Gaussian Mixture Cardinalized PHD Filter for Ground Moving Target Tracking M. Ulmke FGAN - FKIE Neuenahrer Str. 20 D-53343 Wachtberg, Germany
[email protected]
Ozgur Erdinc ECE Department Univ. of Connecticut Storrs, CT 06269-2157, USA
[email protected]
Abstract - The cardinalized probability hypothesis density (CPHD) filter is a recursive Bayesian algorithm for estimating multiple target states with varying target number in clutter. In particular, the Gaussian mixture variant (GMCPHD) for linear, Gaussian systems is a candidate for real time multi target tracking. The present work addresses the following three issues: (i) we show the equivalence between the GMCPHD filter and the standard Multi Hypothesis Tracker (MHT) in the case of single targets; (ii) using a Gaussian sum approach, we extend the GMCPHD filter by employing digital road maps for road constraint targets. The utilization of such external information leads to more precise tracks and faster and more reliable target number estimates; (iii) we model the effect of Doppler blindness by a target state dependent detection probability, leading to more stable target number estimation in the case of low Doppler targets.
Keywords: Probability hypothesis density filter, PHD, multi-target tracking, cardinalized probability hypothesis density filter, CPHD, road targets, ground moving target indicator radar (GMTI)
I. I NTRODUCTION The Probability Hypothesis Density filter for tracking multiple targets in clutter [1] has received a lot of attention recently (see, e. g. [2]–[7]), for it avoids the explicit enumeration of all possible multi target– multi detection assignments that leads to the so-called combinatorial disaster. In numerical application (e. g. particle filters) it indeed beats the curse of dimensionality because the dimension of the PHD problem is equal to the dimension of the single target state. An important step towards practical applicability was made by the Gaussian mixture version of the PHD [6], [8] for linear Gaussian target models. A disadvantage of the PHD, however, is due to the Poisson assumption for the target number (cardinality) distribution that often leads to strong fluctuations in the target number estimators and to the missed detection problem [5]. This problem has been cured, recently, by the so–called Cardinalized PHD, CPHD [9], [10]. A more intuitive “physical state” derivation of the PHD and CPHD filter has been presented in [11]. For the CPHD filter, too, recently a Gaussian mixture variant has been developed, the GMCPHD filter [12], [13].
Peter Willett ECE Department Univ. of Connecticut Storrs, CT 06269-2157, USA
[email protected]
In the present paper we apply the GMCPHD filter to the problem of tracking groups of ground moving targets. In particular, we focus on the issue of integrating digital roadmaps into the algorithm and the problem of modeling the clutter notch of GMTI radar sensors (GMTI: ground moving target indicator) for low Doppler targets. The paper is organized as follows: In Section II we recapitulate the Gaussian mixture cardinalized probability hypothesis density (GMCPHD) filter equation [9]–[12]. In the situation of at most one target, we show in Section III the equivalence to the single target multi hypothesis tracking (MHT) filter [14], [15] with sequential likelihood ratio test for track extraction. The GMCPHD filter is extended by introducing digital roadmaps in Section IV and by modeling the Doppler blindness (clutter notch) in Section V. After briefly discussing some implementation issues in Section VI, we present simulation results for groups of road targets in Section VIII. II. GMCPHD
FILTER
In the Gaussian mixture variant of the CPHD, the PHD at scan k, vk|k (x), is described as the weighted sum of Jk|k normal distributions: Jk|k
vk|k (x) =
X
(j)
(j)
(j)
wk|k N x; mk|k , Pk
j=1
(j)
(j)
(1)
with mk|k and Pk|k being the mean and covariance matrix of component j. The estimated target number is given by the integral of v(x): nk|k =
Z
Jk|k
vk|k (x) dx =
X
(j)
wk|k .
(2)
j=1
In the CPHD, parallel to the PHD, the cardinality distribution pk|k (n) = p(n|Zk ) is estimated iteratively. Equivalently to (2), the target number can also be calculated by: nk|k =
∞ X
n pk|k (n)
(3)
n=1
The prediction and update step for each individual component is performed exactly as in Kalman or MHT filtering supplement by so-called birth and death processes.
Prediction: given the PHD for scan k − 1, the probability of target “death”, d, the target dynamics matrix Fk , and process noise covariance Dk , the predicted components are: (j)
wk|k−1 = (1 − d)wk−1|k−1 (j)
(4)
(j)
mk|k−1 = Fk mk−1|k−1 (j) Pk|k−1
=
+ Dk
n′ =0
min{n,n′ }
′ X ′ n ′ M(n, n ) = pbirth (n − i) dn −i (1 − d)i i
(8)
i=0
pbirth (n) is the probability for n new targets to appear between scan k − 1 and k, derived from the birth model. For time independent pbirth (n), e. g. for a constant scan rate, the matrix M is constant and can be calculated in advance. Update: The general CPHD update equation for the PHD and the cardinality distribution can be written in terms of likelihood ratios [11]: L(Zk |¬D) L(Zk |D) vk|k (x) = (1−Pd) + Pd vk|k−1 (x) (9) L(Zk ) L(Zk ) L(Zk |n) pk|k (n) = pk|k−1 (n) L(Zk )
(10)
where the conditions D and ¬D are short hand notations for target detected and not detected, respectively. The likelihood ratios above are given by: 1 nk|k−1
(j+1)
αk
L(Zk |D) =
(1)
(mk )
=
nk|k−1
(j)
L(Zk ) =
j=0
(j) αk
(16)
(mk − j)! −j λ mk !
(17) (s)
with the single detection likelihood function l(zk |x). pc (m) denotes the probability for m false alarms, where a homogeneous false alarm (clutter) density is assumed for simplicity. The present context of Gaussian mixtures can be preserved in the case of linear measurements with Gaussian measurement errors wk with (potentially detection dependent) covariance (s) (s) (s) Rk : zk = Hk x + wk which results in a normally distributed likelihood function: (s) (s) (18) lk (z|x) = N z; Hk x, Rk
Inserting the Gaussian mixture (1) into (9), we obtain for each original component mk + 1 new components, one from the missed detection part, (1 − Pd ), denoted by s = 0, and one (s) for each detection zk , s = 1, . . . , mk , according to (12). After updating, the total number of components therefore is Jk|k = (mk + 1)Jk|k−1 . In general, the probability of detection, Pd , may depend on the target state, which could be modeled by a component (j) dependent Pd . (See Section V for a more refined treatment.) In that case, we also need the weighted average Pd,k−1 ≡
Jk|k−1
1 nk|k−1
X
(j)
(j)
wk|k−1 Pd
(19)
j=1
which replaces Pd in (15) and (16), see [11]. For a missed detection, the updated component mean and covariance are equal to the predicted ones. Assigning the true (j) detection s to component j, denoted by ak = s, the update follows the usual Kalman filter equations: (j,s)
(j)
(j,s)
mk|k = mk|k−1 + Wk = [1 −
(j,s) Wk
(s)
(j)
(zk − Hk mk|k−1 )
(20)
(j) Hk ] Pk|k−1
(21)
(j,s)
(12)
(j,s) Sk
(j)
(j,s) −1
= Pk|k−1 H⊤ k (Sk =
(s) Rk
+
)
(j) Hk Pk|k−1
(22)
H⊤ k
(23) (j,s)
(j) βk
(j) αk ×
(13)
j=1
(1) (m ) σj−1 ({Lk , . . . , Lk k } mk X
n! pk|k−1 (n)(1 − Pd )n−j (n − j)!
βk ≡ pc (m − j)
Wk
(j)
mk X
(15)
(1) (m ) σj ({Lk , . . . , Lk k })
with
})
l(zk |x) L(Zk |ak = s) 1
∞ X
n=j
j=1
(j) L(Zk |ak = s)
n! (1 − Pd )n−j × (n − j)!
(11)
j=0
(s)
(j)
αk ≡
(j)
σj ({Lk , . . . , Lk mk X
βk
with
(j,s) Pk|k
βk ×
(j)
X j=0
(6)
Additional components are introduced by a birth model which describes potentially new targets appearing in the field of view of the sensor. A simple realization of a birth model is to introduce at each scan one test component with a certain weight wbirth , positioned in the middle of the field of view with zero velocity, and a large covariance Pbirth covering the whole field of view of the sensor. The number of components after prediction, denoted by Jk|k−1 , is equal to Jk−1|k−1 plus the number of new born components. The prediction equation for the cardinality distribution can be written in terms of a “transfer matrix” M: ∞ X pk|k−1 (n) = pk−1|k−1 (n′ ) M(n, n′ ) (7)
L(Zk |¬D) =
L(Zk |n) =
(5)
(j) Fk Pk−1|k−1 F⊤ k
mk X
min{m,n}
\
(j,s=0)
(s) {Lk })
(j) (1) (m ) βk σj ({Lk , . . . , Lk k })
Gk (14)
(j,s)
The component weights are updated by wk|k = Gk where the updating factors are given by:
(j,s>0) Gk
=
L(Zk |¬D) (j) (1 − Pd ) L(Zk ) (j)
(j)
(j)
wk|k−1
P L(Zk |ak = s) (s) (j) (j,s) = d N zk ; zk , Sk λ L(Zk )
(24)
(j)
(j)
with clutter density λ, predicted detection zk = Hk mk|k−1 , (j,s) Sk ,
and innovation covariance see (23). Remark: the above update equations for weights, estimates, and covariances are obtained by using (1) and (18) and applying the product formula for normal densities: N (x; Xy, Y)N (y; z, Z) = N (x; a, A) N (y; b, B) (25) with a = Xz, b = z + W(x − Xz), (26) A = Y + XZX⊤ , B = (1 − WX)Z, ⊤
−1
W = ZX A
=
nk|k−1
where π(n) = p0|0 (n) are the prior probabilities for n targets. Assuming, for simplicity, equal priors π(0) = π(1), the criterion reads: Rk > Rc = nc /(1 − nc ). Applying (10) iteratively, the likelihood ratio can be written as:
(27) Rk =
(28)
where x, y and z denote vectors, and X, Y and Z corresponding matrices. The likelihood ratios above are functions of the weighted single detection likelihoods: Z 1 (s) (s) (s) Pd (x) vk|k−1 (x) lk (zk |x) dx (29) Lk = nk|k−1 1
equivalent to a threshold on the likelihood ratio given target number one and zero: p(Zk |1) π(0) p(1|Zk ) π(0) nk|k Rk ≡ = = (32) p(Zk |0) π(1) p(0|Zk ) π(1) 1 − nk|k
(j) (j) Pd wk|k−1
N
(s) zk ;
(j) zk ,
j=1
(j,s) Sk
In MHT too, the above likelihood ratio serves as the criterion that a track is extracted [14], [15]. For single target MHT, the likelihood ratio can be shown to be equal to the sum of (j) the unnormalized hypothesis weights w ˜k , provided the initial hypothesis weights sum up to one [16]: Jk|k
1≤i1 1), all coefficients αk , see (16), vanish for j > 1, and the GMCPHD filter equations are strongly simplified: (0)
αk = 1 − Pd,k−1 nk|k ;
(1)
αk = nk|k p (m) c (0) (1) βk = pc (m) ; βk = λV (j) L(Zk |ak = 0) = pc (m) pc (m) (s) (j,s) (j) L(Zk |ak = s) = N zk ; Hk xk , Sk λ L(Zk ) = [1 − (Pd,k−1 − Qk ) nk−1 ] pc (m) L(Zk |n = 0) = pc (m) L(Zk |n = 1) = [1 − Pd,k−1 + Qk ] pc (m) mk 1X (s) Qk = L λ s=1 k
(35) (36) (37) (38) (39) (40) (41) (42)
(s)
with the single detection likelihood Lk given in (29) and the size of the field of view of the sensor V . Inserting the above into (24) and (10) we obtain the updated component weights and expected target number: (j)
wk−1 1 − (Pd,k−1 − Qk ) nk−1 (s) (j) (j,s) (j) N zk ; zk , Sk Pd (j,s) (j) wk = w λ 1 − (Pd,k−1 − Qk ) nk−1 k−1 1 − Pd,k−1 − Qk nk = nk−1 1 − (Pd,k−1 − Qk ) nk−1 (j,0)
wk
(j)
= (1 − Pd )
(43) (44) (45)
We note that without any detection (Qk = 0) the updated probability of target existence becomes: pk (1) = nk = nk−1
1 − Pd,k−1 1 − Pd,k−1 nk−1
(46)
avoiding the PHD’s missed detection problem [5]. Inserting (40) and (41), into (33), the likelihood ratio finally reads: Rk =
k Y
[1 − Pd,l−1 + Ql ]
(47)
l=1
In MHT, the corresponding updated weights are iteratively calculated by: (j,0)
w ˜k
(j,s)
w ˜k
(j)
(j)
= (1 − Pd ) w ˜k−1 =
(j) Pd
(s)
(j)
(48) (j,s)
N zk ; zk , Sk
(j)
w ˜k−1 (49) λ Apparently, the weight updating factors are equal to those in GMCPHD, see (43) and (44), besides a scan dependent, but component independent, factor. If we start, at k = 0, with equal relative component weights in MHT and GMCPHD, respectively, they will remain equal scan by scan besides a constant: P (j) ˜ ˜k jw (j) Rk (j) (j) (50) w ˜k = wk P (j) = wk nk w j
k
In other words, component mean, covariances, and relative weights all are identical in single target MHT and GMCPHD. How about likelihood ratios? Inserting the update equations (48) and (49) into (34), and using (50), the MHT likelihood ratio can be written as:
difficulty, in particular in the case of curved roads, lies in the fact that the target motion on road is highly nonlinear in cartesian or sensor coordinates. A flexible approach, suited for winding roads, is given in [17]. There, the target dynamics is modeled in road coordinates, i. e. the target state is described ˙ ⊤ parametrized by ˜ = (l, l) by a two-dimensional vector x the arc length l on road (or mileage), and the speed l.˙ In most areas, road information is provided in relatively high precision in digital vector maps. Typically, a curved road is approximated by a set of linear road segments i = 1, . . . , Nseg , defined by their starting position, si , normalized direction ti , length li and, potentially, certain attributes like road type etc. These linear segments allow for a linear mapping between road and cartesian coordinates. Each road segment i has an uncertainty due to discretization and map errors. These uncertainties are modeled by a Gaussian noise term in the mapping from road to cartesian coordinates, quantified by a covariance Ri . ˜ , i) = N x; tig←r [˜ p(x | x x], Ri (53) using the affine transformation: tig←r [x] = Tig←r x + sig←r with si − li ti ti 0 . (54) Tig←r = , sig←r = 0 0 ti As in Section II, the predicted PHD in road coordinates is written as a Gaussian mixture: Jk|k−1
vk|k−1 (˜ x) =
˜k = R
(1 −
(j) (j) Pd ) w ˜k−1
(51)
j=1
+
Jk−1 mk 1 X (j) X (s) (j) (j,s) (j) Pd N zk ; zk , Sk w ˜k−1 λ j=1 s=1
˜ k−1 [1 − Pd,k−1 + Qk ] =R
(j) ˜ (j) ˜; m ˜ k|k−1 , P pj (˜ x |Zk−1 ) = N x k|k−1
IV. ROAD
pj (˜ x |Z
)≈
Nseg X
(i,j) (i,j) ˜ (i,j) ˜; m ˜ k|k−1 , P fk|k−1 N x k|k−1
i=1
(52)
(56)
(i,j)
(57)
where the normalized weights fk|k−1 correspond to the probability for a target living on road segment i. The calculation of the road segment dependent weight, mean, and covariance (i,j) is given in [17]. Typically, fk|k−1 is sharply peaked about a segment imax (depending on j) and negligable for segments far away from imax . Using (53) and (57), the density in cartesian coordinates, too, is described by a Gaussian mixture over road segments. pj (x |Zk−1 ) =
Nseg X i=1
MAP INTEGRATION
Ground traffic usually is bound to roads. Even military vehicles, in particular convoys, prefer to use roads whenever possible while off-road traffic is the exception. There are different approaches in the literature on how to integrate road map constraints into a Bayesian tracking algorithm. A central
(55)
Each component j of the predicted PHD in road can be decomposed into a Gaussian mixture over road segments: k−1
˜ k obeys the same with Qk given in (42). Apparently, R recursion relation as Rk , see (47). Since, by construction (34), ˜ 0 = 1 and, by definition (32), R0 = 1, it is clear that both R likelihood ratios are identical scan by scan. To summarize: neglecting birth and death processes (which could be incorporated into a MHT algorithm, too), single target MHT and GMCPHD for n ≤ 1, including track extraction by sequential likelihood ratio testing, are identical algorithms. Mixture component reduction by pruning and merging can be performed in the same way within GMCPHD and MHT.
(j)
wk|k−1 pj (˜ x |Zk−1 )
j=1
Jk−1
X
X
(i,j)
(i,j)
(i,j)
(i,j)
fk|k−1 N x; mk
(i,j)
, Pk
(58)
mk and Pk are calculated from their counterparts in road coordinates by Kalman filter like equations, see [17]. In the update step, each component (i, j) is assigned to each detection s and updated according to (20), (21), and (24). The weight update factor depends on road segment i, PHD
component j, and detection s, and is denoted by gk (i, j, s). The new normalized road segment weights are given by: (i,j,s)
fk|k
(j,s) Gk
= =
(i,j,s) (i,j) gk fk|k−1 (j,s) Gk Nseg X (i,j,s) (i,j) gk fk|k−1 i=1
(59)
(60)
(j,s)
The sum Gk plays the role of the update factor for the (j,s) (j,s) (j) component weight (cmp. (24)): wk|k = Gk wk|k−1 , which are, of course, not normalized. For the calculation of gk (i, j, s), the weighted single detection likelihood functions are required, see (30): (s)
Lk =
1 nk|k−1
Jk|k−1
X
(j)
(j)
Pd wk|k−1 . . .
Prediction continuous road
(j) ˜ k|k−1 m (j) P ˜ k|k−1 (j) wk|k−1
(a) −→
(f ) ↑ next k
(j,s) ˜ k|k m (j,s) P ˜ k|k (j) wk|k
(e) ← −
road segments (i,j) ˜ k|k−1 m ˜ (i,j) Pk|k−1 w(j) k|k−1 (i,j,s) fk|k−1 (i,j,s) ˜ k|k m ˜ (i,j,s) Pk|k w(j) k|k (i,j,s) fk|k
(b) − →
Update cartesian (i,j) mk|k−1 (i,j) Pk|k−1 w(j) k|k−1 (i,j,s) fk|k−1 ↓ (c)
(d) ←−
(i,j,s)
mk|k
(i,j,s) Pk|k w(j) k|k (i,j,s) fk|k
Fig. 1. Schema of the transfomation between road and cartesian coordinates.
j=1
Nseg X
(i,j)
(s)
(i,j)
fk|k−1 N zk ; zk
i=1
(i,j,s)
, Sk
(61)
The transformation segment wise back to road coordinates is a simple projection onto each road segment (see [17]). The result, again, is a Gaussian mixture over road segments. This mixture, finally, can well be approximated by a single normal distribution in road coordinates using second order moment matching. Jk|k
vk|k (˜ x) ≈
X
(j) (j) ˜ (j) ˜; m ˜ k|k , P wk|k N x k|k
j=1
(62)
which now can be predicted to the next time step. The algorithm is summarized in Fig. 1: each PHD component itself is decomposed into a Gaussian mixture over road segments (a). For each segment, the mapping to cartesian coordinates is linear, preserving the Gaussian mixture shape of the density (b). The filter update (c) leads to new estimates and new road segment weights, their weighted average is the update factor for the component weight. Each PHD component is mapped back to road segments (d), and before the next prediction step (f), the Gaussian mixture over road segments is approximated by a single Gaussian in continuous road coordinates (e). Other treatments of the Gaussian mixture, avoiding this approximation, are possible, but for realistic model parameters, we have found the second order moment matching approximation to be in excellent agreement with more refined methods [17]. The price of the road segment approach is an additional computational load proportional to the number of relevant road segments which is of the order of 3-6 for typical parameter values. V. D OPPLER BLINDNESS
AND TERRAIN OBSCURATION
As mentioned in Section II, the detection probability Pd , in general, is a function of target state. In particular for GMTI radar, due to the clutter notch of the sensor, Pd sharply drops
to zero when the relative range rate between target and sensor, nc ≡ |r| ˙ =
(˙r − r˙ s )(r − rs ) ||r − rs ||
(63)
falls below a certain threshold, the minimum detectable velocity, vm . In addition, ground targets often become invisible due to terrain obscuration, e. g. by hills, which also depends on the sensor–target geometry. Within the Gaussian mixture framework, such a state dependent Pd (x) can be approximated (j) by a component dependent Pd obtained by averaging Pd (x) over the component distribution, or simply by the value of Pd at the estimated target state: (j) Pd (j)
or Pd
≈
Z
(j) (j) Pd (x) N x; mk|k−1 , Pk|k−1 dx (j)
≈ Pd (mk|k−1 )
(64) (65)
While (64), in general, is difficult to calculate, (65) is a rather crude approximation when the target state covariances increases. In the case of terrain obscuration, the situation is easier for road targets where Pd (x) may well be approximated by its value in the center of each road segment, Pd ((si + si+1 )/2) (i = 1, . . . , Nseg ). For modeling the clutter notch, there is an alternative approach [18], [19] that preserves the Gaussian mixture structure of the PHD. The target sate dependent detection probability is approximated by: Pd (x) = Pd 1 − cm N nc (x); 0, Qm .
(66)
q 2 vm with cm = vm logπ 2 and Qm = 2 log 2 . Far away from the clutter notch, Pd (x) approaches the saturated value Pd , decreases by a factor of two for nc = vm , and vanishes exactly at nc = 0. By linearizing nc (x) around its predicted value
(j)
nc (mk|k−1 ) (for each component j), we obtain: h i (j) (j) (j) Pd (x) ≈ Pd 1 − cm N uk ; Kk x, Qm (j)
(j)
(j)
(j)
with uk = nc (mk|k−1 ) + Kk mk|k−1 ,
(67) (68)
∂nc | . (69) (j) ∂x x=mk|k−1 If we insert (67) into the filter update equation (9), we immediately see that each new component now is split into two. That means, besides the (mk +1)Jk|k−1 components with estimates (j,s) (j,s) (j,s) mk|k (20), covariances Pk|k (21), and weights wk|k (24) we obtain the same number of additional components which are calculated using the product formula (25): (j,s) (j,s) (j) (j) (j,s) ˆ(j,s) w ˆk|k = a(s) (70) m wk|k N uk ; Kk mk|k , Sk (j)
Kk = −
(j,s) (j,s) ˆ (j,s) (u(j) − K(j) m(j,s) ) ˆ k|k = mk|k + W m k k k k|k
(71)
ˆ (j,s) = [1 − W ˆ (j,s) K(j) ] P(j,s) P k k k|k k|k
(72)
with ˆ (j,s) = P(j,s) (K(j) )⊤ (S ˆ(j,s) )−1 W k k k k|k ˆ(j,s) S k
= Qm +
(j) Kk
(j,s) Pk|k
(73)
(j) (Kk )⊤
(74)
where s = 0 stands for the predicted quantities, i. e. (j,s=0) (j) mk|k = mk|k−1 , etc. In this way, the clutter notch serves as a “ficticious” measurement, leading to additional components in the Gaussian mixture. The prefactors in (70) are: ( cm Pd , s=0 (s) am = 1−Pd (75) −cm , s > 0 (0) am ,
Physically, the missed detection hypothesis, puts extra weight into the direction of the clutter notch, in particular (s>0) for large Pd . The detection hypothesis, am , on the other hand, removes weight out of the clutter notch. Although the prefactors and, hence, the weights for s > 0 are negative, there (j,s) (j,s) is no problem in consistency since the sum wk|k + w ˆk|k ≥ 0. In particular, the estimated target number is always positive. VI. I MPLEMENTATION ISSUES Numerically, the cardinality distribution has to be limited to a finite number of terms. Since we do not know the number of targets a priori, a maximum number of targets Nmax is chosen which has to be larger than the expected maximum target number at any time. The number of components Jk|k increases in every time step approximately by a factor equal to the number of measurements. On the other hand, only a limited number of components is really needed to approximate the PHD. Typically, we chose the maximum component number as 10 times Nmax . To reduce the number of components, we apply a heuristic ’pruning’ procedure proposed in [6], [8]. by discarding components with negligable weights and by merging components with overlapping maxima. In constrast to [6], [8], we do not assume to have prior knowledge on
the location of targets sources and drains. For the detection of new targets we introduce a “test component” with small weight, location in the center of the field of view, zero speed and a large covariance given by the size of the field of view and the maximum velocity. For roads targets, it is positioned somewhere on the road with covariance given by the length of the road and the maximum speed. This test component suffices for a quick detection of one or more new appearing target(s). For plotting the history of a track, the actual track (j, s) is connected with its predecessor (j). In the case of merged tracks, the track with the largest weight is appointed predecessor. VII. TARGET
AND
S ENSOR MODELING
We consider a two-dimensional target state at time tk : xk = (x, y, x, ˙ y) ˙ ⊤.
(76)
In road coordinates, the target state is parametrized by the arc ˙ ⊤. ˜ k = (l, l) length and velocity on road, as described above: x For the filtering, the target dynamic is modeled by a linear Markov process: ˜ k+1 = Fk+1|k x ˜ k + Gk+1|k vk+1 x
(77)
We follow the realization given in [16], where the matrices Fk+1|k and Gk+1|k are given by 1 tk+1|k 0 Fk+1|k = , Gk+1|k = (78) Σk+1|k 0 e−tk+1|k /θt with p time difference tk+1|k = tk+1 − tk and Σk+1|k = vt 1 − e−2tk+1|k /θt . For normal, bias free process noise vk+1 , it can be shown that the modeled target velocity is ergodic and given by |tk − tm | 2 ˙ ˙ ˙ E[lk ] = 0, E[lk lm ] = vt exp − (79) θt For off-road targets, the corresponding four-dimensional version of (78) is used. The parameters vt and θt have to be chosen according to the expected dynamics of the targets under consideration. vt limits the velocity standard deviation with vt = 20m/s being a typical value for road targets. The socalled maneuver correlation time θt describes the agility of the targets and typically is chosen between 30 and 90 s. The matching of the parameter values with the true target behavior influences the quality of the track estimates. The measurement, too, is assumed to be a linear function of the target state: zk = Hk xk + wk ,
(80)
with Gaussian, bias free measurement noise wk with covariance Rk . In the case of azimuth ϕk and range rk measurements with independent noise variances σϕ and σr , the measurements can be transformed to cartesian (x, y) coordinates by a rotation about the azimuth angle. Then, the measurement
140 Scans, 8 targets, P =0.8, no road−map information
matrix is strongly simplified, but the noise covariance becomes target state dependent: 1 0 0 0 Hk = , R k ≈ Qk D Q⊤ (81) k 0 1 0 0 Qk =
cos ϕk − sin ϕk
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For simplicity, we consider a static sensor platform. The Clutter notch of the sensor is modeled by a step function: Pd = 0 for radial velocity |r˙k | < vm , and Pd = const. 6= 0, otherwise. VIII. S IMULATION
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In the following we present first results of the implementation of the GMCPHD algorithm for ground moving targets, as described above. Fig. 2 depicts a snapshot of tracks of two group targets, with four single targets each. The groups move at constant speed of 12 m/s and enter the crossing just one after the other. Although the tracks of individual components sometime disappear and reappear, the groups are continuously tracked, and the correct target number is estimated, even without using road-maps. Scan−no.: 78;
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Fig. 3. Scenario and tracks for a group of 8 road targets. Above: without, below: with road-map information. N=8, PD=0.8, λ=20, d=0.01, pB=0.01
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Fig. 2. Snapshot of tracks of two groups with four targets each; crossing roads. (Axis scale in meters.)
Fig. 3 shows the gain in track precision and smoothness if road-map information is used for tracking, again for 8 targets, Pd = 0.8, and 20 false alarms on average. Typical cardinality estimates are provided in Fig. 4 for the same parameters. The sum of weights (2) and the average of the cardinality distribution (3) provide numerically identical results, which serves as a code check. The maximum of the cardinality distribution is an integer fluctuating between 7 and the true value 8. Target number estimation in the case of changing (visible) target numbers is shown in Fig. 5, sampled over 100 Monte Carlo runs. In this scenario, there are three targets on the same road as in Fig. 3. First they are stopping and, hence, undetectable, then one target after the other starts moving along the road. After 60 scans, i. e. 10 min. one after the other stops, and 34 scans later they start to move on, again. The observable target number, therefore, changes between zero
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and three. The CPHD estimates follow the observable target number, however track extraction is faster, typically by 5-7 scans, if road-map information is taken into account. The effect of Doppler blindness is considered in the snapshots, Fig. 6, where the tracks of a 7-target group are plotted at different scan numbers. First, the group moves mostly in radial direction to the sensor, (positions a) and b)), implying a high Pd . The target number is correctly estimated and even the structure of the group is resolved. Later, at positions c) and d), the group rather moves in cross-range direction, implying a low Pd . Taking, the clutter notch into account, the target number estimate is still in good agreement with the true target number,
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reacts faster to changing target numbers. The problem of Doppler blindness of GMTI radar sensors is treated by modeling the target state dependent detection probability. It leads to more stable tracks and a better target number estimation for low Doppler targets. Finally, we have proven that in the single target case (N ∈ {0, 1}), the GMCPHD filter is equivalent to the multihypothesis tracker including a sequential likelihood test for track extraction. In general, the GMCPHD filter, therefore, can be considered as an extension of single target MHT to the multitarget situation. Acknowledgement: We thank H. B¨os for assistance with the implementation of the GMCPHD algorithm.
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Fig. 5. Estimated target number without (above) and with (below) road-map information. Averages and standard deviation from 100 Monte Carlo runs. True detectable target number for comparison.
although the individual targets cannot be resolved anymore.
Fig. 6. Overlay of four snapshots of tracks of a group with seven targets (see text).
IX. C ONCLUSION We have applied the Gaussian mixture version of the Cardinalized Probability Hypothesis Density (GMCPHD) filter to the problem of tracking ground moving targets. A Gaussian sum approach is used to integrate digital road-maps into the Bayesian algorithm. The approach takes road-map and discretization error explicitly into account. Using road-map information, the tracks are not only smoother, but the tracker
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