Modified Maximum Entropy Fuzzy Data Association Filter

Abdolreza Dehghani Tafti Department of Electrical Engineering, Islamic Azad University, Science and Research Branch, Tehran 1477893855, Iran e-mail: [email protected]

Nasser Sadati Department of Electrical and Computer Engineering, University of British Columbia, Vancouver, BC V6T 1Z4, Canada; Department of Electrical Engineering, Sharif University of Technology, Tehran 1458889694, Iran e-mail: [email protected] e-mail: [email protected]

Modified Maximum Entropy Fuzzy Data Association Filter The problem of fuzzy data association for target tracking in a cluttered environment is discussed in this paper. In data association filters based on fuzzy clustering, the association probabilities of tracking filters are reconstructed by utilizing the fuzzy membership degree of the measurement belonging to the target. Clearly in these filters, the fuzzy clustering method has an important role; better approach causes better precision in target tracking. Recently, by using the information theory, the maximum entropy fuzzy data association filter (MEF-DAF), as a fast and efficient algorithm, is introduced in literature. In this paper, by modification of a fuzzy clustering objective function, which is prepared for using in target tracking, a modified maximum entropy fuzzy data association filter (MMEF-DAF) is proposed. The MMEF-DAF has a better performance in case of single and multiple target tracking than MEF-DAF, and the other known algorithms such as probabilistic data association filter and the hybrid fuzzy data association filter. Using Monte Carlo simulations, the superiority of the proposed algorithm in comparison with the previous ones is demonstrated. Simply, less computational cost and suitability for real-time applications are the main advantages of the proposed algorithm. 关DOI: 10.1115/1.4000817兴 Keywords: data/measurement association, target tracking, fuzzy clustering, information theory

1

Introduction

Typically, the target tracking in dense environments, where the probability of having “clutter” or false alarm is not neglected, is a difficult task. Indeed, in addition to the noisy measurements 共hits兲 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 共e.g., a clutter兲 the given measurement is originated. 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 共i.e., position, velocity and so on兲. Consequently, this leads to generating a set of measurement/target associations, and thereby evaluation of such hypotheses, usually in terms of joint probability values. The problems arise in applications such as air traffic control, ocean/ battlefield surveillance, and positioning of enemy targets in military context. A major problem in these applications is also the hit/data to track association. There are two kinds of conventional data association methods based on Bayesian and non-Bayesian approaches, respectively 关1兴. The former includes the probabilistic data association filter 共PDAF兲, in case of single target tracking by Bar-Shalom and Fortmann 关2兴 and the joint probabilistic data association filters in case of multitarget tracking problems 关3–5兴. The latter also includes the nearest neighbor 共NN兲关6兴 and some all-neighbor approaches 共the track splitting and the multiple hypotheses tracking 共MHT兲兲关7–9兴. In recent years, the fuzzy logic theory was used to describe the uncertainty of knowledge or physical phenomena in nonlinear complicated systems 关10–12兴. Because the fuzzy logic provides a flexible framework to couple human judgment with standard mathematical tools, it can provide solutions to engineering problems that are too complex or ill defined to yield the analytical solutions. To solve the target tracking problem, two kinds of fuzzy Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received December 30, 2008; final manuscript received December 1, 2009; published online February 9, 2010. Assoc. Editor: Sheng-Guo Wang.

logic techniques were widely used; fuzzy inference techniques 共IF-THEN rules兲关10–12兴 and fuzzy clustering techniques 关13–15兴. Singh and Bailey 关11兴 have proposed the first fuzzy logic approach for data association in multisensor multitarget tracking problems. In their approach, fuzzified position and velocity errors are used by a fuzzy inference system 共IF-THEN rules兲. Also, defuzzification and fuzzy set decisions are performed to obtain the actual association of measurements to tracks. Unfortunately, the extension of their approach to the case of large number of targets is fairly complex, since a large number of rules are required. Using other techniques, the data association in target tracking is considered as a process to classify a given set of measurements according to some class rules. The class of algorithms, so-called fuzzy ISODATA algorithms, developed mainly by the original works of Bezdek 关16兴, permits performing an unsupervised classification of a set of data given in a multidimensional space, into a given c number of classes. Based on fuzzy c-means clustering, Mourad and De Schutter 关15兴 proposed a fuzzy data association called the hybrid fuzzy data association filter 共HF-DAF兲, in case of single and multiple target tracking. Nevertheless, the cluster centers must be adjusted to ensure eventual convergence to an optimal solution through iteration, thus, the algorithm has a heavy computational load. Liu and Meng 关17兴, therefore, introduced a novel online data driven algorithm from maximum entropy fuzzy clustering to predict the trajectory of a moving target in robot tracking. The algorithm is very fast and efficient in terms of the computational cost and suitability for real-time applications. Liangqun et al. 关18兴, inspired by the work of Aziz et al. 关14兴 and Liu and Meng 关17兴, introduced an effective algorithm based on maximum entropy fuzzy clustering, which is called maximum entropy fuzzy data association filter 共MEF-DAF兲. MEF-DAF has a better performance in comparison with other algorithms 关18兴. Although in order to save its good performance in multiple target tracking, it needs to use multiparallel maximum entropy fuzzy clustering as an additional algorithm. Clearly, this new structure has a high computational cost and increase in complexity, especially when the number of targets increases. In this paper, to improve the performance of the maximum

Journal of Dynamic Systems, Measurement, and Control Copyright © 2010 by ASME

MARCH 2010, Vol. 132 / 021013-1

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entropy fuzzy data association, a modification in maximum entropy fuzzy clustering is introduced, and on this base, a new data association algorithm is reconstructed. The new fuzzy data association algorithm, called the modified maximum entropy fuzzy data association filter 共MMEF-DAF兲, not only has a better tracking performance in case of single and multiple target tracking, but also save the computational cost of maximum entropy clustering without requiring any additional computation in case of multiple target tracking. So the proposed algorithm can be used as a solution to real-time target tracking problems. The remainder of this paper is organized as follows. In Sec. 2, a brief overview of the fuzzy data association filters, such as HFDAF and MEF-DAF, is given. The proposed new fuzzy data association algorithm 共MMEF-DAF兲 is presented in Sec. 3. Simulation results and performance comparisons are given in Sec. 4. Finally, some concluding remarks are provided in Sec. 5.

2 Hybrid /Maximum Entropy Fuzzy Data Association Filter Let us assume the dynamic and measurement models of the target in tracking are considered by the following equations: x共k + 1兲 = F共k兲x共k兲 + G共k兲w1共k兲

共1兲

y共k兲 = H共k兲X共k兲 + w2共k兲

共2兲

where x共k兲 is an n-dimensional state vector and y共k兲 is the measurement vector. F共k兲 and H共k兲 denote the transition matrices pertaining to the state and the measurement models, respectively. G共k兲 is a known noise matrix, and the process noise vector w1共k兲 and the measurement noise vector w2共k兲 are independent, zero mean with known covariance Q共k兲 and R共k兲, respectively. In order to improve the performances of PDAF and joint probabilistic data association filter 共JPDAF兲, as the conventional algorithms in tracking of single and multiple targets, Mourad and De Schutter 关15兴 proposed HF-DAF based on fuzzy clustering techniques. Their approach considers the PDAF and JPDAF algorithms, where the joint probabilities are substituted by membership degrees provided by a modified version of the fuzzy c-mean 共FCM兲 algorithm. Using the FCM algorithm, with a given set of finite data/measurements zi ; i = 1 , 2 , . . . , mk in Rs space 共s-dimensional兲, and c, which stands for the number of classes/ targets, we determine the center v j ; j = 1 , 2 , . . . , c of each class/ target and the mk ⫻ c membership matrix U. The element uij of U, ranging from 0 to 1, represents the degree to which the datum zi agrees with the class/target j supported by the prototype v j. The FCM clustering objective function in hybrid fuzzy data association can be formulated as mk+1 c

J共U,V兲 = K

兺兺共u 兲␣d ij

2 ij

+ 共1 − K兲

i=1 j=1 c

⫻

兺共v − F 共k兲共y 兲 p j

j

j old兲

T

共v j − F pj 共k兲共y j兲old兲

共3兲

the distance from the datum zi , i = 1 , 2 , . . . , mk to prototype center v j , j = 1 , 2 , . . . , c, which can be obtained with the Mahalanobis distance, given as dij2 = 共zi − v j兲TS−1 j 共zi − v j 兲

共5兲

where S j is the innovation covariance of the state estimation. The inverse of the innovation covariance S j, which is a symmetric positive definite matrix, is used to establish a direct link to a normalized distance structure from each measurement to a prototype center. F Pj 共k兲 denotes the part of the matrix F j共k兲 referring to the position components in the state transition matrix pertaining to the target j. Also, 共y j兲old refers to the components of the vector x j共k − 1 兩 k − 1兲 containing the spatial positioning of the target j. So F Pj 共k兲共y j兲old denotes the predicted prototype of the jth target. zi , i = 1 , 2 , . . . , mk stands for the latest set of measurements, where mk is the total number of measurements at the current time k. Also, the mk + 1 datum plays as a noise prototype role 关15兴. Now, by minimizing the objective function 共3兲, the solution can be given as uij =

1

共6兲

mk+1

兺

2 1/共␣−1兲关dij2 /dkj 兴

k=1

and

冋兺冋兺 mk

共uij兲␣S−1 j + 共1 − K兲I

vj = K

i=1

册

−1

c

⫻ K

p 共uij兲␣S−1 j zi + 共1 − K兲F j 共y j 兲old

i=1

册

共7兲

where I stands for the identity matrix with the same size as S j. n Clearly, the matrix 关K兺i=1 共uij兲␣S−1 j − 共1 − K兲I兴 is invertible as a result of S j being invertible. The iteration process can start with initial U共0兲, or equivalently V共0兲. The process then continues by computing V = 关v j兴 and U = 关uij兴 until the convergence criterion is conceptualized, and a sufficiently small difference between U共k兲 and U共k+1兲, or equivalently between V共k兲 and V共k+1兲共value of V at steps k and k + 1兲 is reached. If the Kalman filter is used to estimate the state of a target with dynamic and measurement models as Eq. 共1兲 and 共2兲, the standard PDAF equations, which are used for tracking a target based on validated measurements, zi ; i = 1 , . . . , mk at time k, can be described as follows. The predictions of the state and the measurement at time k are defined as xˆ共k兩k − 1兲 = F共k兲xˆ共k − 1兩k − 1兲

共8兲

yˆ 共k兩k − 1兲 = H共k兲xˆ共k兩k − 1兲

共9兲

where the covariance of the prediction state is given by

j=1

subject to

P共k兩k − 1兲 = F共k兲P共k − 1兩k − 1兲FT共k兲 + G共k兲Q共k兲GT共k兲共10兲 mk+1

兺u

ij

=1

0 ⱕ uij ⱕ 1

共4兲

i=1

where the parameter ␣ has the same role of the fuzzification parameter in the FCM algorithm. It mainly permits to modify the shape of the membership function ascribed to each cluster. In other words, it controls how much the clusters are allowed to overlap. Normally, ␣ ranges from 1.25 to 2 关16兴. The constant parameter K also adjusts the weight for each terms of the objective function. Intuitively, K should be greater than 0.5, which means the current measurements are mostly preferred over the output supplied by the predictions made by the models 关15兴. d2ij is 021013-2 / Vol. 132, MARCH 2010

and the innovation covariance 共for the correct measurement兲 is as follows: S共k兲 = H共k兲P共k兩k − 1兲HT共k兲 + R共k兲

共11兲

The Kalman gain is W共k兲 = P共k兩k − 1兲HT共k兲S−1共k兲

共12兲

and the state update equation of the PDAF is xˆ共k兩k兲 = xˆ共k兩k − 1兲 + W共k兲v共k兲

共13兲

where the combined innovation is Transactions of the ASME


mk

v共k兲 =

兺

共14兲

H共uij兲 = −

i=1

共15兲

Now, the overall covariance associated with the updated state is

再

P共k兩k兲 = P0共k兩k兲 + W共k兲 mk

兺 ␤ 共k兲v 共k兲v 共k兲 − v共k兲v 共k兲 i

T i

i

T

i=1

P 共k兩k兲 = P共k兩k − 1兲 − W共k兲S共k兲W 共k兲 0

T

冎

2 ij ij

兺兺

WT共k兲

mk

+

共16兲共17兲

共18兲

i

i=1

冉

c

∀ uij 苸关0,1兴

c

ij

−1

j=1

冊

ij

· dij2

j=1

共21兲

2

uij =

e−␣idij c

兺e

共22兲

2 −␣idik

k=1

where ␣i and ␭i are the Lagrange multipliers, and varying ␣i adjusts the value of membership of data point zi, with respect to cluster center/target v j, compared with other cluster centers, thus, it is known as the “discriminating factor” 关18兴. The way to pick a proper value, in every cluster for it, is discussed in Ref. 关18兴, where the proposed optimal value is given as

␣opt =

ln共␧兲 2 dmin

共23兲

where d2min is determined as the distance between each data sample zi and the cluster center vc 共i.e., d2min = d2pc ⱕ d2qc; for q = 1 , . . . , mk and q ⫽ p兲. ␧ is a small positive constant. In cases of multiple target tracking, however, the MEF-DAF does not have an acceptable performance in comparison with the HF-DAF, without using an additional reconstructed algorithm. So in Ref. 关18兴, a multiparallel fuzzy clustering structure for the MEF-DAF is proposed. Consequently, complexity and computational cost of the MEF-DAF increases and its optimality is lost.

3 Modified Maximum Entropy Fuzzy Data Association Filter To overcome the shortcoming of MEF-DAF, based on the modification of the clustering method, a new data association filter is proposed in this section. This new algorithm does not require any additional reconstructed computation for an acceptable performance in case of multiple target tracking, and also has a better performance in both cases of single and multiple targets; a tracking algorithm, which saves both simplicity and low computational cost. Carefully studying the HF-DAF, it is shown that the main objective function of this algorithm 共Eq. 共3兲兲 is a convex linear combination of two parts. One part, namely, J1, is the clustering objective function in Dave’s approach 关19兴, which is represented as follows: mk+1 c

is the squared distance between the given measurement where zi and the jth target. The probabilistic constraint is that the summation of all uij 共i = 1 , . . . , mk and j = 1 , . . . , c兲 must be equal to 1, that is =1

兺兺u

By maximizing Eq. 共21兲, the membership function of data zi belonging to target j can be derived as

J1 =

d2ij

ij

c

␣i

i=1

兺␭ 兺u

i=1 j=1

兺u

共20兲

ln uij

mk

uij ln uij −

i=1 j=1

c

兺兺u d

c

mk

␤i共k兲 denotes the probability that the ith measurement comes from the target in track at time k, and ␤0共k兲 denotes the probability that none of the measurements are originated from the target, or equivalently, the probability that the current measurement is a false alarm 共or a clutter兲. The derivation of the probability can be found in Ref. 关1兴. The equation of JPDAF is also similar to the PDAF, however, the only difference is in the computation of the joint probability ␤ij共k兲共i.e., probability of the measurement i belonging to target j兲. In the HF-DAF, the general structure of PDAF and JPDAF is kept unchanged, and only the process of computing joint probabilities 共␤ij兲 is modified and simplified by using the membership degrees uij, as described in Eq. 共6兲. Also, in order to have a better performance, in case of multiple target tracking, it is proposed to keep the idea of feasible joint events involved in the construction of the joint probabilities in JPDAF, although it is almost infeasible in practical applications 共the detailed derivation can be found in Ref. 关15兴兲. Since the cluster centers 共Eq. 共7兲兲 must be adjusted to ensure the eventual convergence to an optimal solution through iterations, the computational load of the HF-DAF is very heavy, and therefore, it cannot be adapted in real-time applications 关18兴. In order to solve the aforementioned problem 关18兴, propose another data association filter based on the maximum entropy fuzzy clustering for real-time target tracking. In this algorithm, which is known as the MEF-DAF, similar to the HF-DAF, the general structure of the PDAF and JPDAF is kept unchanged and only the process of the calculus of joint probabilities is modified and simplified by using the membership degrees uij, where this is provided by using the maximum entropy fuzzy clustering algorithm. The maximum entropy fuzzy clustering algorithm in target tracking is briefly described as follows. To be specific, suppose mk data/measurement set zi ; i = 1 , 2 , . . . , mk is related to the clusters/targets v j ; j = 1 , 2 , . . . , c. The clustering/association process is formulated as an optimization problem, and the corresponding cost function to be minimized is defined as mk

ij

is maximized under the constraints in Eqs. 共18兲 and 共19兲. Now, by using the Lagrange multiplier method, the objective function can be defined as J共U,V兲 = −

where

E=

兺兺u i=1 j=1

vi共k兲 = zi共k兲 − yˆ 共k兩k − 1兲

⫻ ␤0共k兲S共k兲 +

c

mk

␤i共k兲vi共k兲

共19兲

j=1

According to the information theory, the maximum entropy principle is the most unbiased prescription to choose the values of membership uij, for which the Shannon entropy Journal of Dynamic Systems, Measurement, and Control

兺兺 u␣ d

2 ij ij

共24兲

i=1 j=1

and the other part, namely, J2, is represented as c

J2 =

兺共v − F 共y 兲 j

p j

j old兲

T

共v j − F pj 共y j兲old兲

共25兲

j=1

where it comes from the fact that if F pj 共k兲 designates the part of the matrix referring to the position components in the state transition matrix pertaining to the target, then the predicted prototype 共y j兲k兩k+1 is provided by MARCH 2010, Vol. 132 / 021013-3


共y j兲k兩k+1 = F pj 共k兲共y j兲old

共26兲

Therefore, it is natural to assume that in the worst case, at each time increment, the evolution of the class prototypes should not go beyond the assessment supplied by the prediction 共y j兲k兩k+1. So the difference between the unknown prototype and the predicted prototype 共y j兲k兩k+1 should not be large enough. In reality, in clustering the measurements of the targets, the centers of clusters cannot be so far away from the predictions. These predictions, which are typically the predicted positions of targets, are obtained by the dynamic model of targets. This fact mathematically generates an objective function J2 共Eq. 共25兲兲. It should be noted that the core of the objective function in MEF-DAF, which is described in Eq. 共18兲, is similar to part 1 of HF-DAF’s objective function 共24兲, where according to the maximum entropy, the fuzzy clustering principle performs as Eq. 共21兲. According to the description above, which emphasizes the important role of J2 in the clustering of the data in a dynamic problem such as target tracking, the objective function of MEF-DAF can be reconstructed by considering J2 共Eq. 共25兲兲, and using its convex combination with Eq. 共18兲. In the new data association filter, the objective function is defined as mk

E=K

c

兺兺

c

uijdij2 + 共1 − K兲

i=1 j=1

兺共v − F 共y 兲

j old兲

p j

j

T

is considered as the optimal value of ␣. To identify ␣opt in every cluster, substitute Eq. 共30兲 into Eq. 共27兲, and for simplicity, take the variable ␥ to be considered as 1 / ␣ 共i.e., ␥ = 1 / ␣兲. Now, the first derivative of the obtained relationship 共27兲 with respect to parameter ␥ can be written as

⳵E = ⳵␥

mk

兺 i=1

冋

2 dic 2 2 exp共− ␥Kdic 兲 − dic W+ W2

mk

兺u

0 ⱕ uij ⱕ 1

= 1,

ij

共28兲

i=1

where the parameter K is constant and should be greater than 0.5. Similar to HF-DAF, choosing a high value for K 共near 1兲 means that the classification with respect to the distance minimization is favored over a dynamic constraint. Indeed, the latter is based only on predicting some parts of the target, which is not confident enough since the updating part is missing. According to the information theory and using the Lagrange multiplier method, the objective function now can be rewritten as mk

J共U,V兲 = K

c

兺兺u d

2 ij ij

i=1 j=1

c

+ 共1 − K兲

兺共v − F 共y 兲 p j

j

j old兲

T

共v j − F pj 共y j兲old兲

j=1

mk

c

+

兺兺 ␣j

j=1

c

uij ln共uij兲 +

i=1

兺

冉兺冊 mk

␭j

j=1

uij − 1

共29兲

i=1

where ␣ and ␭ are the Lagrange multipliers. The membership value of data zi belonging to cluster center/ target j can be obtained by minimizing Eq. 共29兲

冉冊冉冊

exp uij =

mk

兺 i=1

− Kdij2 ␣j

− Kdij2 exp ␣j

2 exp共− ␥Kdic 兲

册

mk

W=

兺 exp共− ␥Kd

2 ic兲

共32兲

i=1

d2min is defined as the distance between each data sample zi and the cluster center vc, i.e., d2min = d2pc ⱕ d2qc for q = 1 , . . . , mk and q ⫽ p. Thus, ␥opt is obtained if the following criterion is satisfied: 2 兲=␧ exp共− ␥Kd2pc兲 = exp共− ␥Kdmin

共33兲

where ␧ is a small positive constant. Under the above condition, Eq. 共27兲 can now reach its saturation level. So the optimal value of ␣ 共␣ = 1 / ␥兲 can be derived as

␣opt =

2 − Kdmin ln ␧

共34兲

In practice, the choice of an appropriate value for ␣ depends on specific applications. In a specified problem, it is interesting to note that the small value of ␣ causes the larger probability of the measurements close to the cluster center, but this choice may cause losing the track of the target when the density of the clutter is high. So it is natural to say that when the density of the clutter is very low, ␣ should be small, and vice versa. Now, by using Eq. 共5兲 as the distance from the datum zi to the prototype center v j, v j can be derived as

冋兺

vj = K

册冋兺 −1

mk

uijS−1 j

+ 共1 − K兲I

i=1

mk

K

i=1

p uijS−1 j zi + 共1 − K兲F j 共y j 兲old

册

共35兲 where I stands for the identity matrix with the same size as S j. n Clearly, the matrix 关K兺i=1 ␣iuijS−1 j − 共1 − K兲I兴 is invertible, as a result of S j being invertible. The modified objective function, which is considered as the result of using the dynamic model of the target, in determining the cluster center, leads to the superiority of the new algorithm in comparison with the MEF-DAF. The new algorithm, which is now called the MMEF-DAF, with having the least solution time property of MEF-DAF, has better performance in comparison with HF-DAF. The proof of convergence of the MMEF-DAF algorithm to a local optimum is also shown in the Appendix. As a brief overview, the procedure of the MMEF-DAF in the estimation of the target state is summarized by the following steps. Step 1. Starting from estimation x共k − 1 兩 k − 1兲 and P共k − 1 兩 k − 1兲 for each target, determine the prediction using Eqs. 共8兲 and 共9兲. • Step 2. Determine the use of the iterative procedure for the matrix U = 关uij兴共Eq. 共30兲兲, which represents the weight ␤ij 共for i = 1 , . . . , mk and j = 1 , . . . , c兲, where ␤ij = uij. • Step 3. Determine the update state and the covariance estimates using Eqs. 共10兲–共17兲. • Step 4. Repeat steps 1–3 for the next time step. •

共30兲

␣ adjusts the value of membership of data zi with the cluster center/target v j. It should be noted that the membership degree uij is very similar to Eq. 共22兲. Also similar to the MEF-DAF, 1 / ␣ has the role of the discriminating factor 关18兴. The objective function 共27兲 decreases exponentially if ␣ decreases. As a result, when the discriminating factor 1 / ␣ reaches some values, the value of objective function 共16兲 will reach a saturation level, and further decrease in ␣ will only change Eq. 共27兲 very slightly. At this time, it 021013-4 / Vol. 132, MARCH 2010

2 ic

i=1

where

j=1

subject to

兺d

共31兲

共v j − F pj 共y j兲old兲共27兲

mk

4

Simulation Results

In this section, the performance of the proposed algorithm is evaluated and compared with the other algorithms. For doing this, Transactions of the ASME


Fig. 1 The actual position and measurements of a single target

one example from single target tracking and two examples from crossing and paralleled targets in case of multiple target tracking are discussed. To compare the single and multiple target tracking cases, the same examples are simulated, as considered in Refs. 关4,15兴. The targets are modeled as constant velocity objects with white process noise 共Gaussian zero mean兲 that accounts for slight changes in the velocities. The state and the measurement models are given by Eqs. 共1兲 and 共2兲, respectively. By considering Cartesian coordinates, the state vector x containing the positions and velocities in x and y directions, is given by x = 关 px vx p y v y 兴 where

F共k兲 =

冤冥 0 0 1 T

,

0 0 0 1

H共k兲 =

冤冥

F pj =

T 0 2

1 T 0 0

0 1 0 0

共36兲

冋

G共k兲 =

1 0 0 0 0 0 1 0

1

0

0

T 2

0

1

册

Poisson distribution with known parameter ␭ = 2 共number of false measurements per unit of volume 共km2兲兲. Moreover, the initial estimates of the state are obtained by differencing two points of the observations 关1兴. In the PDAF, the detection probability of the true measurement is equal to 1 and the gate probability is 0.99. In case of single target tracking, Fig. 1 represents the actual single target tracking position “o” and the measurements “⫻,” for 30 sample times. In Fig. 2, the root mean square 共rms兲 of the estimation error of the tracking, via various algorithms, is shown. It should be noted that the matrix F pj mentioned in Eqs. 共7兲 and 共35兲 corresponds here to

,

and

共37兲

The process and the measurement noises are considered as zero mean Gaussian with known covariances given as Q共k兲 and R共k兲, respectively. Q共k兲 and R共k兲 are 2 ⫻ 2 matrices. Also, it is assumed that Qii = 0.0004 and Rii = 0.0225 km2 共Rij = Qij = 0; for i ⫽ j兲. The sampling period/interval is also assumed to be 1 s 共T = 1 s兲. The clutter model is assumed to be of uniform distribution and the number of false measurements/clutters is assumed to be of the

冋册 1 0 0 1

and

共y j兲old =

冋

xk−1兩k−1共1兲 xk−1兩k−1共3兲

册

共38兲

Also, the constant K in the objective function 共29兲 is taken to be 0.8. In Fig. 2, it is seen that the MMEF-DAF provides the best performance in terms of the rms of the estimation error. Note that in the present and the following simulations, the figures only correspond to one simulation run, but the generation of random numbers is equally initialized for all algorithms. In order to provide a meaningful comparison, 50 Monte Carlo simulations are performed and the rms of the estimation error is computed, as shown in Fig. 3. It is shown that the MEF-DAF has less rms of the estimation error than the HF-DAF, in cases of a single target tracking; however, the performance of MMEF-DAF is the best among the other algorithms. The average value of the rms of estimation error are 0.7733 km, 0.5918 km, 0.5464 km, and 0.4115 km for PDAF, HF-DAF, MEF-DAF, and MMEF-DAF, respectively. The average percentage of improvement in the rms of the estimation error in MME-DAF, in comparison with ME-

Fig. 2 The rms tracking error in the single target tracking

Journal of Dynamic Systems, Measurement, and Control

MARCH 2010, Vol. 132 / 021013-5


Fig. 3 The rms tracking error of 50 Monte Carlo simulations in the single target tracking

DAF, is 24.69%. Furthermore, the solution time of the Monte Carlo simulations for every algorithm is computed. The times are 6.3659 s, 21.3251 s, 5.3241 s, and 5.8392 s, for PDAF, HF-DAF, MEF-DAF, and MMEF-DAF, respectively. It can be easily seen that, although the solution time of the ME-DAF is the least, the MMEF-DAF with slight increment in solution time has the better performance in target tracking. To compare the results obtained via various algorithms, in cases of multiple target tracking, the tracking of the crossing targets and parallel moving targets are considered. Figure 4 shows the actual positions of two crossing targets with their measurements for 30

sample times. The rms of the estimation error of the Fitzgerald’s JPDAF, as a conventional filter, and also the HF-DAF, MEF-DAF, and MMEF-DAF are all shown in Fig. 5. Regarding the magnitude of the rms of the estimation error in this figure, the superiority of the MMEF-DAF can be seen clearly. Also, for a better illustration of the performance of the algorithms, 50 Monte Carlo simulations are performed. The obtained results are presented in Fig. 6. It is shown that the MEF-DAF has lower performance than the HF-DAF in cases of multiple target tracking. Moreover, the least rms of the estimation error is obtained by using the MMEFDAF. When the MMEF-DAF is used in cases of multiple target

Fig. 4 The actual positions and measurements of two crossing targets

Fig. 5 The rms tracking error in tracking of two crossing targets

021013-6 / Vol. 132, MARCH 2010

Transactions of the ASME


Fig. 6 The rms tracking error of 50 Monte Carlo simulations in tracking of two crossing targets

tracking, the average value of the rms of the estimation error improves with respect to the Fitzgerald’s JPDAF, HF-DAF, and MEF-DAF. These values are 89.15%, 41.73%, and 82.26%, respectively. It is important to mention that, in cases of multiple target tracking, instead of using JPDAF, the Fitzgerald’s JPDAF, as a practical ad hoc of JPDAF, is used. Due to the JPDAF, all feasible joint events must be considered and their probabilities for every joint event must be computed. Meanwhile, the computation burden increases exponentially as the number of targets increases. So, the JPDAF is not practical. Using 50 Monte Carlo simulations, the solution times of the Fitzgerald’s JPDAF, HF-DAF, MEF-DAF, and MMEF-DAF are 9.7370 s, 25.8952 s, 7.1030 s, and 7.9940 s, respectively. In spite of the fact that the best solution time belongs to MEF-DAF, however, its performance is low. The MMEF-DAF with almost the same solution time can generate the best estimate of targets.

As shown in Fig. 7, the actual target positions of three parallel moving targets and their measurements for 100 sample times are illustrated 共third example兲. Figures 8 and 9 show the rms values of the estimation error of the various algorithms for single run and 50 Mont Carlo simulations, respectively. The superiority of the MMEF-DAF is clearly shown in these figures. The average of the rms values of estimation errors of the Fitzgerald’s JPDAF, HFDAF, MEF-DAF, and MMEF-DAF are 4.4167 km, 2.3590 km, 2.6794 km, and 1.8645 km, respectively 共the improved percentages of the MMEF-DAF compared with the Fitzgerald’s JPDAF, HF-DAF, and MEF-DAF are 57.79%, 20.96%, and 30.41%, respectively兲. Also, the solution times of Fitzgerald’s JPDAF, HFDAF, MEF-DAF, and MMEF-DAF are 12.6524 s, 43.3270 s, 8.1559 s, and 9.0250 s, respectively. The above simulations show that by considering the rms of the

Fig. 7 The actual positions and measurements of three parallel targets

Fig. 8 The rms tracking error in tracking of three parallel targets


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Fig. 9 The rms tracking error of 50 Monte Carlo simulations in tracking of three parallel targets

estimation error and the solution time together, the MMEF-DAF has the best performance in cases of single and multiple target tracking

5

be shown in Theorem 3兲. Based on the above theorems, the convergence of MMEF-DAF is then proved in Theorem 4. THEOREM 1. Let zi 苸 R P共i = 1 , . . . , mk兲 be a set of given data/ measurement, and let c 苸兵2 , . . . , mk − 1其 be a real number, as the number of clusters. Then, the membership function of data/ measurements to clusters/targets U = 关uij兴共i = 1 , . . . , mk , j = 1 , . . . , c兲 and the cluster center V = 关v j兴共j = 1 , . . . , c兲 is the local minimum for J共U , V兲 (Eq. 共29兲, i.e., the objective function of the proposed data/measurement clustering/associating algorithm), if uij and v j are obtained by Eqs. 共30兲 and 共35兲, respectively. Proof. In order to solve this optimization problem, the necessary condition is obtained by setting the derivative of J共U , V兲 to zero, with respect to uij. Now we get

⳵J = Kdij2 + ␣ j共ln uij + 1兲 + ␭ j = 0 ⳵ uij where ␭ j is a Lagrange multiplier. So for having optimality

Conclusion

Multitarget tracking problems are theoretically confusing since, unlike the other estimation problems, the origin of the measurements is not identified. This involves hypotheses generation and their evaluation in terms of degree of agreement between the given measurements and the underlying tracks 共i.e., data/ measurement association problem兲. An effective strategy to solve this problem is to consider the data association in target tracking as a process to classify a given set of measurements according to some class rules. In order to deal with fuzzy methods, in comparison with conventional methods, the association probabilities are substituted by the membership degrees provided by a clustering algorithm such as the fuzzy c-means algorithm. Recently, based on the maximum entropy fuzzy clustering, the MEF-DAF is introduced. The MEF-DAF has some advantages over other fuzzy data association filters in terms of efficiency and especially in computational cost, which makes it suitable for real-time applications. However, the MEF-DAF, in cases of multiple target tracking, without using additional reconstructed algorithm, loses its high performance. In this paper, by introducing the effect of the dynamic model of the target in predicting the cluster center, and using this fact that a cluster center cannot be far away from its prediction, a modified objective function for fuzzy clustering is proposed. According to the new objective function and using the maximum entropy principle, a new method for data clustering/ association is presented and called a MMEF-DAF. MMEF-DAF has a better performance than MEF-DAF and other fuzzy data association filters, due to the fact that the data association in a dynamic problem 共e.g., target tracking problem兲 should not be merely based on the classification criterion, but the dynamic constraints must be considered too. The simulation results show that the MMEF-DAF with a slight increment in solution time, in comparison with the MEF-DAF, has the best performance over the other filters in cases of single and multiple target tracking. The superiority of the MMEF-DAF in cases of multiple target tracking is also obtained by saving the maximum entropy clustering property without requiring any additional weight assignment algorithm, where it reduces the computational cost and the complexity. As a result, the MMEF-DAF is suitable for real-time applications.

Appendix To prove the convergence of the MMEF-DAF to a local optimum, at first, in Theorem 1, it is shown that the proposed equations for the membership value of each data point uij 共Eq. 共30兲兲 and the cluster centers v j 共Eq. 共35兲兲 are the necessary conditions for minimizing the object function J共U , V兲共Eq. 共29兲兲. Then, in Theorem 2, it is shown that if V = 关v j兴 in J共U , V兲 is to be fixed, then U = 关uij兴 is a sufficient condition for minimizing J共U , V兲. Also, it is shown that if U in J共U , V兲 is to be fixed, then V 共Eq. 共35兲兲 is a sufficient condition for minimizing the J共U , V兲共as will 021013-8 / Vol. 132, MARCH 2010

共A1兲

mk

⳵J = ⳵␭j

兺u

ij

共A2兲

−1=0

i=1

also from Eq. 共A1兲, we have

冉

uij = exp −

Kdij + ␭ j + ␣ j ␣j

冊

共A3兲

Now, by substituting Eq. 共A3兲 into Eq. 共A2兲

冉

exp −

␭j + ␣j ␣j

冊兺冉冊 mk

exp −

i=1

Kdij2 =1 ␣j

共A4兲

and eliminating ␭ j from Eq. 共A4兲 and using Eq. 共A1兲, Eq. 共30兲 can be obtained as

冉冊冉冊

exp uij =

mk

− Kdij2 ␣j

兺 exp i=1

− Kdij2 ␣j

For keeping the generality of the proof, the Mahalanobis distance is used for d2ij 共Eq. 共5兲兲. The determination of center v j is also accomplished by setting the derivative of J共U , V兲 to zero, with respect to v j. This leads to mk

⳵J p = − 2K uijS−1 j 共zi − v j 兲 + 2共1 − K兲共v j − F j 共y j 兲old兲 = 0 ⳵vj i=1

兺

2

冉冋

册冊册冊

mk

−K

兺u S

−1 ij j zi

冋冉兺

− 共1 − K兲F pj 共y j兲old

i=1

mk

+

K

uijS−1 j + 共1 − K兲I v j

i=1

=0

共A5兲

or the following result, which is equivalent to Eq. 共35兲

冋兺

vj = K

i=1

册冋兺 −1

mk

uijS−1 j

+ 共1 − K兲I

mk

K

i=1

p uijS−1 j zi + 共1 − K兲F j 共y j 兲old

册

where I stands for the identity matrix having the same size as S j. mk Clearly, the matrix 关K兺i=1 uijS−1 j + 共1 − K兲I兴 is invertible, as a result of S j being invertible. THEOREM 2. Assume that the cluster center is fixed (i.e., V 苸 R P⫻c is fixed in J共U , V兲 and ⌽共U兲 = J共U , V兲兲. Then the U = 关uij兴共i = 1 , . . . , mk , j = 1 , . . . , c兲 is a local minimum of ⌽共U兲, if and only if, uij is computed from Eq. 共30兲. Proof. As the necessary condition is proven in Theorem 1, the sufficient condition is proven in Theorem 2. For this purpose, Transactions of the ASME


using the theory of optimization, the H共U兲 = 关hij兴共i = 1 , . . . , mk , j = 1 , . . . , c兲共the Hessian of ⌽共U兲兲 must be computed and shown to be positively definite. According to the definition of ⌽共U兲, as stated above, which is obtained by Eq. 共30兲, or equivalently Eq. 共A1兲, the H共U兲 = 关hij兴 can be obtained as hst,ij共U兲 =

⳵ ⳵ ust

冋册

冦

␣j if s = i, t = j ⳵ ⌽共U兲 = uij ⳵ uij otherwise 0

冧

共A6兲

where uij and ␣ j are positive 共since in Eq. 共30兲 it is clearly shown that uij ⬎ 0, also ␣ j ⬎ 0, because of the maximum entropy prinmk c ciple, in which the term −兺i=1 兺 j=1uij ln uij must be maximized, while in the minimization problem, this term must be considered by a positive coefficient兲. So, according to Eq. 共A6兲, the H共U兲 is a diagonal positive definite matrix, which means Eq. 共30兲 is a sufficient condition for minimizing the ⌽共U兲. THEOREM 3. Assume that the membership function of given data set zi 苸 R P , 共i = 1 , . . . , mk兲 to the cluster center is fixed (i.e., U 苸 Rmk⫻c is fixed in J共U , V兲 and ⌿共V兲 = J共U , V兲兲. Then, the V = 关v j兴共j = 1 , . . . , c兲 is a local minimum of ⌿共V兲, if and only if, v j is computed from Eq. 共35兲. Proof. As the necessary condition is proven in the Theorem 1, the sufficient condition is proven here. For this purpose, the Hessian of ⌿共V兲 must be computed and shown to be a positive definite matrix. According to the definition of ⌿共V兲 above, which is obtained by Eq. 共35兲, or equivalently Eq. 共A5兲, the H共V兲 can be obtained as hi,j共U兲 =

冋册

冦兺 mk

if i = j 2K uijS−1 ⳵ ⳵ ⌿共V兲 j + 2共1 − K兲I = i=1 ⳵ vi ⳵ v j otherwise 0

冧

共A7兲

Since 0 ⬍ K ⬍ 1 in J共U , V兲 and the innovation covariance S j is a symmetric positive definite matrix, the H共V兲 is a positive definite matrix. Therefore, Eq. 共35兲 is a sufficient condition for minimizing the ⌿共V兲. 共l兲共l兲 THEOREM 4. If v j and uij 共l = 0 , 1 , . . .兲 are the generated string by using Eqs. 共30兲 and 共35兲 (i.e., the membership value and cluster center of MMEF-DAF), then the objective function J共U , V兲 converges to the local minimum, that is J共U共l+1兲,V共l+1兲兲 ⱕ J共U共l兲,V共l兲兲

共A8兲

共l+1兲

Proof. As far as V is computed from Eq. 共35兲, for a constant U, according to Theorem 3, we have J共U共l+1兲,V共l+1兲兲 ⱕ J共U共l+1兲,V共l兲兲

共A9兲

共l+1兲

is computed from Eq. 共30兲, for a constant V, Also, as far as U according to Theorem 2, we have


J共U共l+1兲,V共l兲兲 ⱕ J共U共l兲,V共l兲兲

共A10兲

Now, using Eqs. 共A9兲 and 共A10兲, it is clear that J共U共l+1兲,V共l+1兲兲 ⱕ J共U共l兲,V共l兲兲 This shows that the objective function 共29兲, using Eqs. 共30兲 and 共35兲, reduces in each iteration and finally converges to a local minimum.

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