clutter we consider the the scenario with the trajectories in Figure 2 and a time-varying rate. The scenario parameters are as same as in Section 4.1, except for the number of the clutters before scenario with the target trajectories shown in Figure 2 and a time-varying clutter rate. The scenario parameters as same as in Section 4.1, except numbertarget of the clutters from before thresholding. Figure 8are shows the UCR-MBerF-AI output of for the the estimated positions a parameters areFigure as same as in Section 4.1, except foroutput the number of the clutters before thresholding. thresholding. 8 shows the UCR-MBerF-AI of the estimated target positions from a single run. Figurerun. 8 shows the UCR-MBerF-AI output of the estimated target positions from a single run. single radius(m) 2000 radius(m)

90

angle(deg) 120 angle(deg)

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Filter Estimates True Filter tracks Estimates 60True tracks 60

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Figure 8. Position estimations of the UCR-MBerF-AI. Figure Figure 8. 8. Position Position estimations estimations of of the the UCR-MBerF-AI. UCR-MBerF-AI.

Cardinality Statistics Cardinality Statistics

Cardinality Statistics Cardinality Statistics

The averaged cardinality and clutter rate statistics versus time for the 100 MC simulations using The averaged cardinality and clutter rate statistics versus time10, for the 100 MC It simulations the UCR-MBerF-AI and UCR-MBerF are shown in Figures 9 and can be seenusing that The averaged cardinality and clutter rate statistics versus time forrespectively. the 100 MC simulations using the UCR-MBerF-AI and UCR-MBerF are shown in Figures 9 and 10, respectively. It can be seen that UCR-MBerF-AI can estimate the number actualintargets accurately, whereas the UCR-MBerF a the UCR-MBerF-AI and UCR-MBerF are of shown Figures 9 and 10 respectively. It can be seenhas that UCR-MBerF-AI can estimate theperformance number of actual targets accurately, whereas the UCR-MBerF has a positive bias which affects its adversely as the clutter rate increases. Moreover, the UCR-MBerF-AI can estimate the number of actual targets accurately, whereas the UCR-MBerF has positive bias which affectsa its performance adversely as the clutter the UCR-MBerF-AI provides accurate estimation clutter rate rate,increases. while theMoreover, UCR-MBerF a positive bias which affectsmore its performance adversely in as the the clutter rate increases. Moreover, the UCR-MBerF-AI provides a more accurate estimation in the clutter rate, while the UCR-MBerF shows a comparatively accurate estimation when the clutter rate is below 5 but demonstrates UCR-MBerF-AI provides a more accurate estimation in the clutter rate, while the UCR-MBerF shows aa shows a bias comparatively accurate estimation when the becomes clutter rate is noticeable below 5 but a negative the estimation clutter rate is larger; this bias as demonstrates the clutter rate comparativelywhen accurate when the clutter rate is belowmore 5 but demonstrates a negative bias negative bias when the clutter rate is larger; this bias becomes more noticeable as the clutter rate increases. when the clutter rate is larger; this bias becomes more noticeable as the clutter rate increases. increases. 15 15 10 10

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Figure 9. Cardinality statistics for the time-varying clutter rate scenario. UCR-MBerF-AI; (b) UCR-MBerF. Figure 9. Cardinality statistics for the time-varying clutter rate(a) scenario. (a) UCR-MBerF-AI; Figure 9. Cardinality statistics for the time-varying clutter rate scenario. (a) UCR-MBerF-AI; (b) UCR-MBerF. (b) UCR-MBerF.

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ClutterRate Rate Clutter

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(b) (b) Figure True and estimated clutter rates for the time-varying clutter rates scenario. (a) UCR-MBerF-AI; Figure 10. 10. Figure 10. True True and and estimated estimated clutter clutter rates rates for for the the time-varying time-varying clutter clutter rates rates scenario. scenario. (a) (a) UCR-MBerF-AI; UCR-MBerF-AI; (b) UCR-MBerF. (b) UCR-MBerF. (b) UCR-MBerF.

Figure mean estimation errors obtained obtained using metric for for the the 100 mean estimation estimation errors Figure 11 11 shows shows the the mean using the the OSPA OSPA metric 100 MC MC simulations over time. The UCR-PHDF-AI and UCR-MBerF produce over time. The UCR-MBerF-AI, UCR-MBerF-AI, UCR-PHDF-AI and produce UCR-MBerF produce simulations over time. The UCR-MBerF-AI, UCR-PHDF-AI and UCR-MBerF approximately approximately average errors of 25, 36 50 respectively. This is the approximately average errors of per 25, target, 36 and andrespectively. 50 m m per per target, target, respectively. This accurate is due due to toestimation the more more average errors of 25, 36 and 50 m This is due to the more accurate estimation of the number of actual targets and clutter rate of the UCR-MBerF-AI. These accurate estimation of the number actualrate targets and clutter rate of These the UCR-MBerF-AI. These of the number of actual targets andof clutter of the UCR-MBerF-AI. experimental results experimental suggest the outperforms UCR-PHDF-AI, experimental results further suggest that that the UCR-MBerF-AI UCR-MBerF-AI outperforms the UCR-PHDF-AI, further suggestresults that thefurther UCR-MBerF-AI outperforms the UCR-PHDF-AI, which the in turn outperforms which in turn outperforms the UCR-MBerF. which in turn outperforms the UCR-MBerF. the UCR-MBerF. OSPA(m)(c=300,p=1) (c=300,p=1) OSPA(m)

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Figure 11. OSPA metrics the UCR-MBerF-AI and for Figure 11. metrics of theof UCR-MBerF-AI and UCR-MBerF for the time-varying 11.OSPA OSPA metrics ofUCR-PHDF-AI, the UCR-PHDF-AI, UCR-PHDF-AI, UCR-MBerF-AI and UCR-MBerF UCR-MBerF for the the time-varying clutter clutter rates scenario. time-varying clutter rates rates scenario. scenario.

4.2.3. Tracking at Various SNR Levels Multi-Target Tracking at atVarious VariousSNR SNRLevels Levels 4.2.3. Multi-Target Multi-Target Tracking To discuss the effect noise on the 100 MC trials were for To discuss discussthe theeffect effect of the noise on UCR-MBerF-AI, the UCR-MBerF-AI, UCR-MBerF-AI, 100trials MC were trialsperformed were performed performed for To of of thethe noise on the 100 MC for various various SNR values. The same target trajectories shown in Figure 2 with random clutters for each various SNR The values. same target trajectories Figure 2 withclutters random for were each SNR values. sameThe target trajectories shown inshown Figure in 2 with random forclutters each trial trial were also adopted, with an average of 10 clutters per scan after thresholding. The average trial were also adopted, with an average of 10 clutters per scan after thresholding. The average also adopted, with an average of 10 clutters per scan after thresholding. The average OSPA metrics and OSPA corresponding estimated rate the filters the are OSPA metrics metrics and and corresponding estimated clutter rate for for the three three filters versus theinSNR SNR are1shown shown corresponding estimated clutter rate for the clutter three filters versus the SNR areversus shown Tables and 2 in Tables 1 and 2, respectively. As expected, the average OSPA metrics increase and the accuracy of in Tables 1 and respectively. As expected, the average OSPA increase and the accuracy of respectively. As 2, expected, the average OSPA metrics increase and metrics the accuracy of estimated clutter rate estimated clutter rate decreases as the SNR decreases for the three filters. This is because the estimated clutter rate decreases as the SNR decreases for the three filters. This is because the decreases as the SNR decreases for the three filters. This is because the detection probability decreases detection probability decreases with the lower and filters all more sensitive to detection probability decreases with are the all lower SNR, and these these filters are are alllevels, more which sensitive to with the lower SNR, and these filters moreSNR, sensitive to increased noise make increased noise levels, which make detection more difficult. However, the UCR-MBerF-AI still increased more noisedifficult. levels, which make more difficult. However, still detection However, thedetection UCR-MBerF-AI still performs better the thanUCR-MBerF-AI the UCR-PHDF-AI performs better the and UCR-MBerF, as performs better than than thebyUCR-PHDF-AI UCR-PHDF-AI andand UCR-MBerF, as seen seen by by the the lower lower location location and and and UCR-MBerF, as seen the lower location cardinality errors. cardinality errors. cardinality errors. 30400 16 16

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Table 1. Time-averaged OSPA metrics for various SNR with pτFA,k “ 1 ˆ 10´ 4 . SNR

10.50 dB

11.12 dB

11.85 dB

13.00 dB

14.50 dB

pτD,k

0.70 102.63 116.54 129.88

0.80 74.72 91.83 107.48

0.90 49.25 64.61 87.98

0.95 26.39 37.72 63.47

0.99 24.85 34.95 57.84

UCR-MBerF-AI UCR-PHDF-AI UCR-MBerF

Table 2. Time-averaged estimated clutter rate for various SNR with pτFA,k “ 1 ˆ 10´ 4 . SNR

10.50 dB

11.12 dB

11.85 dB

13.00 dB

14.50 dB

pτD,k

0.70 9.65 9.03 8.56

0.80 9.72 9.17 8.73

0.90 9.85 9.26 8.90

0.95 9.94 9.35 9.15

0.99 9.98 9.53 9.32

UCR-MBerF-AI UCR-PHDF-AI UCR-MBerF

5. Conclusions In this paper, we have presented an improved MBerF, named the UCR-MBerF-AI, which not only achieves more accurate and steady estimations of the clutter rate and the number of actual targets, but also relaxes the requirement on the a priori knowledge of the new-born object RFSs. The proposed improved filter retains the mathematical structure of the conventional MBerF, while additionally incorporating the signal amplitude information into the MBerF recursion loop to enhance the discrimination between actual targets and clutters, and utilizing measurements to adaptively generate the new-born object RFSs. Moreover, the SMC implementation of the proposed filter was studied using numerical simulations. The results demonstrate that the UCR-MBerF-AI improves the estimation accuracy of the target number and the corresponding multi-target states, as well as the clutter rate, over previous approaches. Acknowledgments: This study was co-supported by the National Natural Science Foundation of China (Nos. 61171122, 61201318, 61471019, 61501011 and 61501012) and the Fundamental Research Funds for the Central Universities (No. YWF-15-GJSYS-068). Author Contributions: In this study, Changshun Yuan and Jun Wang conceived and designed this novel algorithm and simulation experiment; Yanxian Bi and Zhongsheng Sun provied the advices in the details; Changshun Yuan wrote this paper; Peng Lei modified English errors. Conflicts of Interest: The authors declare no conflict of interest.

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