object-existence-based approach
5.6 Track component control
Without component control, the number of components increases exponentially in time. Denote the number of a posteriori track components at time k− 1 by Ck−1, and the number of selected measurements at time k by mk. Then the number of tentative track components ˜Ckat time k equals
˜Ck = (1 + mk) Ck−1.
5.6 Track component control 185 Track component management processes the set of tentative track components to produce the final a posteriori set of track components within computational capabilities.
As the a posteriori set of track components differs from the Bayesian set of ten-tative track components, this operation contributes to the sub-optimality of object tracking filters presented in this chapter.
The input to the component control operation at time k is:
r the tentative pdf of object trajectory state at time k, p(xk|χk, Yk), which consists of a set of ˜Cktentative track components parameterized by:
– the relative probabilities of tentative track components p(˜ck)= p(˜ξk(˜ck)|χk, Yk); and for each tentative component ˜ck,
– the a posteriori IMM pdf parameterized by probabilities μk|k(˜ck, σ), mean valuesˆxk|k(˜ck, σ) and covariances Pk|k(˜ck, σ) for all models σ.
The output of the component control operation at time k is:
r the a posteriori pdf of object trajectory state at time k, p(xk|χk, Yk), which consists of a set of Cka posteriori track components parameterized by:
– the relative probabilities of track components p(ck)= p(ξ k(ck)|χk, Yk); and for each track component ck,
– the a posteriori IMM pdf parameterized by probabilities μk|k(ck, σ), mean valuesˆxk|k(ck, σ) and covariances Pk|k(ck, σ) for all models σ.
A number of techniques may be used for control management (Blackman, 1986; Salmond, 1990; Blackman and Popoli, 1999; Williams and Mayback, 2003;
Bochardt et al., 2006):
r component merging;
r leaf pruning;
r sub-tree pruning.
5.6.1 Track component merging
Component merging involves merging two or more tentative track components into one. If all tentative track components merge into one a posteriori track component, resulting in the a posteriori object trajectory state pdf approximation by one Gaus-sian probability density function per trajectory model, we obtain the PDA-based algorithms:
r IMM-JITS and IMM-ITS become IMM-JIPDA and IMM-IPDA respectively;
r JITS and ITS become JIPDA and IPDA respectively.
When track components merge, the track component interpretation as the object trajectory state pdf given a measurement sequence may become inappropriate. In that case, the track trajectory state component pdf is the object trajectory state probability density function approximation given a set of measurement sequences.
Track components remain mutually exclusive and exhaustive, and the track trajec-tory state pdf remains a Gaussian mixture.
The track components merging procedure has two parts. The first is the track component merge criteria (the choice of track components to merge into one), and the other part is the actual track component merging.
There are various track components merging criteria, some of which are rather complex and computationally expensive. The recent focus (Salmond, 1990;
Williams and Mayback, 2003; Bochardt et al., 2006) seems to be to merge track components in a manner with least distortion of the a posteriori track trajectory state pdf. The authors have found the following criterion, which is some decades old (Singer et al., 1974), to be very simple and effective:
Merge all tentative track components with common measurement history in last Nm
scans.
The reason for the effectiveness of this criterion is that the measurements lose their “effect” the older they get. As shown in Singer et al. (1974), even a relatively short retained track component history Nm seems to capture measurement infor-mation effectively. A fringe benefit of this criterion is that the track component definition is retained, albeit limited to the window of last Nmscans:
ξk(ck) =2
θk−Nm+1(ik−Nm+1) · · · θk(ik)3 .
In the remainder of this section we describe the merging of one set of track components (of cardinality C ) at time k into one merged track component.
This procedure is valid regardless of the method used to choose the set of track components . Depending on the approach used for track component merging, this procedure may need to be repeated many times, sometimes merging the “merged”
track component with additional track components. In the case of the PDA-based algorithms (PDA, IPDA, IMM-PDA, IMM-IPDA, JPDA, JIPDA, IMM-JPDA and IMM-JIPDA) all track components are merged into one and C = ˜Ck.
Let denote the set of track components (of cardinality C ) complementary to . After the merging operation, all track components belonging to set will be replaced by the merged track component, denoted here by ck= i . The total number of track components is reduced by C − 1.
5.6 Track component control 187 As all track components before merging are mutually exclusive, the merged track component i is mutually exclusive to all track components from the com-plementary set . Relative probability of the merged track component is the sum of relative probabilities of the “constituent” track components
p(i ) =
˜ck∈
p(˜ck) = 1 −
˜ck∈
p(˜ck) . (5.94)
The object trajectory state probability density function of merged track component i is defined by its IMM parameters. The a posteriori probability of the object trajectory modelσ, given the merged component i , is given by
μk|k(i , σ )= p(r k = σ|i , χk, Yk)
= p
rk = σ, i |χk, Yk p
i |χk, Yk
=
˜ck∈
p(rk= σ|˜ck, χk, Yk)p(˜ck|χk, Yk)
p(i ) ,
where the second line is the Bayes’ equation, and the third line is obtained by applying the total probability theorem. Thus,
μk|k(i , σ) =
˜ck∈
μk|k(˜ck, σ)p (˜ck)
p(i ) . (5.95)
Define by β (˜ck, σ, )= p(˜c k|σ, , χk, Yk) the relative probability that the tentative track component ˜ckis correct, given that the set of tentative component is correct, and given that the object trajectory model σ is correct. Given ˜ck∈ , then{˜ck, } = {˜ck} and
β (˜ck, σ, )= p(˜c k|σ, , χk, Yk) = p(˜ck, σ, |χk, Yk)
p( , σ|χk, Yk) = p(˜ck, σ|χk, Yk) p( , σ|χk, Yk), thus
β (˜ck, σ, ) = p(˜ck) p(σ|˜ck, χk, Yk)
˜ck∈
p(˜ck) p(σ|˜ck, χk, Yk) = p(˜ck) μk|k(˜ck, σ)
˜ck∈
p(˜ck) μk|k(˜ck, σ). (5.96)
The a posteriori object trajectory state probability density function, given IMM model (object trajectory model)σ and merged component i , is given by
p(xk|i , σ, χk, Yk) =
˜ck∈
p(xk|˜ck, i , σ, χk, Yk)p(˜ck|σ, i , χk, Yk)
=
˜ck∈
p(xk|˜ck, σ, χk, Yk)β (˜ck, σ, ) ,
which becomes
p(xk|i , σ, χk, Yk) =
˜ck∈
β (˜ck, σ, ) N
xk; ˆxk|k(˜ck, σ), Pk|k(˜ck, σ)
≈ N
xk; ˆxk|k(i , σ), Pk|k(i , σ) ,
with the Gaussian mixture approximated by a single Gaussian with identical mean and covariance,
[ˆxk|k(iθ, σ), Pk|k(i , σ)] = GMix2
ˆxk|k(˜ck, σ), Pk|k(˜ck, σ), β (˜ck, σ, )3
˜ck∈
. (5.97) The merged component i is defined by its relative probability p(i ) (5.94), and for each object trajectory modelσ a posteriori probability μk|k(i , σ ) (5.95) and object trajectory state mean ˆxk|k(i , σ) and covariance Pk|k(i , σ) (5.96)–(5.97).
It is straightforward to verify that after the track component merging operation,
Ck
ck=1
p(ck) =
M σ=1
μk|k(ck, σ) = 1
holds.
5.6.2 Track component leaf and sub-tree pruning
Track components up to and including time k may be graphically represented.
Without loss of generality, assume here that the track was initiated at time = 1 by one measurement. Form a graph with each node representing a track com-ponent, arranged in levels which correspond to scan times. Graph nodes at level
∈ 1, . . . , k are a posteriori track components cat time. Vertices of the graph from level − 1 to level represent measurements (as well as “null” measure-ments) which are paired with track components at the upper level − 1 to form track components at level. It is straightforward to see that this graph is a tree, as the vertices go only from one level to the next, and each node in the tree has only one direct antecedent. An example is depicted in Figure 5.5.