4362 IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016 A Bernoulli Filter for Extended Target Tracking Using Random Matrices in a UWB Sensor Network Abdulkadir Eryildirim and Mehmet Burak Guldogan, Member, IEEE Abstract— In this paper, we propose a new tractable Bernoulli filter based on the random matrix framework to track an extended target in an ultra-wideband (UWB) sensor network. The resulting filter jointly tracks the kinematic and shape parameters of the target and is called the extended target Gaussian inverse Wishart Bernoulli (ET-GIW-Ber) filter. Closed form expressions for the ET-GIW-Ber filter recursions are presented. A clustering step is inserted into the measurement update stage in order to have a computationally tractable filter. In addition, a new method that is consistent with the applied clustering method is embedded into the filter recursions in order to adaptively estimate the time-varying number of measurements of the extended target. The simulation results demonstrate the robust and effective performance of the proposed filter. Furthermore, real data collected from a UWB sensor network are used to assess the performance of the proposed filter. It is shown that the proposed filter yields a very promising performance in estimation of the kinematic and shape parameters of the target. Index Terms— Random finite set, extended target tracking, Bernoulli filter, Gaussian inverse Wishart, inverse Wishart distribution, random matrix, ultra-wideband, sensor network, localization. I. I NTRODUCTION ITH the advent of modern sensors, targets are generally treated as extended targets which can produce multiple measurements per time step [1]–[3]. From a target, a high resolution sensor usually collects multiple measurements per time-step across the target’s surface and/or over target’s boundaries such as edges [4]. For example, Fig. 1 shows an extended target tracking scenario, where a camera is used as a sensor. The dataset (PETS 2000, [5]) includes the video frames of the red car which enters and leaves the scene. By applying a corner detection method, it is possible to collect multiple detections of the red car at each frame. Also note that, as can be seen from the two different frames given in the figure, pixelwise size (i.e. number of possible detections) and orientation of the red car changes over time. Therefore, the car can be considered as an extended target and not only the kinematic parameters but also the shape parameters can be estimated. Recent developments in sensor technologies along with advances in processor speeds have made extended target tracking feasible and this has attracted researchers’ W Manuscript received February 14, 2016; accepted March 13, 2016. Date of publication March 22, 2016; date of current version April 26, 2016. This work was supported by the Scientific and Technological Research Council of Turkey under Grant 114E214. The associate editor coordinating the review of this paper and approving it for publication was Dr. Amitava Chatterjee. The authors are with the Department of Electrical and Electronics Engineering, Turgut Özal University, Ankara 06010, Turkey (e-mail: aeryildirim@turgutozal.edu.tr; bguldogan@turgutozal.edu.tr). Digital Object Identifier 10.1109/JSEN.2016.2544807 Fig. 1. Two images (frame 134 and frame 172) corresponding to two different time steps demonstrating the detections gathered in tracking of the red car using a camera; yellow dots are extended target detections which can be obtained by a corner detection method [5]. (a) Frame 134. (b) Frame 172. attention [1]–[3], [6]–[15]. In the concept of extended target tracking, several different applications are studied in the literature. In [13], a group of closely spaced targets is tracked. Hammarstrand et al. [2], Fortin et al. [3], and Lundquist et al. [7] concern with the tracking using automotive radars. In [16], target tracking is achieved using data sets acquired with a laser range sensor. In [14], a moving car is tracked using the data obtained from a surveillance camera. Baum [1] investigates tracking of a single moving object on a table with a camera that provides RGB and depth images. In [17], tracking using an incoherent X-band radar is considered. Ultra-Wide Band (UWB) is a very promising technology and provides important advantages: high resolution, immunity to multi-paths, small size, and low-power [18]–[20]. These advantages make UWB technology very crucial especially for in-door positioning [18], [21]–[25]. Currently, UWB sensor systems are employed in many applications [24], [26]–[30]. In literature, there exist some works regarding the detection, surveillance and tracking of persons or objects by using UWB sensor network (UWB-SN). 1558-1748 © 2016 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES However, all of these previous works use point-target assumption [24], [28], [31]–[35]. Therefore, the use of an effective extended target tracking filter can play a significant role to enhance the detection, tracking and classification tasks by estimating the target shape jointly with the target kinematics. Consequently, an extended target filter can overcome the challenges associated with the raw UWB sensor measurements and enhance the feasibility of similar systems for further advanced tasks. In this work, we present a new extended target tracking approach to track size and shape parameters along with the kinematic parameters from the raw UWB sensor network measurements. The core of the proposed approach is a Bernoulli filter which can jointly detect and track an extended target in the presence of data association uncertainty, targetmeasurement rate uncertainty, detection uncertainty, noise and false alarms [36], [37]. The Bernoulli filter is the exact Bayes filter which propagates the parameters of a Bernoulli Random Finite Set (RFS) for a single dynamic system which can randomly switch on and off [37]–[41]. The Bernoulli filter provides reliable and computationally inexpensive extraction of state estimates and joint detection and state estimation [42]. Considering extended target tracking using the probability hypothesis density filters [6], [16], [43]–[45], there exists almost no work on extended target tracking using the Bernoulli filter. The novelty of this work is two-fold. Firstly, we present the Gaussian Inverse Wishart (GIW) implementation of the Bernoulli filter, which is capable of estimating target kinematics and target shape, for extended target tracking. The proposed extended target GIW Bernoulli filter (ET-GIW-Ber) assumes random matrix framework for target extent and binomial RFS model of target-originated measurements [4], [37]. Closed form formulas for filter recursions are obtained. Since the proposed tracking filter provides an analytical solution, this makes it a good candidate to easily satisfy real-time demands of a practical system. Furthermore, another contribution is the adaptive estimation of the time varying number of scatters of the extended target, which corresponds to an adaptive target-originated measurement rate estimation scheme. Thus, it increases the feasibility of the filter for realistic scenarios. The proposed scheme uses the output of a clustering step integrated into the measurement update stage of the filter. The proposed filter makes use of the clustering approach in [16] in order to have a tractable implementation. To the best of our knowledge, such a filter has not been presented before. In literature, the research regarding practical extended target tracking using Bernoulli filter is very limited, hence indicating that it is a premature research area. In [14], a Bernoulli filter is presented for extended target tracking that models the extended target state by a vector and the part of the state vector corresponding to the target extent evolves according to random walk. Different than [14], the proposed ET-GIW-Ber filter provides a closed form solution and estimates both the kinematic state and the extent of a target by exploiting the random matrix framework. Secondly, the performance of the proposed filter is evaluated over real-data acquired by an UWB sensor network 4363 system by using an accurate ground-truth data established under controlled experiments. In most extended target tracking works, ground-truth data is not available or its quality is very low. To the best of our knowledge, accurate ground truth data, especially for target extent, are not available in previous works regarding extended target tracking [3], [7], [14]–[16], [43], [46]. Target tracking based on the point-target concept via UWB sensors has been considered in some studies in the literature [27], [28], [30], [47]–[49]. However, this paper presents extended target tracking with UWB sensors for the first time. II. T HE R ANDOM M ATRIX M ODEL In the framework of the random matrix model based tracking, the target extent is modeled by symmetric, positive definite random matrix, Xk of size d × d, which approximates the target extent as an ellipsoid, at each time k [4]. The target extent is assumed to be a part of the target state such that ξk = (xk , Xk ) where ξk represents the augmented state, xk and Xk correspond to the kinematical state vector whose dimension is s × d and target extent state respectively at time k. In the remaining parts of the paper, ξ is used to denote the extended target state which is the augmentation of the kinematical state and target extent state. In this work, the target dynamics is modeled as in [4]: (i) (i) = (Fk+1|k ⊗ Id )xk(i) + wk+1 , xk+1 (i) (i) (1) (i) (i) where wk+1 ∼ N (0, k+1|k ) and k+1|k = Qk+1|k ⊗ Xk+1 . The notation ⊗ defines the Kronecker product of two matrices. (i) (i) The kinematical state xk has the following form: xk = (rkT , ṙkT , r̈kT )T where rk , ṙk and r̈k correspond to the spatial position, velocity and acceleration respectively. Id is a d × d identity matrix, N (m, P) corresponds to the Gaussian probability density function (pdf) with mean vector m and covariance matrix P. In this work, we have s = 3 and d = 2. Thus, the matrix Fk+1|k and Qk+1|k are given by: ⎡ 1 2⎤ T 1 Ts ⎢ 2 s⎥ ⎢ (2) Fk+1|k = ⎣0 1 Ts ⎥ ⎦, 0 0 e −Ts θ Qk+1|k =  2 (1 − e−2Ts /θ )diag([0 0 1]), (3) where Ts is the sampling time,  is the acceleration width and θ is the maneuver correlation time constant. The sensor measurement model is given by the following linear equation: ( j) (i) (i) (4) zk = H k x k + ek , (i) where Hk = Hk ⊗ Id , Hk = [1 0 0] and ek is white Gaussian noise whose covariance is equal to the target extent matrix Xk . This model assumes that the sensor measurements are generated by the target centroid and they are spread over the target extent. Thus, the sensor noise is neglected compared with the target extension. This model leads to analytical Bayesian filter recursions for the estimation of the extension from sensor measurements [4]. Once the target extent is estimated, the target size, shape and orientation can be extracted. 4364 IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016 TABLE I L IST OF M ATHEMATICAL S YMBOLS The Bernoulli filter prediction equations are given as [37]: (6) qk|k−1 = pb (1 − qk−1|k−1 ) + ps qk−1|k−1 , pb (1 − qk−1|k−1 )bk|k−1 (ξ ) sk|k−1 (ξ ) = qk|k−1 ps qk−1|k−1 πk|k−1 (ξ |ξ ′ )sk−1|k−1 (ξ ′ )dξ ′ , + qk|k−1 (7) where pb and ps are the probability of target birth and probability of target survival, respectively. The target state evolves from ξ ′ to ξ with the transitional density πk|k−1 (ξ |ξ ′ ). The PDF bk|k−1 (ξ ) models the target births. In this work, the binomial RFS model of measurements originating from the extended target, which is presented in [37], is used. It is assumed that at time k, the target consists of L k scatters (measurement generating scattering points) with the probability of detection, pd . The corresponding Bernoulli filter update equations are given as [37]: qk|k = 1 − k qk|k−1 , 1 − qk|k−1 k (8) (1 − pd ) L k + ψk W ∈P1:L k (Zk ) sk|k (ξ ) = z∈W 1 − k gk (z|ξ ) λc(z) sk|k−1 (ξ ), (9) where For the sake of clarity and readability, Table I summarizes the notation used in this work. III. T HE B ERNOULLI F ILTER FOR AN E XTENDED TARGET In the RFS framework, both targets and measurements take values as unordered finite sets [50]. Thus, in this work, the target state at time k is modeled by the Bernoulli RFS. The sensor measurements, Zk , at time k are also defined ( j ) Nz,k {zk } j =1 as the RFS Zk = where Nz,k is the number of sensor measurements at time k. ξ and z denote the extended target state and sensor measurement respectively. The finite set statistics probability density function (FISST PDF) [37], [50] statistically characterizes an RFS. The ’FISST PDF’ is abbreviated by the term PDF in this work for simplicity. For the Bernoulli filter considered in this work, the Bernoulli RFS, which models the target state, assumes that at most one target can be present at time k. The Bernoulli RFS can either be empty set, denoted by ∅, with probability 1 − q or it can have one element with probability q distributed over the state space X with PDF s(ξ ). The PDF of a Bernoulli RFS X is given by [50]:  1 − q if X = ∅, f (X) = (5) qs(ξ ) if X = {ξ }. For tracking an extended target, the Bernoulli filter propagates the following two quantities recursively [37]: the posterior probability of target existence qk|k = P(|X| = 1|Z1:k ) and the posterior spatial PDF sk|k (ξ ) = p(ξ |Z1:k ). k = 1 − (1 − pd ) L k − ψk  gk (z|ξ )sk|k−1 (ξ )dξ z∈W W ∈P1:L k (Zk ) λc(z) , (10a) z∈W |W | pd Lk ! ψk = . (L k − |W |)! (1 − pd )|W |−L k (10b) The notation P1:L k (Zk ) corresponds to the set of all subsets of Zk with cardinalities equal to 1, 2, . . . , L k . gk (z|ξ ) is the likelihood function and c(z) denotes the spatial distribution of the clutter, which is assumed to be known. λ is the average rate of the clutter measurements modeled by a Poisson distribution at time k. The parameter L k must be set in order to update the filter. In most practical applications, L k is unknown. Therefore, we will present a method for adaptive estimation of L k in the following section. IV. T HE E XTENDED TARGET G AUSSIAN I NVERSE W ISHART B ERNOULLI (ET-GIW-BER) F ILTER In [4], under certain assumptions which are justified for most scenarios and ellipsoidal target extent model, a Bayesian recursive filter which describes the posterior density as a product of Gaussian and Wishart-related densities is presented. Thus, considering that the Bernoulli filter is a Bayesian optimal filter, the filter prediction and update results given in [4] are also preserved for the extended target Bernoulli filter [43]. Therefore, in this work, the results presented in [4] are adapted into the extended target Bernoulli filter framework accordingly. ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES ( j) ηk−1|k−1 and ( j) k−1|k−1 4365 are the inverse Wishart degrees of freedom and inverse scale matrix of the j -th component. The notation IW(X; η, ) corresponds to an inverse Wishart pdf defined over the variable X with degrees of freedom η and inverse scale matrix [51]. In order to shorten the equations, here we define the following variable:   ( j) ( j) ( j) ( j) ( j) △ (12) , P , η , = m k|k k|k k|k k|k . k|k Fig. 2. Extended target measurements with the ellipsoidal extension model: Measurements correspond to the x-y coordinates of an extended target at two different time instances. Gray squares represent the measurements collected by a sensor, red squares correspond to the centroid of the extended target and red ellipses represent the true shape of the target. Bearing, semi-major and semi-minor axes lengths are seen for two time instances. An extended target and measurements generated at different times according to the models used in this work are illustrated in Fig. 2. From an extended target, a high-resolution sensor in general collects multiple measurements originating from the scatters of the target at each time. Assuming that sensor measurements consist of position coordinates x and y of the target, an extended target might generate multiple measurements at each time as shown in Fig. 2. Both cardinality and spatial spread of extended target measurements at each time are assumed to be random. The measurements originated from the scattering points vary with time depending on the properties of the target extent, sensor-to-target geometry and properties of sensor. Therefore, binomial RFS model for extended target measurements is a very appropriate model. In this work, the target shape is approximated by an ellipsoid represented by the symmetric positive definite matrices as in [4]. As shown in Fig. 2, target shape parameters including the bearing φ, the semi-major axes length A1 and semiminor axes length a1 can be extracted from the random matrix model used in this work. Ellipsoidal shape representation is appropriate in many practical target tracking scenarios [4], [43]. This is due to the fact that in many practical cases, the target-to-sensor range is neither high enough to generate only a single measurement nor is it low to allow target features to be distinguished in detail. It is assumed that the spatial PDF sk−1|k−1 (·) at time k −1 is given by a GIW mixture of the form, Assuming the birth PDF has a mixture of GIW distribution, the form of the prediction equation (7) results in a mixture of GIW distribution with the parameters Mk|k−1 , wk|k−1 , k|k−1 according to the model and assumptions for prediction step in [4]. Thus, applying the equations (24), (25), (28), (29) in [4], we obtain the predicted components corresponding to existing targets as: ps qk−1|k−1 ( j ) ( j) wk−1|k−1 , (13a) wk|k−1 = qk|k−1 ( j) = j =1 ( j) k−1|k−1  , (11) ( j) where Mk−1|k−1 is the number of components, wk−1|k−1 is ( j) = (13c) ( j) k|k−1 = = ( j) T Qk|k−1 + Fk|k−1 Pk−1|k−1 Fk|k−1 , −Ts /τ ( j ) e ηk−1|k−1 , ( j) ηk|k−1 − d − 1 ( j ) k−1|k−1 , ( j) ηk−1|k−1 − d − 1 ( j) k|k−1 ( j) the weight of j -th component; mk−1|k−1 and Pk−1|k−1 are the Gaussian mean and covariance of the j -th component; (13d) (13e) = ( j) b,k . (15) The probability of existence is predicted according to (6). The updated spatial distribution is again a GIW mixture; D sk|k (ξ, W ), ND sk|k (ξ ) = sk|k (ξ ) + (16) W ∈P1:L k (Zk ) where ND and N stand for no detection and detection cases, respectively. No detection part of the updated spatial N D (ξ ), is updated by distribution, sk|k ( j) wk|k = ( j) k|k   ( j) ( j) ( j) wk−1|k−1 N x; mk−1|k−1 , Pk−1|k−1 ⊗ X  ( j) ×IW X; ηk−1|k−1 , (13b) ( j) Pk|k−1 ( j) ηk|k−1 where τ is a temporal decay constant. The parameter, τ , is used to accommodate the variation of the target extent over time. We assume that the birth model, bk|k−1 (ξ ), has exactly the same form as sk−1|k−1 (·) with parameters Mb,k , wb,k , b,k . This assumption can be validated for many realistic scenarios. Then, the contribution of birth model to the prediction is given by: pb (1 − qk−1|k−1 ) ( j ) ( j) wb,k , (14) wk|k−1 (ξ ) = qk|k−1 sk−1|k−1 (ξ ) Mk−1|k−1 ( j) mk|k−1 = (Fk|k−1 ⊗ Id )mk−1|k−1 , = (1 − pd ) L k ( j ) wk|k−1 , 1 − k ( j) k|k−1 . (17) (18) In order to obtain the detection part of the updated distribution, D (ξ, W ), which is also a GIW mixture, the likelihood of the sk|k measurements in each subset W is to be multiplied with the predicted spatial PDF components. Assuming uniform clutter distribution, c(z k ) = V1 where V is the surveillance volume, we have the following structure for the likelihood part:  gk (zk |ξ )  −|W | = αc,k N (zk ; (Hk ⊗ Id )x, X), (19) λc(z k ) zk ∈W zk ∈W 4366 IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016 where αc,k = Vλ is the average number of clutter measurements per surveillance volume per time. Therefore, we obtain the following form for the product of sk|k−1 (ξ ) and (19): ⎞ ⎛  −|W | αc,k ⎝ N (zk ; (Hk ⊗ Id )x, X)⎠ sk|k−1 (ξ ) zk ∈W =   ( j,W ) ( j,W ) x; mk|k , Pk|k ⊗ X   ( j,W ) ( j,W ) , ×IW X; ηk|k , k|k −|W | ( j,W ) αc,k Lk N (20) ( j,W ) where Lk represents a kind of likelihood. Then, using the filter update results derived in [4], the updated components are obtained as,   ( j,W ) ( j,W ) ( j) ( j,W ) (21a) mk|k = mk|k−1 + K k|k−1 ⊗ Id E k|k−1 , ( j,W ) Pk|k ( j,W ) ηk|k ( j,W ) k|k ( j) ( j,W ) T ( j,W ) ( j,W ) = Pk|k−1 − K k|k−1 Sk|k−1 (K k|k−1 ) , = = ( j) ηk|k−1 + |W |, ( j) ( j,W ) k|k−1 + Nk|k−1 (21b) (21c) + Z kW . (21d) The respective intermediate variables in the above equations are given by, z̄kW = 1 |W | z(i) k , (22a) (i) zk ∈W T (i) W W (z(i) k − z̄k )(zk − z̄k ) , Z kW = (22b) (i) zk ∈W 1 ( j,W ) ( j) Sk|k−1 = Hk Pk|k−1 Hk T + , |W |   ( j,W ) ( j) ( j,W ) −1 K k|k−1 = Pk|k−1 Hk T Sk|k−1 , ( j,W ) E k|k−1 ( j,W ) = z̄kW  ( j,W ) ( j,W ) Lk (22d) ( j) − (Hk ⊗ Id )mk|k−1 ,    ( j,W ) −1 ( j,W ) ( j,W ) T Nk|k−1 = Sk|k−1 The likelihood term Lk (22c) E k|k−1 E k|k−1 (22e) (22f) . in (20) has the following form [43]: 1 =   ( j,W ) d/2 π |W | |W |Sk|k−1 | ( j) k|k−1 | | ( j,W ) k|k | ( j) ηk|k−1 2 η ( j,W ) k|k 2 ( j,W ) (ηk|k ) ) 2 . ( j) ηk|k−1 Ŵd ( 2 ) Ŵd ( (23) where | | corresponds to the determinant of the matrix and |W | is the number of measurements in the subset W . The weights of the updated GIW mixture are obtained as: ( j) wk|k = 1 ( j,W ) −|W | ( j ) αc,k wk|k−1 , ψk Lk 1 − k (24) where k given in (10a) has the following form: k = 1 − (1 − pd ) L k Mk|k−1 −|W | − ( j,W ) Lk ψk αc,k W ∈P1:L k (Zk ) j =1 ( j) wk|k−1 . (25) The estimate of the target extent is obtained from the inverse Wishart parameters by using the following formula: X̂ = η−2d−2 [4]. In order to have a tractable number of components, pruning and merging of GIW components is achieved similarly as in [43]. The computational complexity and required memory size for the ET-GIW-Ber filter are both proportional to the total number of subsets, P1:L k (Zk ), generated from the measurements Zk . Thus, total number of subsets, P1:L k (Zk ), exponentially increases as cardinality of measurements and L k corresponding to the true number of scattering points of extended target at time step k increases. Even if L k is small, increased clutter rate leads to huge number of subsets. Considering augmented vector and matrices used in the measurement update step, required memory size might become infeasible to do the computation for even powerful computers. Therefore, gating of measurements is commonly applied to alleviate the computational burden for extended target tracking [37]. However, target-originated measurements must be preserved in order to have a good extended target tracking performance. Thus, high number of target-originated measurements leads to intractable solution even if gating is used to eliminate the clutter. Considering this problem, we insert a clustering step into the measurement update stage of the proposed filter in order to have a computationally tractable filter. The clustering method approximates the set of all subsets, P1:L k (Zk ), generated in the measurement update stage with a subset of partitions. Assume that the output of the clustering is denoted by P Zk . The notation P Zk means that P partitions the measurement set Zk into non-empty clusters . Then, P Zk replaces P1:L k (Zk ). Thus, partitions or clusters, P Zk , are processed in the measurement update instead of using all subsets, P1:L k (Zk ). Therefore, the proposed filter can handle large number of target-originated measurements per time, which is often the case in many real world scenarios. Having said that, we note that the parameter L k still influences the tracking performance of the proposed filter as it is embedded in the measurement update equations. Thus, we still need to set the value of L k in the measurement update stage of the proposed filter. In [14], a fixed formula which uses the cardinality of current measurements, probability of detection and known clutter rate is given for the estimation of L k . In order to estimate L k , we propose a new method which is consistent with the applied clustering method. Consequently, we cluster the measurement set received at time step k spatially with the distance partitioning method presented in [43] by exploiting the knowledge of the spatial distribution of the extended target measurements. At the output of this clustering process, typically, one large cluster corresponding to the extended target and other clusters are obtained. Also, the cluster corresponding to the extended target in general contains more measurements than the other clusters. To sum up, in our method, the parameter, L k , is adaptively estimated using the output of the applied clustering method, which is distance partitioning. Assume that the clustering step which employs the distance partitioning method generates n number of clusters, P Zk , at time step k such that each cluster and the corresponding ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES 4367 cardinality is represented by (i) and |(i) | where i = 1, . . . , n. Then, the parameter L k is estimated such that: Mk,max = max {|(1)|, |(2) |, . . . , |(n) |}, Mk,max  ⌉. Lk = ⌊ pd (26) (27) where ⌊·⌉ corresponds to the nearest integer operation and  Lk represents the adaptive estimate of L k . V. S IMULATION R ESULTS We demonstrate the ET-GIW-Ber filter performance on a scenario involving time-varying target extent and time-varying number of target scatters. For nearly all kinds of sensors from laser range finders to cameras, quality and statistical characteristics of target-originated measurements strongly depend on target-to-sensor distance, which is also called as target range. In general, as target range increases over time, target occupies fewer number of resolution cells and signal-tonoise ratio decreases. Hence, the number of target-originated measurements (which is also called as detections) as well as the detected target size reduce. However, it is desired that a tracking filter maintains its track while the target range increases. Therefore, the considered scenario is frequently encountered in real-world applications. The surveillance area is taken as 600 m2 . The simulation duration is 100 s where each time k corresponds to 1 s (i.e. the sampling time is T = 1 s). The target is born at time k = 5 and dies at k = 72. The target-originated measurements consisting of Cartesian x-y coordinates are generated according to (4). The true target extents are given by Xk = Mk diag([ A1,k a1,k ])(Mk )T where Mk is the rotation matrix which determines the bearing (orientation), A1,k and a1,k are the semi-major and semi-minor axes lengths of the extended target at time k respectively. In the considered scenario, the target extent is time-varying such that the lengths of the principal axes reduce to their halves with constant rate over time assuming initial values (A1,birt h , a1,birt h ) = (30 m, 10 m). Besides that, the bearing of the target linearly increases up to 75 degrees from 15 degrees over time. The true number of scatters of the target L k,true linearly reduces to 10 over time starting from the value 20 such that L k,true = ⌊dk + e⌉ for 5 ≤ k ≤ 72 where d = −0.1493 and e = 20.7463. Thus, the target extent undergoes rotation and scaling which often occur in practical applications. The scenario might correspond to a target moving far away from a possibly radar sensor such that as the target-tosensor distance increases, its size detected by the radar sensor reduces and fewer number of measurements are received by the radar. The true target trajectory, which is shown in Fig. 3a, is generated according to the dynamic model in [4]. The average clutter rate is taken as λ = 10. The clutter is uniformly distributed over the surveillance area. For most practical applications, there is not a-priori knowledge on the locations where the target is likely to appear. Thus, assuming that we lack a-priori information about target birth, we use the following birth model: mb,k = [250 250 3 3 0 0], Fig. 3. A single run of the ET-GIW-Ber filter: (a) All measurements generated, true target (centroid) trajectory (blue line), estimated target (centroid) trajectory by the ET-GIW-Ber filter (red dots). (b) The corresponding targetoriginated measurements and tracking results at k = 7, 11, 20, 30, 35, 40, 46, 50, 54, 64, 71: the target-originated measurements (gray dots), blue ellipses representing true extended target shape, red ellipses representing estimated extended target shape. Pb,k = diag([1502 502 502 ]) ηb,k = 7, b,k = diag([1 1]). The birth model that we use here is generally the case in most real applications and therefore makes the scenario more realistic [39], [52]. The main parameters of the ET-GIWBer filter used are: ps = 0.99, pb = 0.01, pd = 0.9,  = 0.15 m2 /s, θ = 0.5 s, and τ = 2 s. The pruning/merging parameters are selected as Ttrunc = 10−4 , U = 20 and Jmax = 100 where Ttrunc is the truncation threshold for weights, U is the merging threshold, and Jmax is the maximum allowable number of GIW mixture components. Clustering of measurements is achieved by using the distance partitioning method. The threshold τ for the target detection is chosen as 0.5 (i.e. a target is declared as detected if qk|k ≥ τ ). The performance of the ET-GIW-Ber filter is evaluated via Monte Carlo simulations. For each Monte Carlo run, random sensor measurements including the ones originating from the target and clutter are generated accordingly. For performance evaluation, optimal subpattern assignment (OSPA) error is 4368 IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016 Fig. 4. The average OSPA errors of the ET-GIW-Ber filter. used with cut-off parameter, c = A1,birt h = 30 m and p = 1 [53]. The sum of the localization error and the cardinality error is both involved in OSPA error. The relative weighting of the costs assigned to localization and cardinality errors depends on the selection of c. We define four different OSPA errors: OSPA position, bearing, size, and shape error. The OSPA position error corresponds to the OSPA between the true target centroid coordinates and the estimated ones. The OSPA bearing, size and shape error are computed according to the true shape parameter vectors [φ], [ A1 a1 ] and [ A1 a1 φ] respectively where A1 , a1 , and φ are the semi-major axes length, semi-minor axes length (in meters) and the bearing angle of the target (in degrees) respectively. All OSPA errors are averaged over 1000 Monte Carlo runs. The results of a single Monte Carlo run of the proposed ET-GIW-Ber filter are shown in Fig.3. As seen in Fig.3a, the ET-GIW-Ber filter provides accurate tracking of the extended target position. The estimates of the extended target shape as well as the bearing and size variation of the target over time are both shown in Fig.3b at certain time steps. Fig.4 shows the OSPA error performances of the proposed filter. According to Fig.4, the ET-GIW-Ber filter provides a very promising accuracy in estimating both the centroid position and the size/bearing of the target. At around k = 5 when the target is born, both the average OSPA position and shape/size errors rise up to some value which is lower than the cut-off value c. This is a desired result indicating that the ET-GIW-Ber filter can initiate the track accurately and immediately following the target birth. Similar to the target birth, the ET-GIW-Ber filter is sensitive to the death of the target at k = 72 as the corresponding OSPA errors very quickly reduce to zero after k = 72. In [4], it is claimed that time-varying target extents, whose shape and number of scatters change over time, are tractable within the presented framework. Consequently, the simulation results suggest that target with time-varying extent can be estimated with low error by using the proposed ET-GIW-Ber filter. Furthermore, the proposed filter performs well with unknown target-originated measurement rate and lack of a-priori information about target births. VI. E XPERIMENT R ESULTS W ITH THE UWB S YSTEM We used UWB Real-time Location System (RTLS) provided by Ubisense to perform extended target tracking experiments [54]. Ubisense RTLS is an in-door UWB sensor Fig. 5. The UWB-SN experimental setup. (a) Indoor environment where the experiments take place, a person moves inside the area whose corners are assumed to be the UWB sensors shown by the red circles. (b) UWB tags mounted on a person: The tags are shown as numbered from one to six corresponding to their IDs. (c) The UWB sensor (Ubisense). network (UWB-SN) which uses active tags mounted on moving objects for positioning. Therefore, the UWB-SN system installation is composed of a network of UWB sensors that are installed in a room and small radio-emitting tags that can be mounted on a person as shown in Fig. 5 [54]. The measurement data collected by the UWB-SN are (x,y,z) spatial coordinates of the tags and their related acquisition time slot. The UWB-SN allocates time slots to each tag for emitting its signal in order to eliminate interference between tags. Position refreshment frequency, i.e. measurement rate, differs for each tag. The localization accuracy of the system is claimed to be 0.15 m with 99% of errors being within 0.30 m [55] in an ideal setting. However, in reality, due to the environmental effects, its positioning accuracy degrades. In [56], it is claimed that for dynamic positioning where the UWB tag is moving, they obtain a horizontal positioning mean error of 0.37 m and a standard deviation of 0.24 m for the accuracy of the UWB-SN system. For the physical set-up of the UWB-SN system, the sensors are fixed at a height of 1.74 m from the ground. Four sensors are placed in the four vertices of the bounding rectangle ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES covering the area where experiments take place as seen in Fig. 5a. Different scenarios including different trajectories and human movements are achieved inside this area. In all experimental scenarios, the elevation (height) of the tag remains fixed. Hence, the tags attached to the humans keep a constant distance to the floor and tag movement is considered to be only horizontal. Therefore, we have a 2D tracking problem. Data collected by the UWB-SN system bring several challenges [35], [56], [57]. First, due to lost packets or low signalto-noise ratio (SNR), missing values occur at certain time steps. This means that detection uncertainty exists for the UWB-SN system. Second, a certain percentage of the collected data has high error - which can be called outliers or clutter. Lastly, mobility rate of the tag, non-line-of-sight conditions as well as obstruction due to the human body and other factors further degrade the accuracy of the UWB-SN system compared to the ideal accuracy values. In [56], Ubisense tags are mounted on wheel chairs of the athletes and a Leica TS-30 robotic total station is employed for establishing the ground-truth. In this work, an average horizontal positioning error of 0.37 m (with standard deviation 0.24 m) is reported for athletes tracking. For our experiments, the conditions of our environment do not favor the Ubisense system. Furthermore, the geometry and placement of Ubisense sensors cause non-line-of-sight for some sensors at certain time steps during the movement of the person. Consequently, in this work, considering the quality of the acquired data, it is assumed that the Ubisense positioning accuracy is 0.4 m. This assumption also means that assuming identical conditions for both axes, the standard deviation of the measurement noise is 0.4 m. 6 UWB tags are mounted on the body of a person as shown in Fig. 5b. During the experiments, the person with the UWB tags follows pre-defined trajectories with relatively constant velocity. The parameters of the proposed filter as well as the values of the parameters in the target dynamics and sensor model are adjusted according to the motion of the human, measurement statistics of the UWB-SN system and other conditions of the experiments. Since the person can start its movement at any location inside the surveillance area, it is assumed that we do not know the target birth locations a-priori. Thus, the spontaneous birth density is defined in the center of the surveillance area with an appropriate covariance such that target appearance can occur in any location of the area. The birth density used in the experiments is given by: bk|k−1 = 0.2N (x; mb , Pb ), (28) mb = [3, 6, 0, 0, 0.0], Pb = di ag([32, 52 , 52 ]). (29) where Consequently, accommodating all scenarios, this approach is realistic for practical considerations. The clutter is assumed to follow a uniform spatial distribution over the rectangular surveillance area whose corners are determined by four UWB sensors. 4369 Fig. 6. The UWB-SN measurements and ground truth trajectory for Scenario I: The UWB-SN measurements corresponding to different UWB tags are denotes with different color coded markers. Solid blue line represents the ground truth. Pruning stage of the proposed filter is achieved with the same parameter values used in the simulations such that Ttrunc = 10−5 , U = 4 and Jmax = 100. The threshold τ for the target detection of the ET-GIW-Ber is chosen as 0.5. In our tracking problem, due to the non-line-of-sight (NLOS) and signal attenuation problems, unreliable measurements are collected at certain time instants. Also, no measurement might be collected at some time instants for a corresponding tag. Thus, clutter rate is taken as λ = 1 with clutter intensity distributed uniformly over the surveillance area. We present the results from the experiments achieved with the UWB-SN system. Two different scenarios, which we call as Scenario I and II respectively, were considered in order to evaluate the performance of the proposed filter. In order to obtain a definite measure of target tracking performance, ground truth data for the state of the target and its extent are required. Since the experiments were achieved under our control, the ground-truth for the trajectory of the person were obtained in a reasonable and relatively accurate manner. The ground-truth trajectories for both scenarios are shown in Fig. 7a and 9a respectively. While it is relatively an easy task to build the ground-truth accurately and reliably for the target position, it was relatively difficult to obtain the ground truth data in a very accurate manner for the target extent, especially for the bearing. However, by examining the geometric configuration of the tags mounted on the person, true trajectory and accuracy of the UWB-SN system, the true target size and bearing were established. Scenario I involves a person walking over a straight line along the y direction. The person follows a trajectory starting from coordinates (3.0, 3.5) m and ending in coordinates (3.0, 7.2) m along a straight line. Besides that, the person moves in the middle of the surveillance area. This enhances the line-of-sight condition for all sensors. As the number of sensors seeing the UWB tag directly increases, the probability of detection of the system increases and the accuracy of the localization performance improves [55], [56]. Moreover, due to the target trajectory, the bearing of the person, which 4370 Fig. 7. Scenario I, extended target tracking results for the UWB-SN measurements: (a) Solid blue line represents the ground-truth for the target trajectory and red-diamonds represent the estimates of the ET-GIW-Ber filter. (b) Extended target size tracking results: Estimates of the major and minor axes lengths, A1 and a1 and the corresponding true values, respectively. (c) Extended target bearing tracking results: Solid blue lines represent the lower and upper bounds for the ground-truth and blue-crosses correspond to the bearing estimates. is considered as an extended target, is expected to remain approximately constant with some amount of fluctuations. This is due to the placement of the tags. There exist two tags mounted close to the knees of the persons. The movement of legs of the person causes some amount of bearing change over time. Thus, while it is difficult to obtain an accurate groundtruth for the bearing of the person, a reasonable assumption is made. Considering the spatial spread of the leg movement is IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016 limited, the variation of the true bearing should be within some bounds. An investigation of the trajectory, which is along y direction only, and geometric configuration of the tags leads to the conclusion that the true bearing is around zero degree. The random matrix model ignores the contribution of the measurement noise [4]. Thus, it assumes that all targetoriginated measurements are contained inside the target extent. Therefore, the proposed filter estimates the target extension plus sensor accuracy [58]. Thus, in order to properly define the true target size, i.e. the ground-truth for the semi-major and minor axes lengths, the accuracy of the UWB-SN system is involved to obtain the ground-truth for the target size. Considering the geometric configuration of the tags and the trajectory in Scenario I, it is assumed that the extent of the target along y direction is approximately determined by the average accuracy (error) of the UWB-SN. Based on the collected data for our mobile measurements, the standard deviation of the measurement noise in both directions is assumed to be 0.4 m, as mentioned previously. Consequently, assuming that the width between the two shoulders of the person is 0.4 m, we have the following true values for the semi-major and minor axes lengths: A1,true = 0.6 m and a1,true = 0.4 m. The collected measurements are shown in Fig. 6. The corresponding target position estimates provided by the proposed filter are shown in Fig. 7a. The differences between the blue line representing the ground-truth positions and the red diamonds corresponding to the filter estimates are very small in this scenario. Thus, the results show that the proposed filter can estimate the position of the person accurately in this case. The tracking performance for the target extent is shown in Fig. 7. The accuracy of target size tracking appears very good as shown in Fig. 7b. As seen in Fig. 6, the UWB-SN measurements contain more distortions at the beginning of the trajectory. This also affects the size estimation performance of the filter for the initial measurements. In general, the proposed filter provides smooth and accurate estimates which are close to the ground-truth. Besides that, the proposed filter provides reasonable bearing estimates consistent with the true values as shown in Fig. 7c. We think that the movement of person includes some small gait changes as it is not easy for the person to preserve the same posture during the whole trajectory. Due to this and other factors, the fluctuations of the bearing estimates as well as the true bearing values have different behavior during different portions of the trajectory. However, over-all, the bearing estimates are good and consistent with the expected true bearing values. The second scenario (Scenario II) corresponds to a more challenging target trajectory with target maneuvers in a small area, as shown in Fig. 8. The trajectory of the person starts at the position (2.3, 3.5) m and ends at the position (3.5, 3.5) m. As seen in Fig. 8, the trajectory is in the shape of a rectangular with one missing edge. Therefore, the trajectory followed by the person in Scenario II can be divided into three segments, Segment 1, 2 and 3 respectively. The person first moves along Segment 1 which is a straight line segment along y direction. Then, the person does a maneuvering-like action in order to enter Segment 2 which is a shorter straight line segment ERYILDIRIM AND GULDOGAN: BERNOULLI FILTER FOR EXTENDED TARGET TRACKING USING RANDOM MATRICES 4371 Fig. 8. The UWB-SN measurements and ground truth trajectory for Scenario II: The UWB-SN measurements corresponding to different UWB tags are denotes with different color coded markers. Solid blue line represents the ground truth. along x direction. Finally, the person follows Segment 3 in reverse direction with respect to the one in Segment 1 and finishes its trajectory. At certain locations on the considered trajectory, non-line-of-sight conditions occur due to the orientation of the sensors and their limited field of view. Furthermore, the human body parts might obstruct and weaken the signals of some tags depending on the geometry. Therefore, at certain locations, the UWB-SN measurements degrade more severely as exhibited by the plot shown in Fig. 8. However, it is desired that the proposed filter maintains track during these unfavorable and challenging conditions. Similar to the previous experiment, the red diamonds in Fig. 9a are very close to the real positions of the person and the proposed filter performs very well for this difficult scenario. Similar effective performances are also exhibited for the size and bearing tracking provided by the proposed filter as shown in Fig. 9b and 9c. The filter successfully captures the sudden bearing changes which occurs when the person moves from Segment 1 to 2 and from Segment 2 to 3. This is exhibited by jumps in the bearing estimates in the interval between k = 30 and k = 50 approximately as shown in Fig. 9c. This time interval corresponds to the movement of person in Segment 2 including the maneuvering actions while entering and leaving Segment 2. Thus, the results indicate that the proposed filter can track the bearing of an extended target. The size estimation performance of the proposed filter for this scenario is also good as indicated by the small differences between the ground-truth and estimates as shown in Fig. 9b. The two scenarios above are promising evaluations of the ET-GIW-Ber filter and they present a proof of concept study for indoor person tracking using an UWB sensor network system. In [35], it is claimed that although the accuracy of Ubisense system is about 0.15 m in ideal setting, however, in practice, the quality of Ubisense raw measurements makes human action recognition challenging. Being an extended target tracking filter capable of the estimating the shape of target, the proposed filter provides extra information and inherently can handle the noise, occlusions, distortions and Fig. 9. Scenario II, extended target tracking results for the UWB-SN measurements: (a) Solid blue line represents the ground-truth for the target trajectory and red-diamonds represent the estimates of the ET-GIW-Ber filter. (b) Extended target size tracking results: Estimates of the major and minor axes lengths, A1 and a1 and the corresponding true values, respectively. (c) Extended target bearing tracking results: Solid blue lines represent the lower and upper bounds for the ground-truth and blue-crosses correspond to the bearing estimates. anomalies associated with UWB systems. Thus, we think that the proposed filter can be useful for tasks like human action recognition or behavior analysis of humans. VII. C ONCLUSION In this work, a tractable extended target Bernoulli filter using random matrix model is presented. The proposed filter is 4372 IEEE SENSORS JOURNAL, VOL. 16, NO. 11, JUNE 1, 2016 capable of estimating both the target kinematics and the nonstationary target extent parameters. Analytical formulas are presented for filter recursions based on GIW approximation. A method which uses the output of the integrated clustering method is also applied in order to adaptively estimate the timevarying number of scatters of the extended target. Simulation results show that the proposed filter provides accurate and robust performance in extended target tracking. Moreover, performance of the proposed filter is tested in an indoor person tracking scenario using real UWB data. For future work, we intend to improve the performance of the proposed filter by considering different models developed in the random matrix framework for different scenarios. For instance, we plan to achieve experiments which involves the scaling of the target size. Furthermore, we are planning test the performance of the proposed filter in a tag-less experiment using UWB radar sensors. R EFERENCES [1] M. Baum, “Simultaneous tracking and shape estimation of extended objects,” Ph.D. dissertation, Dept. Fakultät für Informatik, Karlsruhe Inst. Technol., Karlsruhe, Germany, 2013. [2] L. 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World J., vol. 2014, Jan. 2014, Art. no. 796279. [58] M. Feldmann, D. Fränken, and W. Koch, “Tracking of extended objects and group targets using random matrices,” IEEE Trans. Signal Process., vol. 59, no. 4, pp. 1409–1420, Apr. 2011. Abdulkadir Eryildirim was born in Turkey in 1984. He received the B.S. degree in electrical and electronics engineering from Middle East Technical University, Turkey, in 2007, the M.S. degree in electrical and electronics engineering from İ.D. Bilkent University, Turkey, in 2009, the master’s degree in space communication systems from ISAE (SUPAÉRO), France, in 2012. He is currently pursuing the Ph.D. degree with Turgut Özal University, Turkey. His research interests are signal processing, detection and estimation, UWB systems, radar systems, localization, and target tracking. Mehmet Burak Guldogan received the B.S., M.S., and Ph.D. degrees from Bilkent University, Turkey, in 2003, 2006, and 2010, respectively, all in electrical and electronics engineering. From 2010 and 2012, he was a Post-Doctoral Fellow with the Automatic Control Group, Linkoping University, Sweden. He joined Turgut Özal University, Ankara, Turkey, in 2012, where he is currently an Associate Professor with the Department of Electrical and Electronics Engineering. His current research interests are in statistical signal processing, sensor fusion, estimation theory, and target tracking. He has been an Associate Editor of Digital Signal Processing (Elsevier) since 2012.