Hybrid fuzzy probabilistic data association filter and joint probabilistic data association filter

Information Sciences 142 (2002) 195–226 www.elsevier.com/locate/ins

Hybrid fuzzy probabilistic data association filter and joint probabilistic data association filter Mourad Oussalah a

a,*,1

, Joris De Schutter

b

Centre for Software Reliability, City University, 10 Northampton Square, London EC1VOHB, UK b K.U. Leuven, PMA, Celestijnenlaan 300B, Heverlee, Belgium

Abstract Multitarget tracking problems are theoretically interesting because, unlike other estimation problems, the origins of the measurements are not identified. This involves hypothesis generation and their evaluation in terms of degree of agreement between the given measurements and the underlying tracks. Typical algorithms to deal with such problems are the probabilistic data association filter (PDAF) in the case of single target tracking and joint probabilistic data association filter (JPDAF) in the case of multiple target tracking proposed by Bar-Shalom and his team. The basis of JPDAF is the calculus of the joint probabilities over all targets and hits. The algorithm assigns weights for reasonable hits and uses a weighted centroid of those hits to update the track. In this paper, we propose a new weight assignment based on fuzzy c-means methodology. Particularly, in order to take account for the false alarms (clutter) where none of the measurements is target originated, a new noisy fuzzy c-means algorithm is elaborated. The latter contrasts with that provided by Dave regarding the location of the noise prototype as well as the meaning of the universality of the noise class. The treatment of conflictual situations where, for instance, more than one hit fail in a target extension gate is accomplished using some weighted based procedure with respect to all feasible joint matrices involved in the construction of joint probabilities in JPDAF. In the meantime, the general methodology of PDAF and JPDAF remains unchanged. This leads to Hybrid Fuzzy PDAF in the case of single target tracking and Hybrid Fuzzy

*

Corresponding author. Tel.: +44-207-477-8424; fax: +44-207-477-8585. E-mail addresses: [email protected] (M. Oussalah), [email protected] (J. De Schutter). 1 This work was performed when the author was in PMA K.U. Leuven as Research Associate in robotics unit of the department. 0020-0255/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 2 0 - 0 2 5 5 ( 0 2 ) 0 0 1 6 6 - 4

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JPDAF in the case of multiple target tracking. This investigation shows a fruitful combination between fuzzy and probabilistic approaches in order to accomplish target tracking tasks. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: Target tracking; Joint probabilistic data association filter; Fuzzy c-means

1. Introduction Typically, target tracking in dense environments where probability of presence of ‘‘clutter’’ or false alarms is not neglected is a difficult problem. Indeed, in addition to the noisy measurements (hits) supplied by the source, there is an additional uncertainty concerning the origin of the measurement. In other words, we do not know exactly from which sensor or source (a clutter is also a source) the given measurement is originated from. This induces a risk of updating the target model by a wrong measurement, which obviously leads to a wrong estimation of a target state vector (position, velocity and so on). Consequently, this leads to a generation of a set of measurement/target associations and, thereby, evaluation of such hypotheses, usually in terms of joint probability values. Such problems arise in applications like air traffic control, ocean/battlefield surveillance, positioning of enemy targets in military context. The field of multiple target tracking has been investigated intensively since the 1970s and the pioneer works of Singer and Sea [16] on tracking in clutter environment, see also recent papers in [1,6]. A significant problem in multiple target tracking is the hit-to-track data association. The aim is to perform the estimation of some unknown parameters, described in terms of the state vector and its associated variance–covariance matrix. This estimation is performed such that each hit is related to its appropriate target including a clutter for false alarms with a high value of joint probability or confidence factor in more general terms. Bar-Shalom and Tsee [2] have first proposed ‘‘Probabilistic Data Association Filter’’ (PDAF) as a method for associating hits where only one target is available. That is, it assumes that all hits are in particular target extension gate and originated either from a target or clutter. If another target is persistently in this target extension gate, the results are poor and may be wrong. To account for the problem of more than one target in the cluttered environment, an extension of PDAF called ‘‘joint probabilistic data association filter’’ (JPDAF) was investigated by Fortmann et al. [12]. Their proposal acts as a natural extension of PDAF where all the feasible joint events are considered. The process was simplified by considering just the feasible joint events, those for which there is at most one hit per target and no two tracks associated with the same hit.

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These algorithms, among others, have successfully demonstrated their feasibility in high cluttered situations including air traffic management and military applications [3]. However, the calculation of joint probabilities in JPDAF seems complicated even if the used formulas are well established in probability theory, while the non-optimality still occurs regarding the number of hypotheses and restrictions governing the construction of PDAF or JPDAF. This motivates some authors to investigate simplifications of these formulations. This includes, for instance, Fitzgerald’s ad hoc proposal [11], which seems to work well in some cases of crossing targets. The class of algorithms, so-called fuzzy ISODATA algorithms developed mainly by original works of Dunn and Bezdek (see [4] and references therein), permits performing an unsupervised classification of a set of data given in multidimensional space into a given c number of classes. The algorithm assigns at each datum a grade of membership to which it deemed to be in agreement with each of the c classes such that the datum belongs certainly to all the classes in the probabilistic sense. The latter arises from the constraint, which asserts that, for each datum, the sum of the membership grades over the c classes is equal to one. Dave [9] introduced a robust clustering by incorporating an additional imaginary class that takes account for all noisy data. The idea developed in this paper is the possibility of incorporating the membership grades supplied by the fuzzy algorithm as a direct counterpart of the joint probabilities in PDAF or JPDAF while keeping the main structure of both PDAF and JPDAF. This leads to Hybrid Fuzzy PDAF in the case of single target tracking and Hybrid Fuzzy JPDAF in the case of multiple target tracking. The fuzzy classification algorithm is inspired from Dave’s approach. Particularly, in order to take account for the false alarms (clutter) where none of the measurements is target originated, a new noisy fuzzy c-means algorithm is elaborated. The latter contrasts with that provided by Dave regarding the location of the noise prototype as well as the meaning of the universality of the noise class. Strictly speaking, the fact of substituting joint probabilities by other values even outside the framework of probability in the same process is not quite strange as it may sound at first glance. Indeed, the justification of such substitution can be carried out from different viewpoints: (i) The existence of different proposals like Fitzgerald’s ad hoc formulations for the joint probabilities proves the non-uniqueness of the solution. (ii) The interpretation of the joint probability quantities as degrees of agreement between measurement and targets means that both joint probabilities and membership grades have the same interpretative setting. (iii) The general constraints governing the construction of the joint probabilities are kept preserved in the fuzzy setting. This induces a fruitely combination of probabilistic and fuzzy approaches. Section 2 of this paper deals with a short description of PDAF and JPDAF

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algorithms including their physical constraints, requirements and formulations. In Section 3 the methodology of fuzzy c-means is detailed. Then, a modified approach, which takes account for PDAF/JPDAF requirements, is investigated. Section 4 emphasizes some simulation results where the performances of the elaborated hybrid fuzzy PDAF/JPDAF are compared with the original version of PDAF/JPDAF as well as with the Fitzgerald’s ad hoc approach. The simulations consist of a constant velocity moving targets with the same or different directions. Single target moving makes a link to PDAF (and alternative approach) and multiple targets build a bridge to JPDAF (and alternative approach).

2. Probabilistic data association filter (PDAF) and joint probabilistic data association filter (JPDAF) 2.1. PDAF A detailed derivation of PDAF and JPDAF can be found in [3] while it is just briefly described here. Notice that, in the mathematical viewpoint, the JPDAF is very similar to its broader algorithm PDAF used in the case of single target tracking [2,3]. The only difference between JPDAF and PDAF is summarized in the calculus of the weights, attached to the innovations, providing the degree to which the underlying association target–measurement is deemed to be correct. Formally, let us assume a linear system of one target and measurement models described in discrete case according to the following: X ðkÞ ¼ AðkÞX ðk  1Þ þ n1 ðkÞ;

ð1Þ

Y ðkÞ ¼ CðkÞX ðkÞ þ n2 ðkÞ:

ð2Þ

Both the state vector X ðkÞ and the measurement vector Y ðkÞ are pervaded by additive, Gaussian, zero mean noise with known covariance respectively Q and R. AðkÞ and CðkÞ stand for the transition matrices pertaining respectively to the state and the measurement models. h i h i E n1 ðkÞðn1 ðjÞÞT ¼ QðkÞdkj ; E n2 ðkÞðn2 ðjÞÞT ¼ RðkÞdkj ;  ð3Þ 1 if k ¼ j; dkj ¼ 0 if k 6¼ j: In the case there is no uncertainty about the target–measurement association, then Kalman filter (KF) is an optimal estimator (in the sense of variance minimization) that supplies the best estimate for X ðkÞ. We denote by P ðk j k  1Þ, P ðk j kÞ respectively the state predicted covariance and the state

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updated covariance pertaining respectively to the prediction and updated state vector X ðk j k  1Þ and X ðk j kÞ. Similarly, Y ðk j k  1Þ refers to the predicted measurement given the predicted estimate of the state. The update stage makes use of a filter gain KðkÞ and the innovation covariance SðkÞ. The quantity ½Y ðkÞ  Y ðk j k  1Þ refers to the filter innovation. KF equations are summarized in expressions (4)–(10): P ðk j k  1Þ ¼ Aðk  1ÞP ðk  1 j k  1ÞAT ðk  1Þ þ Q;

ð4Þ

X ðk j k  1Þ ¼ Aðk  1ÞX ðk  1 j k  1Þ;

ð5Þ

Y ðk j k  1Þ ¼ CðkÞX ðk j k  1Þ;

ð6Þ

T

SðkÞ ¼ CðkÞP ðk j k  1ÞC ðkÞ þ R; T

1

ð7Þ

KðkÞ ¼ P ðk j k  1ÞC ðkÞS ðkÞ;

ð8Þ

X ðk j kÞ ¼ X ðk j k  1Þ þ KðkÞ½Y ðkÞ  Y ðk j k  1Þ ;

ð9Þ

P ðk j kÞ ¼ ½I  KðkÞCðkÞ P ðk j k  1Þ:

ð10Þ

However, in the case of the presence of target–measurement association uncertainty, KF leads obviously to a wrong estimate since the prediction will be updated by wrong measurements. This motivates the use of conditional probabilities for evaluating the measurement–target association in PDAF. For this purpose, the number of rational measurements is constrained by a statistical validation test such that only those failing in some ellipsoid probabilistic region determined by a threshold gate are validated. That is, the set of validated measurements at kth time ZðkÞ is determined by ZðkÞ ¼ fY ðkÞ; Y~ T ðkÞS 1 ðkÞ  Y~ ðkÞ 6 cg

ð11Þ

provided that Y~ ðkÞ is the innovation induced by the measurement Y ðkÞ, i.e., Y~ ðkÞ ¼ Y ðkÞ  CðkÞ  X ðk  1 j kÞ ðsimilarly; Z~i ðkÞ ¼ Zi ðkÞ  CðkÞ  X ðk  1 j kÞÞ and the threshold c is fixed by the gate probability PG such that PG ¼ P ðv2 6 cÞ where the Khi-square distribution has nz ¼ m (dimension of the measurement vector) degrees of freedom. The previous region forms an ellipsoid of probability concentration – a region of minimum volume that contains a given probability mass. The inequality shown in (11) is equivalent to, say that, the normalized innovation is below the threshold c. Now assume that there are mk number of validated measurements at time k. Let us denote by bi ðkÞ the probability that the ith hit (measurement) comes from the target in track. So, b0 ðkÞ corresponds to the probability that none of the measurements is originated from the target, or equivalently, to the probability that the current measurement is a false alarm (or clutter). Then, PDAF comes down to the following equations: X ðk j kÞ ¼ X ðk j k  1Þ þ W ðkÞ  mt ðkÞ

ð12Þ

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and mt ðkÞ ¼

nv X

bi ðkÞ  mi ðkÞ ¼

nv X

i¼1

bi ðkÞ  ½Zi ðkÞ  C  X ðk j k  1Þ ;

ð13Þ

i¼1

where m is referred to as a weighted innovation. The covariance is given by P ðk j kÞ ¼ b0 ðkÞ  P ðk j k  1Þ þ ½1  b0 ðkÞ  PG ðk j k  1Þ þ P~ðkÞ; ð14Þ where

"

P~ðk j kÞ ¼ W ðkÞ 

nv X

# T

bi ðkÞ  mi ðkÞ  ðmi ðkÞÞ  mðkÞ  ðmðkÞÞ

T

W ðkÞ;

ð15Þ

i¼1

PG ðk j kÞ ¼ ½I  W ðkÞ  CðkÞ  P ðk j k  1Þ:

ð16Þ

The derivation of the probabilities bi is given by the following: First the volume of the ellipsoidal region is given by V ðkÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pm=2 c  S i ðkÞ: Cð1 þ m=2Þ

ð17Þ

Then, let k be the density of false alarm and Pd the probability of target detection in the given track and m=2 2p 1  Pd Pg b¼ k  V ðkÞ ; ð18Þ c Pd bi ¼ exp½Z~iT ðkÞS 1 ðkÞZ~i ðkÞ=2 :

ð19Þ

So, b0 ðkÞ ¼

bþ

b Pmk

i¼1

and

bi

bi ðkÞ ¼

bþ

b Pi mk

i¼1

bi

ði 6¼ 0Þ:

ð20Þ

Note that the weights bi ðkÞ fulfill the constraint mk X

bi ðkÞ ¼ 1 and

0 6 bi ðkÞ 6 1:

ð21Þ

i¼0

2.2. JPDAF In this case the tracking is carried out over multiple targets, with possibly different state and measurement equations. That is, X t ðkÞ ¼ At ðkÞX t ðk  1Þ þ nt1 ðkÞ;

ð22Þ

t

ð23Þ

Y ðkÞ ¼ CðkÞ  X ðkÞ þ

nt2 ðkÞ:

It may also happen that there are several transition matrices, one associated to each target, i.e., C t ðkÞ instead of CðkÞ.

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It is also supposed that the perturbations nt1 and nt2 are independent and with known covariance. That is, h i h i E nt1 ðkÞðnt1 ðjÞÞT ¼ Qik dkj ; E nt2 ðkÞðnt2 ðjÞÞT ¼ Rik dkj ;  ð24Þ 1 if k ¼ j; dkj ¼ 0 if k 6¼ j: As mentioned in Section 1, from the algorithmic point of view, JPDAF differs from PDAF only in the calculation of the joint probabilities bti ðkÞ – the probability that the ith hit (measurement) comes from a target t in track (the superscript t stands for a target). So, bt0 ðkÞ corresponds to the probability that none of the measurements is originated from the target t. Consequently, Eqs. (12)–(16) allowing the determination of the state and its covariance estimates are applied for each target t. It suffices to add a superscript t referring to the target t to the above equations. Similarly, Eqs. (4)–(10) allowing the determination of filter gain and innovation covariance used in (12)–(16) still are held. A crucial aspect in JPDAF methodology is the determination of the weights bti ðkÞ. To this end, validation gates are first used to determine the feasible joint event matrix. The latter is a non-conflicting association of current targets with hits. It assumes that: (i) no two tracks are associated with the same hit; (ii) there is at most one hit per target; (iii) hits which are not associated with any target are supposed clutter or false alarms. Additional requirements pertaining to JPDA construction are summarized in: (iv) there exists some detection probability PD less than one that target will be detected; (v) the measurement-totarget association probabilities are computed jointly across the targets; (vi) the association probabilities are computed only for the latest set of measurements; (vii) the states of the targets conditioned on the past observations are assumed independent; (viii) the past is summarized by an approximate Gaussian sufficient statistic – specified by state estimate and covariance for each target. The determination of whether a given measurement Y ðkÞ lies within the validation gate is described in PDAF section assuming some threshold c, which is based on the gate probability PG . The latter is taken very close to 1 in order to ensure that the validation gate for each target coincides with the entire surveillance region. This enables us to have the same probability density function for each false measurement, i.e., uniformly distributed in the entire surveillance region [3]. Suppose that hðkÞ corresponds to the feasible joint event, Y k is the history of hits up until time k, i.e., Y k ¼ fY ð1Þ; Y ð2Þ; . . . ; Y ðkÞg. The probability of an event given its hits’ history is P ðhðkÞ j Y k Þ ¼ P ðhðkÞ j Y k ; Y k1 Þ ¼

1  P ðY ðkÞ j hðkÞ; Y k1 Þ  P ðhðkÞÞ: r

ð25Þ

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The parameter r is the normalization constant factor, determined by summing over all joint events (the resulting probability is equal to one). Given the independence hypothesis, the conditioned probability in the right-hand side may be rewritten as P ðY ðkÞ j hðkÞ; Y k1 Þ ¼

n Y

P ðYi ðkÞ j hi ðkÞ; Y k1 Þ;

ð26Þ

i¼1

where hi is the assignment of the ith hit to either a current track or clutter. The density probability (PDF) of each hit is given by  N½mi ðkÞ if hit i is assigned to a track; k1 P ðYi ðkÞ j hi ðkÞ; Y Þ ¼ V 1 if hit i is assigned to a clutter or P ðYi ðkÞ j hi ðkÞ; Y k1 Þ ¼ V / 

n Y

s

½N½mi ðkÞ i ;

ð27Þ

i¼1

where V corresponds to the volume of the extension gate, given by (m is the dimension of the measurement) V t ðkÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pm=2 c  S t ðkÞ: Cð1 þ m=2Þ

ð28Þ

N ½mi ðkÞ is the normal distribution with zero mean and covariance equal to a covariance matrix of mi ðkÞ for the target to which hit i is assigned, / is the number of clutter points (false measurements), si is a binary hit indicator, which takes a value one if the ith hit is assigned to a track and zero otherwise. Let dt be a target indicator indicating whether there is a hit associated with a target t (dt ¼ 1), or not (dt ¼ 0). Finally, the probability of the joint event (see [3,12] for a full proof) is given by P ðhðkÞ j Y k Þ ¼

mk Nt Y Y

j dt 1dt 1  /!V /  ½ N ½mi ðkÞ si  PD 1  PDj ; c i¼1 j¼1

ð29Þ

where Nt is the number of targets, PDt is the probability of detection of target t. Thus, if xðhðkÞÞ indicates whether track t is associated with hit j in hðkÞ, then X btj ðkÞ ¼ P ðhðkÞ j Y k Þ  xðhðkÞÞ: ð30Þ hðkÞ

This forms a basis of JPDAF as pointed out by Bar-Shalom and Fortmann [3]. Now let us see how a new methodology based on fuzzy c-means algorithm can be developed.

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3. Fuzzy c-means methodology 3.1. Basics of fuzzy c-means algorithm Basically, the problem in fuzzy c-means (FCM) is the following [4]. Given a finite set of data X ¼ fx1 ; x2 ; . . . ; xn g in Rs space (s-dimensional) and let c be a real integer standing for the number of classes. How to split up the elements of X into c classes (2 6 c < nÞ? This permits us to determine both the centre Vj (1 6 j 6 c) of each class and the n c matrix U . The element uij of U whose value ranges from 0 to 1 represents the degree to which the datum xi agrees with the class j supported by the prototype Vj . It is also supposed that the full degree of agreement ‘‘1’’ is shared between the c classes. That is, for all i from 1 to n, c X

uij ¼ 1:

ð31Þ

j¼1

More formally, in a FCM algorithm, the functional J ðU ; V Þ given by expression (32) is minimized: J ðU ; V Þ ¼

n X c X i¼1

a

ðuij Þ dij2

ða P 1Þ

ð32Þ

j¼1

Subject to c X

uij ¼ 1 and

uij P 0:

ð33Þ

j¼1

dij stands for the distance from the datum xi to the prototype Vj given up to some symmetric positive-definite matrix F :  2 T dij2 ¼ xi  Vj  ¼ ðxi  Vj Þ F ðxi  Vj Þ: ð34Þ The parameter a refers here to a parameter that models the degree of fuzziness. It mainly permits to modify the shape of the membership function ascribed to each cluster. The solution of (34) and (35) is provided by the following: Pn a 1 i¼1 ðuij Þ xi P ; V ¼ 1 6 i 6 c; 1 6 k 6 n: ð35Þ uki ¼ j h i n a 2=ða1Þ Pc dki i¼1 ðuij Þ j¼1

dkj

The iteration process starts by initial U ð0Þ or, equivalently, V ð0Þ and processing respectively expressions of Vj , dij and uij until the convergence criterion conceptualized by the fact that the difference between U ðkÞ and U ðkþ1Þ (value of U at steps k and k þ 1) is sufficiently small (up to some threshold).

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In order to accommodate different application contexts, many variants of FCM algorithm have been put forward and investigated by many authors. Bezdek et al. [5] proposed c-variety algorithm in order to detect straight line or hyperplane classes. Gustafson and Kessel [13] investigated the case of ellipsoidal classes by letting the matrix F be determined for each class and whose determinant is kept unchanged. Dave and Fu [10] and Dave [8] proposed spherical c-means clustering in order to deal with spherical classes. Krishnapuram and Keller [14] investigated the case where the constraint (31) is no longer satisfied and proposed possibilistic clustering where the objective function is modified. Dave [9] investigated robust clustering by introducing an imaginary ðc þ 1Þ class referring to the noise prototype. Typically, a noise prototype is defined as a universal entity such that it is always at the same distance from every point in the data set. That is, if Vp stands for the noise prototype, then dip2 ¼ d

for all i ¼ 1; n

ðp ¼ c þ 1Þ:

ð36Þ

This permits ensuring rather the constraint cþ1 X

uij ¼ 1:

ð37Þ

j¼1

So, the new functional J 0 ðU ; V Þ becomes J 0 ðU ; V Þ ¼

n X cþ1 X i¼1

m

ðuij Þ dij2 :

ð38Þ

j¼1

The resolution of the noisy FCM algorithm is very close to the one described in the FCM algorithm (expression (35) still holds) except that the calculus will be extended up to c þ 1 classes following (36) and assuming that the distance d is specified. However, the noise distance d is usually a critical parameter in the above algorithm, and would be different for different problems. Ideally as pointed out in [9] it is rather based on statistics of data set. The additional c þ 1 class is referred to as the cluster noise. Notice that in classical FCM algorithm, the total number of classes should be greater than or equal to 2. Unlike FCM algorithm, data with only one class are allowed in the case of Dave’s approach of FCM with noise prototype. 3.2. Application of fuzzy c-means to data tracking Clearly, the meaning of the grade uij as a degree of agreement between the datum i and the class j, as well as the fulfilment of a constraint like (31) or (37), makes it a potential candidate and an alternative to btj in JPDAF or PDAF (where Nt ¼ 1).

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However, in order to accommodate PDAF constraints some further restrictions are required. Particularly, the elaborated FCM algorithm should take account for (i) the dynamic structure of the system; (ii) the centre of the class is not a central point as the value of the grade uij is; (iii) the clutter (noise prototype) is rather attached to the measurement space. Now let us explain the above additional constraints and how they might be translated into their consistent and rational meanings in the framework of Dave’s approach of noise prototype. The requirement (i) of the dynamic behaviour of the system, which is also valid in the case of JPDAF (so the reasoning is elaborated for the general JPDAF case, Nt P 1), induces some computational requirements for the elaborated FCM approach. This means that some compromized situations between the convergence aspect of FCM approach, which basically requires several iterations, and the real time constraint should be reached. For this approach, it is evident that the goal of full convergence should be avoided. However, it is interesting that the process, when truncated because of real time constraint, is as close as possible to the convergence state. To this end, a rational idea is to take account for the dynamic behaviour of the system. In other words, we do know the model of evolution of the targets, so why not include this important knowledge in the problem formulation. Assume Zi (i ¼ 1 to n) stands for the latest set of measurements, where n is the total number of measurements generated at a current time k. Of course, n is not a priori fixed but is a random variable as suggested in initial construction of PDAF and JPDAF. Here the centre Vj of the jth class can also be assimilated to a sort of a vector position of the target j (j ¼ 1 to Nt where Nt is the known number of targets). As it sounds from tracking vocabulary, the state vector of each target should include its spatial co-ordinates. Consequently, the information about the estimated position of each target is available at each time. Let Yj be the part of the state vector X j , referring to the jth target, which contains the spatial position of the target. Rationally, the prototype Vj should be close to vector Yj . Let ðYj Þold be the previous knowledge about the prototype of the jth class. Typically, if the current state is k, then ðYj Þold refers to the components of the vector X j ðk  1 j k  1Þ containing the spatial positioning of the target j. So, if Ajp ðkÞ designates the part of the matrix Aj referring to the position components in the state transition matrix pertaining to the target j, then the predicted prototype ðYj Þk j k1 is provided by ðYj Þk j k1 ¼ Ajp ðkÞðYj Þold :

ð39Þ

It is therefore natural to assume that in the worst case, at each time increment, the evolution of the class prototypes should not be beyond the assessment supplied by the prediction ðYj Þk j k1 .

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Consequently, a natural way to incorporate this knowledge into our proposal is to assume that globally the difference between the unknown prototype Vj and the predicted prototype ðYj Þk j k1 should not be large enough. This globality is understood in the sense of the sum over all classes, as it is usually the case in FCM algorithms and their ramifications. Consequently, one may incorporate this knowledge as an objective function J0 to be minimized. That is, J0 ¼

Nt X T ðVj  Ajp ðYj Þold Þ ðVj  Ajp ðYj Þold Þ:

ð40Þ

j¼1

The requirement (ii) means that the centre of classes should not be used as counterparts of the state vector X . In other words, the general scheme of PDAF and JPDAF is kept unchanged and only the process of the calculus of the joint probabilities is considered. Strictly speaking, since the centre of the class refers to a target positioning, then this estimate should not be far enough from the positioning information supplied by the state vector X pertaining to that target. Interestingly is the requirement (iii) referring to the noise prototype. Clearly some conceptual differences can be pointed out in view of expressions (21) and (33) when we focus on the sum in both cases. Indeed, regarding Dave’s methodology about noise prototype, (21) looks like a reversal behaviour of Dave’s assumptions. That is, in the latter (Dave’s noise prototype) it is assumed that the additional class referring to a noise prototype is rather related to targets. More formally, for a given measurement i, N t þ1 X

uij ¼ 1;

ð41Þ

j¼1

where the ðNt þ 1Þth datum stands for noise prototype. However, the appropriate translation of requirement (ii) leads to the constraint m k þ1 X

uij ¼ 1:

ð42Þ

i¼1

In (42) the noise prototype is rather supported by the ðmk þ 1Þth datum. Clearly, this induces two main differences compared to Dave’s approach. First, the noise class is rather associated to some unobserved element for a given class. This means that the n elements of the class may be not enough representative and pertinent to characterize that class. Second, the universality concept of the noise prototype is understood differently. Indeed, this subsumes that the imaginary ðn þ 1Þth datum plays the same role within the different Nt classes. Therefore, this viewpoint can be seen as a complementary aspect to Dave’s noise

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prototype in the sense that the latter permits us to complete the lack of agreement between the data and the classes where all data are taken for granted and the doubt refers to the representativity and the pertinence of the different classes. So, the lack of agreement or conflict is transferred to an additional class. However, in the second case, the classes themselves are taken for granted while the doubt occurs in the representativity of the elements of each class. Intuitively, this is equivalent to assume Nt classes for noise, i.e., one additional sub-class for each class. While the universality property of the noise restricts to only one noise class, which is supposed to be in agreement with all classes (targets). Similarly to Dave’s approach it is assumed that the ðn þ 1Þth datum is equally situated from each class. That is, 2 dðnþ1Þj ¼d

for j ¼ 1 to Nt :

ð43Þ

A critical point in Dave’s approach is the calculus of the noise distance d. For this end, the following rational requirements are assumed: (i) d is dependent on the density of false alarms used in previous PDAF and/or JPDAF formulations. Pn PNt 2 1 (ii) d is dependent on the total average of distances nN j¼1 dij . i¼1 t The intuitive justification of requirement (i) arises from the fact that typically, the divergence or the loss of a target is caused by false alarms, so it is natural to take this into account. While the requirement (ii) is put forward by Dave himself in his approach. In the latter it has been proposed that d be proportional to the total average of distances. To determine d, let us notice first that smaller the distance larger the corresponding membership grade, and greater the number of elements belonging to a given class higher the probability of target loss. So, it is natural to require that when the density of false alarms is very small, then d should be in the same order of magnitude than the greatest available distance, i.e., maxi¼1;n;j¼1;Nt dij2 . This induces a lower value of membership grade. Inversely, high value of density of false alarms enables d to be in the same order of magnitude as the smallest distance, i.e., mini¼1;n;j¼1;Nt dij2 . Consequently assuming k to be the density of false alarm, d can be approximated in the following way: d ¼ /  min dij2 þ ð1  /Þ max dij2 ; i¼1;n j¼1;Nt

ð44Þ

i¼1;n j¼1;Nt

where /¼k

v

Pn

maxj¼1;Nt dij2 PNt 2 : i¼1 j¼1 dij

i¼1

Pn

ð45Þ

v is a non-zero constant permitting to normalize / according to the range of values allowed to k. / takes its values in the interval ½0; 1 .

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It is easy to see from (44), (45) that in the boundary cases k ¼ 0 leads to d ¼ maxi¼1;n;j¼1;Nt dij2 . While greater values of k make d tend towards the smallest distance. In practice, the values of the smallest and the greatest distances are slightly augmented up to some threshold for security purposes. Consequently, the application of (43), (44) is performed only at the beginning step, i.e., according to the available a priori knowledge. Indeed, the fact of maintaining the value of d invariant during the iterative process is in full agreement with the above optimization process, since otherwise, its derivative cannot be treated as if it is a constant parameter. Note that in Dave’s approach, the value of d may change between two different iterations. This, of course, violates the considered optimization problem where it is also deemed to be a constant parameter. Another interesting remark arises from the rich structure of available a priori knowledge. Indeed, using the dynamic of the system it permits predicting the new locations of the class prototypes, therefore, the distance matrix and the different membership grades. This justifies why the restriction to the beginning step in estimating d will not cause drastic errors in the whole algorithm. While, typically, this kind of knowledge is no longer available in classical classification problems, and thereby the restriction to the starting point for estimating d may not be meaningful at all. Now keeping the objective function in Dave’s approach, we have J1 ¼

Nt nþ1 X X i¼1

uaij dij2 :

ð46Þ

j¼1

The distance from the datum Zi to the prototype centre Vj , for i ¼ 1 to n and j ¼ 1 to Nt , can be obtained using dij2 ¼ ðZi  Vj ÞT Sj1 ðZi  Vj Þ:

ð47Þ

If i ¼ n þ 1, then (44) is used. The reason for using the inverse of the innovation covariance Sj , which is symmetric positive definite matrix, is to establish a direct link to a normalized distance structure from each measurement to a predicted hit. This also makes a bridge to the feasible joint event matrix, as it will be seen later on. Consequently, the main objective function J can be constructed as a convex combination of J0 and J1 : J ðU ; V Þ ¼K

Nt nþ1 X X i¼1

T

uaij ðZi  Vj Þ Sj1 ðZi  Vj Þ

j¼1

þ ð1  KÞ

c X j¼1

T

ðVj  Apj ðYj Þold Þ ðVj  Ajp ðYj Þold Þ

ð48Þ

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Subject to nþ1 X

uij ¼ 1;

ð49Þ

0 6 uij 6 1:

i¼1

The constant K stands for a parameter, which makes the objective function as a convex combination of J0 and J1 , and then induces a compromise situation between distance minimization requirement and the known dynamic of the system. Intuitively, K should be greater than 0.5, which means that the current measurements are mostly preferred over the output supplied by the prediction according to the models of the targets. The matrix S j is by construction symmetric, positive and definite matrix, which makes it suitable for distance structure. Roughly, the effect of the noise distance d can be compensated by a slightly different choice of the parameter K in the objective function in (48). So, in order to solve the above optimization problem (48), (49), the augmented Lagrangian system is the following: J ðU ; V ; kÞ ¼ K

Nt nþ1 X X j¼1

i¼1

þ ð1  KÞ þ

Nt X

T

uaij ðZi  Vj Þ Sj1 ðZi  Vj Þ

" kj

c X j¼1

nþ1 X

j¼1

ðVj  Ajp ðYj Þold ÞT ðVj  Ajp ðYj Þold Þ #

uij  1 :

ð50Þ

i¼1

This is solved by setting the derivative of J ðU ; V ; kÞ with respect to each parameter to zero. That is, oJ 2 ¼ aKua1 ij dij þ kj ¼ 0; ouij

ð51Þ

nþ1 X oJ ¼ uij  1 ¼ 0: okj i¼1

ð52Þ

From (51), " uij ¼

kj aKdij2

#1=ða1Þ :

ð53Þ

Putting (53) in (52) and eliminating kj lead to uij ¼

Pnþ1 k¼1



1 dij2 2 dkj

1=ða1Þ :

ð54Þ

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The determination of the centres Vj is accomplished by setting the derivative of J with respect to Vj to zero. This leads to n X oJ ¼ 2K uaij Sj1 ðZi  Vj Þ  2ð1  KÞðVj  Ajp ðYj Þold Þ ¼ 0 oVj i¼1 " # n n X X j a 1 a () ð1  KÞAp ðYj Þold  K uij Sj Zi þ K uij Sj  ð1  KÞId Vj ¼ 0

" () Vj ¼ K "

i¼1 n X

#1

i¼1

uaij Sj1  ð1  KÞId

i¼1

K

n X

# uaij Sj1 Zi

 ð1  KÞAjp ðYj Þold Þ

;

ð55Þ

i¼1

where Id stands for the identity Pnþ1 amatrix with the same size as Sj . Clearly, the matrix ½K i¼1 uij Sj  ð1  KÞId is invertible since Sj is. 3.2.1. Initialization of U As it is the case for all algorithms of FCM family, the above algorithm also requires an initialization step. Particularly, the importance of such step arises from the restricted number of allowed iterations due to dynamic behaviour constraint. Consequently, a good initial guess is highly preferable. For this purpose, we use the knowledge about the system dynamic. More formally, we will consider that a good candidate for the centres Vj (j ¼ 1 to Nt ) is provided by the predicted measurement. That is, for each class, Vj ¼ CX j ðk j k  1Þ ¼ C  Aj  X j ðk  1 j k  1Þ ðj ¼ 1; Nt Þ:

ð56Þ

Using (56), the distance matrix dij2 is constructed for i ¼ 1 to n and j ¼ 1 to Nt . While, for i ¼ n þ 1, the calculus of the noise prototype is accomplished via (44), (45). Finally, the initial matrix U is obtained via (54). Clearly, the use of predicted measurements as estimates for the centre prototypes is not the unique solution. Another rational way is to use the predicted target position. That is, the components of the vector X j ðk j k  1Þ referring to the positioning. However, practical considerations have shown that the first proposal seems to be more appropriate in terms of the performance of the tracking. Basically, this superiority can be justified by facts from state estimation theory claiming that the innovation sequences are less corrupted [1]. 3.2.2. Traitment of conflictual situation Conflictual situations occur particularly in the case of multiple targets. A characterization of a conflictual situation is, for the same given measurement, high probabilities are attached to two different targets. This contradicts with feasible joint event where it is required that at most one measurement is

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associated to a given target, and there are no two targets sharing the same measurement. Consequently, if after the above calculus, we obtain high values for uij and uik (j 6¼ k), then at least one of the previous values is wrong and therefore some re-distribution of the total amount of mass is necessary. A rational way to solve this inconsistency is to keep the idea of feasible joint events involved in the construction of the joint probabilities in JPDAF initial scheme. That is, from the distance matrix, which corresponds to a normalized distance from each measurement to the predicted measurement ascribed to each target, all possibilities that conflict with JPDAF’s hypothesis are excluded. To illustrate this approach, let us assume the following examples. 3.2.2.1. Example 1. Let us assume that the distance matrix corresponds to the following binary matrix: 2 3 1 0 F ¼ 4 1 1 5: 0 1 The matrix F means that there are three hits, say 1, 2 and 3 (from top to down rows) and two targets, say k1 and k2 (from left to right column). Then, one may argue that target k1 is viewed by both hits 1 and 2 while the target k2 is viewed by both hits 2 and 3. The assertion of link between a given hit and a given target follows from the fact that the normalized distance from that hit to the predicted measurement associated to the target is less than the threshold c mentioned in (11). More formally, for hit 1 ðz1 Þ and track k1 ðX 1 Þ, we have T

ðz1  C  X 1 ðk j k  1ÞÞ S11 ðz1  C  X 1 ðk j k  1ÞÞ 6 c: Clearly, the situation described by a matrix F is conflicting in the sense that there exist hits failing into more than one target and there exist targets, which are associated to more than one hit. Feasible joint matrices that agree with F are 2 3 2 3 2 3 1 0 1 0 0 0 F 1 ¼ 4 0 1 5; F 2 ¼ 4 0 0 5; F 3 ¼ 4 1 0 5: 0 0 0 1 0 1 Note that in each of the above matrices, each target is associated with only one hit and each hit fails in the probabilistic region of only one target. Now the new repartition of the mass according to the preceding, which leads to a new matrix U 0 of membership grades, is given by the following: First given a matrix U constructed from (54), each feasible joint event F i can be characterized by a mass mF i given by Y mF i ¼ uij : ð57Þ F s ði;jÞ¼1

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Strictly speaking, mF i corresponds to the total mass assigned to the joint feasible matrix F i considering all the possible measurement/target associations supported by F i . Note that mF i is zero valued as soon as one of the targets is not detected, situation in which all the components of one of the columns are zero valued. Now the new distribution of the mass is determined considering a weighted mass with respect to all situations supported by feasible joint matrices. That is, P F s ði;jÞ¼1 mF s 0 uij ¼ P : ð58Þ F s mF s In other words, u0ij corresponds to a weight of the mass of those feasible joint event matrices allowing the association of the ith hit to the jth target with respect to the total masses of all feasible matrices. 3.2.2.2. Example 2. Considering the situation described in Example 1 and assuming a membership grade matrix U determined via (54), then applying (57) and (58) leads to a new matrix U 0 such that mF 1 þ mF 2 u11 u22 þ u11 u32 ¼ ; u011 ¼ mF 1 þ mF 2 þ mF 3 u11 u22 þ u11 u32 þ u21 u32 0 u012 ¼ ¼ 0; mF 1 þ mF 2 þ mF 3 mF 3 u21 u32 u021 ¼ ¼ ; mF 1 þ mF 2 þ mF 3 u11 u22 þ u11 u32 þ u21 u32 .. . 3.2.3. Discussion Clearly, the fact of using a weighted approach to elevate the conflictual situation in the case of JPDAF is not completely new when we consider the literature dealing with this subject or related general cases of pattern recognition [17]. For instance, Fitzgerald’s ad hoc approach [11] fails in this viewpoint. His proposal consists in formulating the joint probabilities btj (t ¼ 1 to Nt and j ¼ 1 to n) as follows: btj ¼

Gtj ; St þ Sj  Gtj þ B

ð59Þ

where

 Gtj ¼ N mj ðkÞ ;

St ¼

m X j¼1

Gtj ;

Sj ¼

Nt X

Gtj

ð60Þ

t¼1

and B is a constant, which depends on clutter density (usually with B ¼ 0, the algorithm works well).

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213

Also Roecker and Philips [15] have proposed a suboptimal JPDAF using a weighted methodology with respect to feasible joint events. The main idea in these approaches is heavily weighting hits in only one target extension gate and lightly weighting hits in an area where several target extension gates overlap and are contending. Now considering the above construction, one may notice the following: (i) Regarding the formulation of the matrix U 0 , the mass assigned to the clutter (noise prototype) is taken into account only at the construction of the matrix U , while U 0 ðn þ 1; jÞ ¼ 0 for all j ¼ 1 to Nt where n stands for the total number of measurements. This follows from the fact that it always holds n X

u0ij ¼ 1 for i ¼ 1 to Nt :

ð61Þ

i¼1

Consequently, the standard calculus of JPDAF is reduced in the sense that the factor bt0 ¼ 0 for all targets t. This reduction is highlighted in the matrix expression where (14) comes down to P ðk j kÞ ¼ PG ðk j k  1Þ þ P~ðkÞ:

ð62Þ

Intuitively this can be explained by the fact that when inconsistency occurs, it is better to keep the covariance matrix as small as possible in order to avoid further complications arisen from the possibly large number of false measurements failing in more than one target extension gate, and thereby, the algorithm may assign too high weighting to incorrect hit(s) and causes a divergence of the track. (ii) If F s ði; jÞ ¼ 0 for all s, then u0ij ¼ 0. This follows from the fact there is no mF s 6¼ 0 such that F s ði; jÞ ¼ 0. This has obviously a strong intuitive ground in the sense that if a measurement/target association is no longer valid by any of the feasible joint event matrices, then the degree of membership attached to such association should vanish or at least tends towards zero. (iii) In the case of a complete confusion situation, it leads to uniform values for u0ij , those for which the measurement/target associations are allowed by feasible joint matrices. Indeed, in the case of complete confusion, the components of the distance matrix have almost the same values. Consequently, the use of the modified version of the FCM also leads to same values for different components of the matrix U . Applying this to (57) and (58) leads to same values for the masses mF i ascribed to different feasible joint matrices, and, therefore, induces similar values for grades u0ij , which should be equal to 1=n. This remark is in complete agreement with the principle of total ignorance in probability theory where uniform probability is used.

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Note that in practice, when reasoning in terms of membership values, the feasible joint event constraint may not be necessary. Indeed, when restricting to the latest set of measurements, the probability of finding two high membership grades pertaining to the same measurement is quite small except in the cases of target loss or high divergence. This makes the use of the conflict treatment procedure only for safety purposes. Finally the structure of the fuzzy based PDAF/JPDAF for the calculus of probabilities bti can be summarized in the following steps: Step 1. Compute the initialization using the dynamic of the system and the preceding descriptions. Step 2. Determine the centres using (55). Step 3. Compute the distance matrix using (47) and (44). Step 4. Compute the new matrix U using (54). Step 5. Check consistency by looking for possible conflictual situation (used only in the case of Nt > 1, i.e., JPDAF) and use (57) and (58) considering all feasible joint matrices. Step 6. Repeat steps 1–5 for few iterations. Now we are able to address the whole algorithm mixing PDAF/JPDAF scheme and the new elaborated modified FCM algorithm called fuzzy JPDAF algorithm. Step 1. Starting from previous estimation X t ðk  1 j k  1Þ and P t ðk  1 j k  1Þ for t ¼ 1 to Nt , determine the prediction using (4), (5). Step 2. Determine using the above procedure the matrix U , which represents the weights bti , where bti ¼ uit (for i ¼ 1 to n) bt0 ¼ uðnþ1Þt . Step 3. Determine the updated state and covariance estimates using (12)–(14). Step 4. Repeat Steps 1–3 for different time increments.

4. Simulation examples The aim in this section is to test the performance of the elaborated fuzzy PDAF/JPDAF versus the standard PDAF/JPDAF algorithms. For this purpose, we will consider here a target tracking example and multiple targets tracking examples with parallel and crossing targets. For the sake of comparison, we will reproduce here the same example as that considered by Chang and Bar-Shalom [7] in an attempt to tackle crossing targets. The targets are modelled as constant velocity objects in a plane with process noise (Gaussian zero mean) that accounts for slight changes in the velocities. More specifically, T let a state vector X ¼ ½x; x_ ; y; y_ support both the x–y co-ordinates and velocities of a given target. Then all targets have the same state transition and measurement transition matrices such that the state and the measurement models are given by

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2

1 60 X ðk þ 1Þ ¼ 6 40 0

1 1 0 0

0 0 1 0

3 2 0 1=2 6 1 07 7X ðkÞ þ 6 4 0 15 1 0

3 0 0 7 7n ðkÞ: 1=2 5 1 1

215

ð63Þ

Mathematically, expression (63) induces no major difference compared to expression (1) except that state noise covariance Q will be now rather GT QG where G is the matrix

T 1=2 1 0 0 G¼ : 0 0 1=2 1

1 0 Y ðkÞ ¼ 0 0

0 1

0 X ðkÞ þ n2 ðkÞ; 0

ð64Þ

where n1 ðkÞ and n2 ðkÞ are zero mean Gaussian noises with known variance– covariance matrix respectively Q and R. The initial estimate is assumed random with mean X ð0 j 0Þ and covariance P ð0 j 0Þ. Initial estimates of the state were obtained by two points differencing of the observations with a corresponding covariance matrix [3]. The noise covariance matrices are such that Rii ¼ 0:0225 km2 and Qii ¼ 4 104 , where R is 2 2 matrix and Q is 4 4 matrix (Rij ¼ Qij ¼ 0 for i 6¼ j). The set of measurements is created in the following way. First the true measurements are created using the true target position added with zero mean Gaussian perturbation of covariance R. Clutter measurements, whose number is Poisson distributed with parameter k ¼ 1 (number of false measurements per unit area (km2 )), are generated uniformly within the ellipsoid region centred in the predicted measurement for each target and with volume V ðkÞ as given by expression (17). First we will consider the case of single target tracking, which makes direct link to PDAF. Fig. 1 represents the single target tracking using standard Kalman filter. In Fig. 1(b) is represented the estimated target position ‘o’. Also are represented in the same plot, the true target position ‘*’ and the measurements ‘+’. In Fig. 1(b) is represented the normalized estimation error of the tracking. That is, we compute at each sample k, the quantity qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eðkÞ ¼ ðX p ðk j kÞ  XðkÞÞT ½P p ðk j kÞ 1 ðX p ðk j kÞ  X ðkÞÞ; ð65Þ where X p ðk j kÞ, X p ðkÞ and P p ðk j kÞ stand for the parts of respectively X ðk j kÞ, X ðkÞ and P ðk j kÞ containing only the x–y components of the target position. For instance, X p ðkÞ refers to vector containing the first and third components of the vector X ðkÞ.

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Fig. 1. Single target tracking using standard Kalman filter. (a) Normalized estimation error, (b) estimated target positioning ‘o’.

Remark that for the use of the Fuzzy Hybrid PDAF (or JPDAF), the matrix Ap mentioned in (40) corresponds here to



Xk1 j k1 ð1Þ 1 0 and ðYj Þold ¼ Ap ¼ : Xk1jk1 ð3Þ 0 1 The constant K in the objective function is taken equal to 0.9. This means that classification with respect to distance minimization is favoured over dynamic

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Fig. 2. Single target tracking using Hybrid Fuzzy PDAF.

Fig. 3. Normalized estimation error using PDAF and Hybrid Fuzzy PDAF.

217

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constraint. Indeed, the latter is based only on predicting part of the target that should not be too confident since the updating part is missing. Notice also that for a better illustration, we often use, throughout the forthcoming figures, different scale in for y-axis. Note that in this simulation and all the forthcoming results of this section, the figures correspond to the one single simulation run but where the generator of random numbers is equally initialized. In Fig. 2, we represent the same simulation while using instead of KF the Hybrid Fuzzy PDAF. Note the results pertaining to the use of PDAF are not represented because of their similarity to that of Hybrid Fuzzy PADF.

Fig. 4. Multiple target tracking using standard JPDAF algorithm.

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219

However, in order to provide a meaningful comparison, we represent in Fig. 3, the normalized estimation error using both algorithms. It is easily checked that both PDAF and Hybrid Fuzzy PDAF provide better performance in terms of normalized error estimation than that supplied by standard KF. Besides, slight superiority of Hybrid Fuzzy PDAF can be noticed from Fig. 3. Now we will consider the case of multiple target tracking either crossing targets or parallel moving targets where comparison to results obtained via JPDAF is supplied. Note that the use of standard KF in this case leads to

Fig. 5. Multiple target tracking using the Hybrid Fuzzy JPDAF method.

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completely wrong results since there is very strong confusion about the measurement–target associations. Figs. 4 and 5 represent respectively the results obtained using JPDAF and the Hybrid Fuzzy JPDAF. Also 50 Monte Carlos runs have been performed in each case. The superiority of the elaborated Hybrid Fuzzy JPDAF is clearly highlighted. Strictly speaking, even Bar-Shalom has mentioned the lack of JPDAF for dealing with crossing targets. This justifies the investigation of JPDAF with possibly unresolved measurements proposed by Chang and Bar-Shalom [7]. Note that even in the case of targets with parallel motion as pictured in the

Fig. 6. Target tracking using Fitzgerald’s approach.

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221

bottom part of Fig. 4, the algorithm leads to a loss of 2 targets after few time increments. In Fig. 6 are represented the results obtained using the above Fitzgerald’s ad hoc approach for the same example of crossing targets and parallel moving targets. It is clear that the performances of such tracking in terms of the closeness of the estimated position to the true target positions are acceptable and better than standard JPDAF. While these results seem also to be close to those obtained via Hybrid Fuzzy JPDAF. Also, for a better illustration of the performance of Hybrid Fuzzy JPDAF and Fitzgerald’s approach, we compute the normalized estimation error

Fig. 7. Normalized estimation error squared ascribed to the example of two crossing targets.

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squared in the case of crossing targets and targets with parallel velocities, see Figs. 7 and 8. Clearly, from Figs. 7 and 8 a slight superiority of the fuzzy based approach over Fitzgerald’s ad hoc approach can easily be checked regarding the magnitude of the normalized estimation error considering either the crossing targets or the parallel targets.

Fig. 8. Normalized estimation error squared ascribed to the example of tracking three parallel targets.

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223

Table 1 provides some values of the joint probability weights corresponding to the crossing target example and determined according to the three algorithms, JPDAF, Fitzgerald’s approach and Hybrid Fuzzy JPDAF methods. These results correspond to only one simulation run and for three different time increments. The two columns in each result presented in Table 1 correspond to the weights associated to target 1 and target 2. While the last row in each set of measurements corresponds to the weights allocated to the clutter (false alarms). For example, in the case of Hybrid Fuzzy JPDAF, at time k1 , 0:7093 corresponds to the degree of compatibility between the target 1 and hit 1, and 0.0117 for the compatibility between the first measurement and target 2. While the clutter is assigned zero weight. This is due to the use of conflict handling procedure previously described. However, at time k2 , where the conflict procedure is not fired, there is non-zero weight attached to the clutter. In each case, the matrix of weight assignments is generated. In the latter, the number of measurements generated at a given time, which is Poisson randomly distributed, is equal to the total number of rows minus one. For example, there are five measurements generated at time k1 . Besides, it is easily checked that in each case, the sum over the columns of each matrix is equal to one, which ensures the satisfaction of the additivity constraint when considering additional weight Table 1 Comparison of weight factors pertaining to measurement–target association between the three approaches Timen method

JPDAF approach

Hybrid Fuzzy JPDAF method

Fitzgerald’s approach

Target 1

Target 2

Target 1

Target 2

Target 1

Target 2

Time k1

0.0421 0.0113 0.0006 0.0011 0.0280 0.0001 0.9167

0.0006 0.0000 0.0191 0.0269 0.0073 0.0295 0.9167

0.7093 0.0442 0.0704 0.0496 0.0852 0.0366 0.0047 0.0000

0.0117 0.0178 0.0133 0.6007 0.2243 0.1295 0.0025 0.0000

0.7976 0.1819 0.0204 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0006 0.4501 0.1532 0.3961 0.0000

Time k2

0.5473 0.0027 0.4500

0.0027 0.3473 0.6500

0.8301 0.0103 0.1596

0.0120 0.6050 0.3830

1.0000 0.0000 0.0000

0.0000 1.0000 0.0000

Time k3

0.2200 0.0282 0.7500 0.0018

0.0180 0.2218 0.7500 0.0102

0.8163 0.0910 0.0927 0.0000

0.0547 0.8640 0.0813 0.0000

1.0000 0.0000 0.0000 0.0000

0.0000 1.0000 0.000 0.000

The results correspond to a one-simulation run of crossing target example.

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due to the clutter. From Table 1, it is easy to notice the following differences in the general structure of the weight assignment. In Fitzgerald’s approach, the weights tend usually to the boundary limits 1 and 0 much more faster than in other algorithms. In other words, the consideration of the smoothness behaviour is less taken into account. In the standard JPDAF algorithm, the weights attached to the clutter (last row at each set of measurements in each sample) tend usually to become preponderant. This is mainly due to the splitting effect caused by the search for the feasible joint matrix where many solutions are considered, while a lot of them seem to be less meaningful and perhaps not realistic. This justifies in some ways the divergence of the filter for particular cases of crossing targets or parallel targets as it has been pointed out in the preceding. Furthermore, as previously mentioned JPDAF is less suited for crossing targets, this is why Chang and Bar-Shalom have developed a modified, but much more heavy, algorithm, called joint probabilistic data association for possibly unresolved measurements [7]. Consequently, regarding the outputs of these algorithms in terms of the weights ascribed to the joint probability associations, it seems that the Hybrid Fuzzy JPDAF provides quite meaningful and rational values in the sense that the most appropriate hit is assigned a larger value.

5. Conclusion In this paper, we have focused on a fruitely combination of probabilistic and fuzzy approaches in order to perform a tracking task in the light of PDAF and JPDAF schemes. The approach developed in this paper consists in considering the PDAF and JPDAF algorithms where the joint probabilities are substituted by membership grades provided by a modified version of FCM algorithm. Particularly, the new fuzzy classification algorithm is mainly inspired from Dave’s FCM algorithm approach where an imaginary class pertaining to the noise and equally situated from all the data is added. However, in order to accommodate PDAF and JPDAF requirements, our approach differs from Dave’s methodology from both conceptual and structural viewpoints. Indeed, in order to take account for the dynamic behaviour of the system and real time requirements, the objective function is augmented by a factor referring to the available knowledge concerning the model of the targets. Further, to take account for the clutter prototype as described by PDAF expressions, a new noise model structure is elaborated. In the latter, given n available measurements, an imaginary ðn þ 1Þth datum plays the same role within the different Nt classes. Therefore, this viewpoint can be seen as a complementary aspect to Dave’s noise prototype in the sense that the latter permits to complete the lack of agreement between the data and the classes where all data are taken for granted and the doubt refers to the representativity and the pertinence of the

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different classes. So, the lack of agreement or conflict is transferred to an additional class. However, in the second case, the classes themselves are taken for granted while the doubt occurs in the representativity of the elements of each class. Intuitively, this is equivalent to assume Nt classes for noise, i.e., one additional sub-class for each class. While the universality property of the noise permits restricting to only one noise class, which is supposed to be in agreement with all classes (targets). Also, a weighted procedure has been developed in the case where inconsistencies occur, in the sense that more than one hit fails in the extension gate of a given target or the same hit fails in extension gates of two different targets. At the meantime, the general structure of PDAF and JPDAF is kept unchanged and only the process of the calculus of joint probabilities is modified. This leads to Hybrid Fuzzy PDAF in the case of one single target tracking and Hybrid Fuzzy JPDAF in the case of multiple target tracking. Finally, the elaborated algorithm has been successfully tested using the same simulation setting as that pointed out by Chang and Bar-Shalom using constant velocity motion target. The results show some similarities between the elaborated Hybrid Fuzzy PDAF/PDAF and the Fitzgerald’s ad hoc approach and superiority over standard PDAF and JPDAF schemes.

Acknowledgements This work is supported by K.U. Leuven’s GOA’99 Concerted Research Action for Active Sensing for Intelligent Machines.

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