Track state

Here we consider a hybrid track state. Track state has a discrete component, which is the object existence at time k, and a continuous component, which is the object trajectory state pdf at time k.

5.3.1 Object existence

Two models for object existence propagation have been identified in Muˇsicki et al. (1994). In one, termed Markov Chain One, object existence has two possible states:

r the object either exists; or r the object does not exist.

If it exists, the object is detectable (generates a measurement) with the probability of detection P_D. This is the default model which is used in this chapter. The other object existence model is termed Markov Chain Two in Muˇsicki et al. (1994), and has three possible object existence states:

5.3 Track state 143 r the object exists and is detectable; or

r the object exists and is temporarily not detectable; or r the object does not exist.

The object may exist and be temporarily not detectable if, for example, the object gets temporarily concealed by an obstacle between the sensor and the object. If the object exists and is detectable, its measurements are present in each scan with the probability P_D. The Markov Chain Two model is adaptable to unknown or fluctuating probability of detection (Wang and Muˇsicki, 2007; Muˇsicki and Wang, 2004), at the expense of a slightly more complex model. For reasons of clarity and simplicity, the rest of this chapter follows the Markov Chain One model.

The Markov Chain One object existence model defines two mutually exclusive and exhaustive events modeled by a random variable Ek:

χk = E k = 1 the event that an object exists,

¯χk = E k = 0 the event that the object does not exist.

The probability of propagated object existence is obtained by applying the Markov chain propagation formula,

with the elements of matrix γ being the transitional probabilities between object existence states:

γi j 

= p(Ek = 2 − j|Ek−1= 2 − i), i, j ∈ {1, 2}, and

γ11+ γ12 = γ21+ γ22 = 1.

This operation denotes the Markov Chain One model for object existence predic-tion (propagapredic-tion), and its pseudo-funcpredic-tion is

p(χk|Y^k−1) = TEXP[p(χk−1|Y^k−1), γ ].

As discussed in Muˇsicki et al. (1994, 2007), the value of γ21 should be zero, from which γ22 = 1. The value of γ21 is the transitional probability of a false track becoming a true track. While such an event may happen in practice when the false track starts to follow an object, it also renders the object trajectory state pdf p(xk|χk) meaningless. The authors recommend using γ21= 0, and treating the

emergence of new tracks as part of the track “birth” process or track initialization, detailed in Section 9.4. Equation (5.9) therefore reduces to

p(χk|Y^k−1) = γ11p(χk−1|Y^k−1), (5.10) whereγ11 denotes the probability that an object will continue to exist at time k, given that it exists at time k− 1. Value of γ11is calculated as (Bar-Shalom and Li, 1993)

γ11 = 1 − tk

T_χ ,

where T_χ denotes the average object lifetime in the surveillance region, andtk

denotes the time between measurement scans k− 1 and k, with the assumption thattk Tχ.

5.3.2 Object trajectory state

The object trajectory state at time k is denoted by xk. The a priori track state probability density function is given by

p(xk, χk|Y^k−1) = p(xk|χk, Y^k−1)p(χk|Y^k−1), (5.11) and the a posteriori track state probability density function is given by

p(xk, χk|Y^k) = p(xk|χk, Y^k)p(χk|Y^k). (5.12) The object trajectory state pdf, p(xk), is always calculated conditioned on the object existence eventχk. The object trajectory state pdf conditioned on the object non-existence event is undefined and, indeed, does not make sense.

Denote byθk(ik), ik≥ 0, the event that measurement yk(ik) is the detection of the object being tracked by the track. Event θk(0) denotes the event that no selected measurement is the detection of the object, which can happen because either the object does not exist, or the object was not detected at time k, or its detection was not selected. Assuming that the track was initialized at time k= 1, each measurement sequence

ξk(ck) = {i1, . . . , ik} ; i = 0, . . . , m;  = 1, . . . , k, (5.13) denotes one possible object detection sequence; the index ck is the past measure-ment sequence index at time k, ck= 1, . . . , Ckand

Ck =

*k

=1

(1 + m). (5.14)

5.3 Track state 145 Table 5.2 Measurement sequences/track components.

c1 ξ1(c1) c2 ξ2(c2) c3 ξ3(c3)

1 {1, 0} 1 {1, 0, 0}

2 {1, 0, 1}

1 {1} 2 {1, 1} 3 {1, 1, 0}

4 {1, 1, 1}

3 {1, 2} 5 {1, 2, 0}

6 {1, 2, 1}

T i m e 1

T i m e 2

T i m e 3

1

1 0 2

0 1

2 1 3

1 2

3

4 5 6

Figure 5.2 Measurement sequences/track components.

This is illustrated in Figure 5.2 and Table 5.2. At time k = 1 there is only one mea-surement which starts the track. There is only one meamea-surement sequence ξ1(c1) at time k = 1. At time k = 2, there are two measurements, indexed as 1 and 2. The

“null” measurement indexed as 0 corresponds to object non-detection or object non-existence events. Each measurement at time k= 2 associates with the mea-surement sequence at time k = 1 to create a measurement sequence ξ2(c2) at time k = 2. The process repeats at time k = 3 where we have one measurement indexed as 1 and one “null” measurement indexed as 0. Each measurement sequenceξ2(c2) at time k = 2 associates with each measurement at time k = 3 to create one mea-surement sequenceξ3(c3) at time k = 3.

The object trajectory pdf conditioned onξk(ck), or in other words assuming that ξk(ck) is the sequence of object detections of measurement set Y^k, is denoted by shorthand p(xk|ck):

p(xk|ck)= p(x^ k|ξk(ck), χk, Y^k),

and

p(ck)= p(ξ^ k(ck)|χk, Y^k)

is the a posteriori probability that measurement sequenceξk(ck) consists entirely of object detections. Events{ξk(ck)|χk, Y^k} are mutually exclusive, as there is one and only one measurement sequence, which consists of object detections:



c_k

p(ck) =

c_k

p(ξk(ck)|χk, Y^k) = 1.

Given a linear system with a known object trajectory model, p(xk|ck) is obtained by applying measurement sequence ξk(ck) to the Kalman filter with corresponding object trajectory model parameters. In that case p(xk|ck) is a Gaussian pdf defined by its mean and covariance. In this chapter we consider a more general case of M possible object trajectory models, and use the interacting multiple models (IMM) estimator. Therefore, p(xk|ck) is a Gaussian mixture of M Gaussian pdfs

p(xk|ck) =

M σ=1

p(xk|ck, σ)μk|k(ck, σ), (5.15) where σ indexes the trajectory models, and μk|k(ck, σ) is the a posteriori prob-ability of object trajectory model σ at time k conditioned on the event that the measurement sequenceξk(ck) is correct, and

p(xk|ck, σ) = N Each trajectory state pdf at time k, given measurement historyξk(ck), is called a track component, indexed by ck. The number of track components, as defined by (5.14), grows exponentially with k. This soon exhausts the available reasonable computational resources. Any practical implementation has to limit the number of components using the various methods discussed in Section 5.5.5. Due to this component management, the strict definition of components as defined by (5.13)