Multitarget tracking with the IMM and Bayesian networks: Empirical studies Sampsa K. Hautaniemi, Jukka P.P. Saarinen Digital and Computer Systems Laboratory Tampere University of Technology P.O. Box 553 33101 Tampere Finland Email:
[email protected] Tel: +358 3 3653878 Fax: +358 3 3653095
ABSTRACT This paper concentrates on multitarget tracking (MTT) simulation. The purpose of this paper is to simulate 11 targets in the noisy environment. The sensors used in the simulations are passive. First, we use the interactive multiple model (IMM) algorithm with the probabilistic data association (PDA) algorithm. The PDA is not capable to process attribute observations (i.e. observations of features such as the form of wings, radio frequency etc.). Therefore we have applied Bayesian networks to our tracking system, since they are capable to process attribute observations. The main gain of using the Bayesian networks is that the type of the target is possible to determine. In this paper we briefly recapitulate the most important features of the IMM, PDA and Bayesian networks. We also discuss how the establish attribute association probabilities, which are possible to fuse with the association probabilities computed by the PDA. We have executed the simulations 30 times. In this study we show one typical example of tracking with IMM and PDA as well as tracking with IMM, PDA and Bayesian networks. We conclude that tracking results with IMM and PDA are quite satisfactory. Tracking using the Bayesian networks produces slightly better results and identified the targets correct. Keywords: Bayesian networks, multitarget multisensor tracking, type identification.
1. INTRODUCTION Tracking is a complex procedure containing many algorithms. In this paper we use the concept “tracking system” to refer all blocks of the system. Therefore, a tracking system contains blocks such as track initialisation/deletion, track updating, type identification etc. In this study we simulate a tracking system, which is able to handle the attributes, i.e. type depended features. We describe theories of the algorithms briefly. The purpose of a tracking system is to identify targets and then to estimate where the target is heading. Unlike in the civilian tracking, in the military tracking the targets are not co-operative. Thus, a military tracking system requires more efficient methods to identify and solve the heading of a target. The tracking methods should also tolerate noise and jamming situations. The order of this paper is as follows. First we describe essential track estimation algorithms. After that, we describe how attributes could be processed. Then, we discuss Bayesian networks and type identification using Bayesian networks. This is followed by simulations and discussion the simulation results. Finally, we summarize the paper and discuss what to do in order to make tracking and identification perform better.
2. A TRACKING SYSTEM Word “tracking” is closely related to estimation. Therefore, mathematical estimation methods are powerful also in tracking. The generalised one cycle of tracking is as follows. 1. A posteriori state estimate from the last moment is available (the best comprehension of the state of the target). 2. Using the state model, the predicted state is computed. 3. Measurements are received from the sensors. 4. Measurements are combined (e.g. PDA) or one measurement is chosen (e.g. nearest neighbor). The result of this phase is one vector, which is either combined quantity from all the measurements or e.g. the nearest measurement. 5. Predicted state (2) and (processed) measurement from (4) are used in computing new a posteriori estimate and updating the covariances. The procedure above is almost one to one to the process of the Kalman filter.2 Thus, it can be said that the heart of the tracking is the Kalman filter. One cycle of tracking is illustrated in Figure 1. (1)
(2)
A posteriori state estimate
Prediction
(3)
(4)
Measurements
PDA, NN, etc.
(5) Combination: New a posteriori state estimate
Figure 1. General tracking procedure.
The update equations are derived e.g. in 2. Here they are briefly recapitulated. We assume that the state and measurement equation are (respectively): x t + 1 = F t x t + q t and
(2.1)
zt = Ht xt + wt ,
(2.2)
where F is state transition matrix, x is state, q is process noise whose mean is 0 and covariance Q, H is measurement matrix and W is measurement noise whose mean is 0 and covariance R. If measurement and process noises are uncorrelated and the covariance of the state is P, we obtain following equations (the hat refers to an estimate) 7: Prediction: xˆ t + 1
t
= F t xˆ t t and
(2.3)
T
Pt + 1 t = Ft Pt t Ft + Qt .
(2.4)
So called Kalman gain is defined by: T
T
–1
Kt + 1 = Pt + 1 t H t + 1 ( H t + 1 Pt + 1 t Ht + 1 + Rt + 1 ) .
(2.5)
New a posteriori estimate and covariance: xˆ t + 1
t+1
= xˆ t + 1 t + K t + 1 ( z t + 1 – H t + 1 xˆ t + 1 t ) and
(2.6)
Pt + 1 t + 1 = ( I – Kt + 1 Ht + 1 ) Pt + 1 t .
(2.7)
One of the assumptions in the Kalman filter is linearity of the process. However, tracking is a non-linear procedure and results using just the Kalman filter are inaccurate. In order to overcome this problem one can try to linearize the state equation, this approach leads to extended Kalman filter (EKF).7 2.1 Handling kinematic measurements Kinematic measurements refers to measurements of position, velocity, angle (in the passive sensor case) etc. However, not all the measurements origin from the target. Therefore measurements being most likely false should be eliminated so that they do not affect to the estimate. The coarse elimination is done in the phase called validation (g-σ-gating). In the validation procedure distance between measurements and the estimate are compared. If the distance is more than predetermined threshold, the measurement is rejected. Figure 2 illustrates the validation phase. Measurement
Estimate Validation circle (gσ-gate)
Figure 2. Validation scheme.
The nearest neighbor algorithm (and its derivatives) chooses the nearest validated measurement and abandon all the others. As this method leads to decreased tolerance of noise, we do not discuss it further. One of the most used algorithm today is probabilistic data association (PDA) algorithm. It is derived in 2 and here we only briefly repeat the main points. The PDA is a Bayesian approach that computes the probability that each validated measurement is actually originated from the target and the probability that none of the validated measurements is correct. PDA assumes that there is a single target present and at most one measurement is actually target oriented and all the others are due to clutter. 1 While discussing the PDA we must define the following notions. v is called the innovation (or residual) of the actual measurement and the predicted measurement. S is the covariance of the innovation. Thus, the validation procedure is possible to 2
T
–1
express using v and S: d = v S v . We assume that before tracking has been started, the gate probability PG has been determined. Using PG and χ2-tables we are able to obtain a quantity g, which is the threshold such that if d < g, the current measurement is accepted to update phase.1 For all validated measurements the PDA filter results association probabilities (β) for the target oriented measurements: ej β j = --------------------- , j = 1,..., m m b+
∑ el l=1
and none of the measurements is target oriented:
(2.8)
b β 0 = ---------------------, m b+
(2.9)
∑ el l=1
1 –P D P G 1 T –1 where e j = exp – --- v j S v j , b = λ det ( 2π S ) ⋅ -------------------- , λ is the spatial density of the clutter (assumed known), PD is 2 PD the detection probability (assumed known). Later on βi:s are referred as βPDA in order to distinguish them from attribute β:s. When the PDA is used the equation Eq. 2.6 and Eq. 2.7 are modified:2 m
xˆ t + 1
t+1
= xˆ t + 1 t + K t + 1 ∑ βi z˜i , where z˜ = ( z t + 1 – H t + 1 xˆ t + 1 t )
(2.10)
i=1 c P t + 1 t + 1 = β 0 P t + 1 t + ( 1 – β 0 ) P t + 1 t + 1 + P˜ t + 1 , where
(2.11)
c Pt + 1 t + 1
(2.12)
= ( I – K t + 1 H t + 1 ) P t + 1 t and m
T T ˜ ˜ i z˜ i – z˜ z˜ K Tt + 1 . P t + 1 = Kt + 1 ∑ βi z
(2.13)
i=1
2.2 Interactive multiple model algorithm The well-known fact is that the Kalman filter (and EKF) performs poor when the model is not linear. Therefore new methods to solve this problem are developed. One approach is hybrid systems (i.e. systems having both discrete and continuous uncertainties), which include the method used in this paper. The idea of the hybrid estimation procedure is that when the target is flying direct with constant velocity, the tracking is performed by a filter, which is designed to estimate constant velocities. When target is manoeuvring, the best result is gained by using the filter, which is designed to cope manoeuvres, etc.1 This sort of outlook leads to interactive multiple model (IMM) filters.1 Basically, IMM method consists of multiple (say r) models. We assume that the true model is one of these r models. In addition, we need the following a priori quantities: mean and covariance of the state and initial model probability (µj) of the model j. µj tells the probability that j:s model is correct. The procedure of the IMM is similar to the Kalman filter except that the IMM computes also the model probability as well as model likelihoods. The model likelihoods are used in updating the model probabilities. The output of the IMM algorithm is:1 r
xˆ t + 1
t+1
=
ˆj
∑ µj ⋅ x t + 1 t + 1
and
(2.14)
j=1 r
Pt + 1 t + 1 =
j
ˆj
ˆj
∑ µj ( P t + 1 t + 1 + [ x t + 1 t + 1 – xˆ t + 1 t + 1 ] [ x t + 1 t + 1 – xˆ t + 1 t + 1 ]
T
),
(2.15)
j=1
where xj and Pj refer to the state and covariance computed by j:th model. The precise description of the IMM algorithm can be found in 1,3.
3. HANDLING ATTRIBUTE MEASUREMENTS Signal processing with the kinematic measurements is studied in many publications. Nowadays it is possible to obtain other quantities in addition to kinematic measurements, such as frequency of the radio emitter, form of wing etc. These type depended features are called attributes. The problem of processing the attributes is that they do not share dimensions. That is,
PDA and other algorithms are incapable to combine measurement vector consisting frequency (Hz), position (m) and velocity (m/s). In this paper we process attribute observations with the Bayesian networks. Bayesian networks are developed to cope with a situation where a system receives multiple observations which are uncertain. The goal is to make as reliable inference as possible. Bayesian network is a graph having arcs and variables. Arcs define causal-consequence relations between two variables, which refers to attributes. If we have a simple model, where we have two attributes: type and identification friend, foe, or neutral (IFFN). Thus, we are able to form the Bayesian network illustrated in Figure 3. 9
A = type A
B
O
B = IFFN O = observation
Figure 3. A simple Bayesian network with an observation variable.
The observation variable differs from the variables B and A. The variable O depicts the observation and the link between O and B corresponds to sensor’s mixing matrix (i.e. by what probability the sensor mixes the correct value to incorrect ones). The arcs define a joint distribution between two variables. The structure of the graph is called directed acyclic graph (DAG) if there is no cycles and all arcs are directed. A joint distribution together with a corresponding DAG constitutes Bayesian networks if the following three conditions hold8: 1. Specified conditional probability distributions determine a joint probability distribution of the variables in the DAG. 2. Specified conditional distributions are indeed the conditional probabilities, relative to that joint distribution, of every variable given its parents. 3. Specified joint distribution together with the DAG satisfies the conditional independence assumptions in a Bayesian network. We do not give detailed explanation of the theory of Bayesian networks, but interested reader is advised to read 8,9. An application of Bayesian networks in tracking can be found e.g. on 6. When the application is tracking, the output of the Bayesian networks is a type probability function. The type probability function tells the probability of each possible target type in the surveillance area when the observations are received at a certain moment of time. This implies that all possible targets must be known before the tracking begins. 3.1 Comparison of the attribute distributions In every moment of time, probability function of every attribute is computed. It does not matter whether attribute is discrete or continuous. Our task is to compare the probability distributions of the estimate to the ones of the measurement and yield a value, which tells the closeness of the distributions in question. The value, which is a result from the comparison, indicates closeness of these two distributions. In this study we compare only the type distributions. That is, we do not need to compare the probability distributions of every attribute. There are many ways to compute the closeness. The general requirements for the used method are: 1. If functions are the same, their closeness should be zero. 2. As the difference between the functions increase, so should the value of the used distance. 3.1.1. Hausdorff distance The Hausdorff distance (h(A,B)) where A and B are arbitrary distributions, is defined:
d
min
d
max
( a i, B ) = min ( d ( a i, b j ) ), b j ∈ B ( A, B ) = max ( d
h ( A, B ) = max ( d
max
min
(3.16)
( a i, B ) ), a i ∈ A
(3.17)
max
(3.18)
( A, B ), d
( B, A ) )
here d(*,*) is some distance. We have used weighted Euclidean norm as d(*,*). The weighted Euclidean norm takes on account the position of the probability mass. When we have computed distance to each validated measurement, the distances are called attribute association values. Later on, they are referred with a symbol βattribute. 3.1.2. Kullback-Leibler distance The other method used to compute the closeness between two distributions is the Kullback-Leibler, which is defined:5 p(x)
, ∑ p ( x ) log ---------q(x)
D ( p, q ) =
(3.19)
x
where p(x) is the distribution of the estimate and q(x) the distributions of the measurement. It is good engineering practise to actually code the Kullback-Leibler because situations where observation contains very large, small or zero entries must be considered separately. 3.2 Combining association probabilities In Chapter 2.1 the association probabilities were computed by the PDA. In this phase we have computed closeness of the estimate and the attribute observations. Thus, next thing is to combine association probabilities from the PDA and closeness values. In order to perform the combination, we have applied the method used in 4 and multiply the association values (the attribute association values must be normalized first): β = β PDA • β ATTRIBUTE .
(3.20)
The type distribution of the estimate is updated as follows. Validated measurements’ type estimates are combined according to association probabilities. In mathematical terms, the combined type estimate is as follows: m
∑ βi pi ( x ) i=1 , p ( x ) = ------------------------------------n m
(3.21)
∑ ∑ βi pi ( zk ) k = 1i = 1
where m is the number of validated measurements, n is the number of targets, β i are association probabilities, pi(*) represents the type distribution of each observation. The type distribution of the estimate is updated using the Bayes’ formula. If the existing type estimate is t(x), the updated type estimate is: p ( x )t ( x ) . t ( x ) = ------------------------------n
(3.22)
∑ p ( z i )t ( zi ) i=1
When attributes are taken in account, the tracking scheme is altered a little bit. In Figure 4 is illustrated the tracking scheme when attributes are along.
(1)
(2)
A posteriori state estimate
Prediction
(5)
(5)
Combination: New a posteriori state estimate
Kinematic measurements
(3) Measurements
PDA, NN, etc. (7) (6)
Attribute observations
Combination of association probabilities
Bayesian networks Figure 4. Tracking scheme when attributes are also used.
3.3 Type identification As mentioned before, the type of the current target is impossible to conclude using the kinematic measurements only. In the previous section one can notice that the output of the method we have used to process attribute observations is type probability function. According to its name, the type probability function tells the probabilities of each type. The needed modifications to the theory of Bayesian networks in order to model sensors are presented in 6. Bayesian networks assume that attributes are modelled before tracking. Natural way to do that (and sensible too) is to model attributes as a dependency tree. For example, let us suppose that we are able to model three features such as identification friend, foe or neutral (IFFN), the radio frequency and form of the wing, we can obtain graph like in Figure 5. In addition, each of these features is directly dependent on the type of the target (e.g. Falcon is friendly, JAS is neutral etc.).
A B
C
A = type, B = IFFN, C = radio frequency, D = form of wings D
Figure 5. An attribute tree.
The structure of the Bayesian networks is exactly same as the structure of an attribute tree. That is one of the major reasons to use Bayesian networks in the identification and thereby in tracking.
4. SIMULATIONS The main goal of this study is to illustrate how IMM and PDA methods are able to track multiple targets in the noisy environment. The sensors used are passive, so they observe angles. Then Bayesian networks are applied to the tracking system. With this extension the tracking system is able process both angle measurements and attribute observations. The validation procedure is the same in both cases: all measurements are validated using g-σ-gate. When Bayesian networks are applied, the association probabilities are computed with the Hausdorff distance.
4.1 Simulation specifications Simulation proceeds as follows. We assume that there are 11 targets in the surveillance area. Each target has 12 attributes. There are five sensors, which are able to detect different attributes and angles of the targets as well as different sets of attributes. We assume that all targets have been modelled using the attribute structure. The structure is illustrated in Figure 6, where the most important node is named with an alphabet A depicting the type of the target, which is our main interest. The other nodes represent an arbitrary discrete feature. A
B
C
E
F
G
H
I
J
D
K
L
Figure 6. The attribute tree of the simulation.
As there are 11 targets, the variable A has 11 values. The values of the variables are presented in Table 1. Table 1. Values of the variables.
Variable
Number of values
Variable
Number of values
A
11
G
5
B
2
H
3
C
3
I
6
D
2
J
2
E
4
K
3
F
2
L
4
In this scenario we assume that we have three sensor classes, which are able to measure different amount of attributes. We have used five sensors, which have different accuracies. The sensors are specified in Table 2. Table 2. Sensor specifications.
Sensor
Observed attributes
Accuracy
1
B, C, D, F, J
0.9
2
F, I, J, K
0.95
3
C, G
0.99
4
B, C, D, F, J
0.85
5
C, G
0.95
The simulation consists of 11 targets, which are all of different type. We tried to make the simulation contain formations, manoeuvres and crossing targets. These properties are known to be hard to tracking systems. The true trajectories with their identification numbers and five sensors are illustrated in Figure 7. The overall duration of the simulation is 100 seconds.
. x 10
10
4
11
9
10 9
8
7
6
8
5
7
1
4
3
2
1
2 34 0
0
1
2
3
4
5
6
5
6
7
8
9
10
Figure 7. True trajectories and sensors.
4.2 Tracking with IMM and PDA algorithms In this section we simulate the scenario using the IMM with the PDA. We have used three models in the IMM: nearly constant velocity and two models for manoeuvres. The gate probability was set to three (Bar-Shalom suggest that the value should be between one and five). We performed simulation 30 times. The results were very similar. A typical result is illustrated in Figure 8. 4
10
x 10
9
8
7
6
5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
10
Figure 8. Tracking results (IMM with PDA).
The tracker had difficulties to distinguish formations. Especially crossing targets cause troubles. However, the IMM handled maneuvering targets very good. We can consider that as the tracking performance is satisfactory, there is no need to use attribute association. The situations where attribute association could be used is the crossings of the targets. However, if all the targets in the formation are dissimilar, their attributes differs from each other too. Thus, the use of attribute association might yield better tracking results.
4.3 Tracking with the attributes Next we simulate the same scenario, but in this time we use Bayesian networks beside the PDA and IMM. The number of the simulations was also 30. A typical result is depicted in Figure 9. As expected, the results are quite similar to results of using just PDA and IMM. The main purpose of using the Bayesian networks is to determine the type of the target. 4
10
x 10
9
*
8
7
6
* 5
4
3
2
1
0 0
1
2
3
4
5
6
7
8
9
10
Figure 9. Tracking results with Bayesian networks.
Tracking results were very similar compared to tracking without Bayesian networks. The main difference is that the formation was tracked better with than without then Bayesian networks. However, there is few false tracks (marked with *). Both extra tracks are formed to the intersection points. This is because attributes of the targets 9 and 10 (as well 8 and 7) are very similar. Therefore, if the tracker initializes a false track, it is updated a couple of times. When the false track is too far and it is not updated, its uncertainty will increase and the track is deleted. There is not extra tracks in the formation of the targets 1,2,3 and 4 because their attribute structure is variant. The identification succeeded excellent. All the targets were identified correctly. The types of the false tracks were systematically the type of the closest real track when the track was initialized. An example of the progressing of the one type is illustrated in Figure 10, where the type in question is 11. Each target was identified correctly not later than in 10 seconds. 1 0.9 0.8 0.7
probability
0.6 0.5 0.4 0.3 0.2 0.1 0 10
20
30
40
50
60
70
time
Figure 10. An example of the one target type probability.
80
90
100
5. SUMMARY In this paper we have briefly illustrated the tracking procedure with IMM and PDA. One of the main objectives was discussion of the attribute association. Data association is one of the most critical phases of tracking because if too many non-target originated measurements are validated and then associated, the value of the estimate will be next nothing regardless of the used tracking algorithms. The simulations showed that the method, which we have used in order to compute the attribute association probabilities and to combine them to the association probabilities computed by the PDA, functions well. On the basis of the simulation results, it can be stated that in some cases it is enough to do association with kinematic measurements only. However, when there are targets near each other, the attributes may produce better tracking results. The attributes are mostly needed when trying to identify the target. Identification is possible to execute with the Bayesian networks. As the simulations showed, the Bayesian networks were able to yield accurate identifications, even in the cases where there are multiple targets flying in the formation. Although in this simulation the PDA succeeded quite well, its drawback is the assumption that there is only single target present. Therefore, if the tracking system is observing wide area with multiple targets, the PDA may not be sufficient. There exists an extension of the PDA: the joint probability data association (JPDA) algorithm. The tracking may be better if association would be performed by the JPDA. In these scenarios we do not used track-to-track fusion. That is, the features of the tracks were not compared and fused, if two tracks are close enough each other. The use of track-to-track fusion should reduce the number of the false tracks. It can be noticed that as the number of the methods and algorithms increases, so does the number of parameters, which affect to the tracking results. This may cause troubles in adjusting the parameters when tracking.
ACKNOWLEDGMENTS The authors thank persons working in Data Fusion Group at the Tampere University of Technology. Especially, the work made by Petri Korpisaari and Mikko Pekkarinen has been helpful while doing this paper.
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