Multiple Sensor Multiple Object Tracking With GMPHD Filter - ISIF

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ange tracking, and multiple speaker tracking are presented in the section V. II. PROBLEM FORMULATION The multi-sensor multi-object tracking problem can be modelled by random nite set (RFS) framework. Let X be the single object state space then multiple object state at time k is presented by X k = fx k;1 ; x k;2 :::; x k;Nk g 2 F(X ), where F(X ) denotes the collection of all nite subsets of the space X . For a multi-object state X k 1 at time k 1, each x k 1 2 X k 1 can continue to exist at time k with probability p S;k or die at time k with probability (1 p S;k ). Let S k (x k 1 ) denote the object that is the transition from x k 1 at time k on condition that the object is survived and let B kjk 1 (x k 1 ) be objects spawned at time k from an object with previous state x k 1 . Let k be RFS of spontaneous births at time k and can be determined by using the assumption of spontaneous birth models. Given a multi-object state X k 1 at time k 1, the multi-object state X k at time k is given by union of the surviving objects and new objects, X k = h[ h[ i Sk (x k 1 )i [ Bkjk 1 (x k 1 ) [ [ k] (1) The RFS X k encapsulates all aspects of multi-object tracking problem, such as time varying number of objects, object motion. Similarly, let Z i be the measurement space of single object at sensor i then measurements collected at sensor i time k is Zk i 2 F(Zi ). A given object state x k 2 X k is either detected with probability p D or missed with probability (1 p D ). Conditional on detection, the measurement from x k at sensor i is dened by the RFS i k (x k). The sensor i also can receive a set of clutters Ck i . So, given a multi-object state X k at time k, the measurement set from sensor i at time k is formed by the union of object generated measurements and clutters, " # [ Zk i = i k(x k ) [ Ck i (2) x k 2X k Assuming that we have Q sensors, the RFS of measurements at time k is modelled by h i Z k = Zk; 1 Zk; 2 : : : ; Z Q k (3) The RFS Z k encapsulates all sensor characteristics such as measurement noise, sensor eld of view, clutter. The multi-sensor multi-object tracking can be posed as follows: given set of measurement Z 1:k collected from sensors up to time k, the problem is to nd ^X k is expectation or maximization of the posterior density function p(X k jZ 1:k ). III. PROBABILITY HYPOTHESIS DENSITY APPROACH In multiple object tracking problem, we usually need to obtain the posterior density p(X k jZ 1:k ). When the number of object increases, the multiple object state space become large. Hence, it is difcult to obtain the posterior density function. Fortunately, this density function can be approximately recovered from the probability hypothesis density (PHD) [9]. The PHD is dened as follows. For a random nite set X on X with probability distribution P , the PHD is the density v(x) such that for each region S X , the integral of v over region S gives the expected number of elements of X that are in S, Z Z j X \ S j P (dX) = v(x)dx; (4) Thus, instead of estimating states of objects from posterior density, we can estimate them by investigating peaks of PHD. It helps to reduce from searching in multiple object state space to single object state space. IV. GAUSSIAN MIXTURE PROBABILITY HYPOTHESIS DENSITY FILTER IN MULTI-SENSOR MULTI-OBJECT A. Assumptions TRACKING First, there are some assumptions. The transition function of each object follows a linear Gaussian model, i.e., f kjk 1 (xj) = N(x; F k 1 ; Q k 1 ) (5) where N(:; m; P ) denotes a Gaussian density with mean m and covariance P , F k 1 is the state transition matrix, Q k 1 is the process noise covariance. There are Q sensors, the likelihood function at each sensor is also a linear Gaussian model, i.e., g i k(zjx) = N(z; H i kx; R i k) (6) where H i k is the observation matrix of the sensor i, and Ri k is the observation noise covariance of the sensor i. The survival and detection probabilities are S p S;k (x) = p S;k (7) p D;k (x) = p D;k (8) The intensity of the spontaneous birth RFS is J ;k X k (x) = i=1 w (i) ;kN(x; m(i) ;k ; P (i) ;k ) (9) where J ;k is the number of birth Gaussian components at time k, and w (i) ;k is the weight for i-th Gaussian component. The posterior intensity at time k 1 is a Gaussian mixture of the form v k 1 (x) = J X k 1 i=1 w (i) k 1 N(x; m(i) k 1 ; P (i) k 1 ) (10) where J k 1 is the number of Gaussian components of posterior intensity at time k 1, and w (i) k 1 is the weight for i-th Gaussian component. B. GMPHD lter with one sensor Vo [12] proposed a closed form expression of the PHD lter for linear Gaussian multi-object tracking, called the Gaussian mixture probability hypothesis density lter. Under assumptions in IV-A, the initial prior intensity is a Gaussian mixture, the posterior intensity at any subsequent time step is
1 (x) with measurement set Z1 k by equation (13) to obtain also a Gaussian mixture. In the case there is one sensor, this the PHD at time k sensor 1, vk 1(x). Because v k 1(x) is a method can be employed. Gaussian mixture, vk 1 (x) is also a Gaussian mixture and has In what follows, we outline the GMPHD lter with assumptions the form vkjk 1 Jk 1 = (J k 1 + J ;k )(1 + jZkj) 1 (20) there are no spawning objects. (In the case there JkX 1 are spawning objects, the prediction equation is modied vk(x) 1 = w (i) 1;kN(x; m(i) 1;k ; P (i) 1;k ) (14) by adding Gaussian components representing for spawning i=1 objects. The details is in [12]). Now, at the sensor 2, we use vk 1 (x) as the predicted PHD for Under assumptions in IV-A, the predicted intensity to time the sensor 2 and in the similar way to (13), we have k is given by v v kjk 1 (x) = v S;kjk 1 (x) + k (x) (11) k(x) 2 = (1 p D;k )vk(x) 1 + X v D;k (x; z) (15) z2Zk 2 where So, v J X k 1 k 2 (x) also have the Gaussian mixture form. v S;kjk 1 (x) = p S;k w (j) k 1N(x; m(j) S;kjk 1 ; P (j) S;kjk 1 ); JkX 2 j=1 vk(x) 2 = w (i) 2;kN(x; m(i) 2;k ; P (i) 2;k ) (16) m (j) S;kjk 1 = F k 1 m (j) k 1 ; i=1 P (j) S;kjk 1 = Q k 1 + F k 1 P (j) k 1 F k T We repeat this process with Q sensors. At the Qth sensor, we 1: obtained v Q k (x), and it has the form Because v S;kjk 1 (x) and k (x) are Gaussian mixtures, J v kjk 1 (x) can be expressed as a Gaussian mixture of the form kX Q v Q k (x) = w (i) Q;kN(x; m(i) Q;k ; P (i) Q;k ) (17) J kjk X 1 v kjk 1 (x) = w (i) kjk 1N(x; m(i) kjk 1 ; P (i) i=1 kjk 1 ) (12) The PHD for the multi-sensor multi-object posterior density i=1 will be Then, the posterior intensity at time k is also a Gaussian v k (x) = v Q k (x) (18) mixture, and is given by v k (x) = (1 p D;k )v kjk 1 (x) + X The number of objects is estimated by v D;k (x; z) (13) Z z2Z k ^N kjk = v k (x)dx where J kjk X 1 Z JkX Q v D;k (x; z) = w (j) k (z)N(x; m(j) kjk ; P (j) kjk ); = w (i) Q;kN(x; m(i) Q;k ; P (i) Q;k )dx j=1 i=1 w (j) k (z) = p D;k w (j) kjk 1 q(j) k (z) JkX Q P = w (i) Jkjk k (z) + p 1 D;k l=1 w (l) kjk 1 q(l) k (z); Q;k i=1 (19) q (j) k (z) = N(z; H km (j) kjk 1 ; R k + H k P (j) kjk 1 HT k ); So, the properties of the Gaussian mixture in the case of m (j) kjk = m (j) kjk 1 + K(j) k (z H km (j) kjk 1 ); multi-sensor is similar with one sensor case. This means in the multi-sensor multi-object tracking problem, under assumptions P (j) kjk = [I K (j) k H k]P (j) kjk 1 ; in IV-A, the initial prior intensity of multi-sensor multi-object tracking is a Gaussian mixture, the posterior intensity for K (j) k = P (j) kjk 1 HT k (H k P (j) kjk 1 HT k + R k ) 1 : asynchronous sensor fusion method at any subsequent time C. GMPHD lter with multi-sensor step is also a Gaussian mixture. When there are many sensors, we propose a method to D. Implement issues solve by using GMPHD lter sequentially at each sensor. The The state estimations of objects are the means of Gaussian algorithm is described as follows. components that have high weights (above 0.5) in v With assumptions in IV-A, at time k 1 we have k (x). This estimation method is more efcient than particle PHD lter. J X k 1 Because in particle PHD lter, we obtain the number of objects v k 1 (x) = m (i) k 1N(x; w(i) k 1 ; P (i) k 1 ) ^N kjk then partition particles into ^N kjk clusters. If ^N kjk is not i=1 First, we used assumptions on state equation (5), measurement corrected, then the tracking performance will be affected. Now, we investigate the number of Gaussian components in equation (6) and v k 1 (x) to predict the intensity vkjk 1 1 (x) v k (x). At the rst sensor, the number of Gaussian components at the sensor 1 by using the equation (11). Then we update is
Page 1: Multiple Sensor Multiple Object Tra
Page 5 and 6: delay of arrival measurement (TDOA)
Page 7: Fig. 9. Probability of correct spea

Multiple Sensor Multiple Object Tracking With GMPHD Filter - ISIF

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