6.3 Generalized PHD Intensity Filters Applied to BML
6.3.2 Approximation of the Update Equation
generalized iFilter consists of the union of the target spaceX, which is equal to the target space of the generalized PHD filter, and the hypothesis spaceXφ, which is used to model clutter by scatterers.
Note that the generalization of the target–oriented measurement model implies that a scatterer with a state inXφcan also generate multiple clutter measurements per scan.
This is equivalent to the arbitrary clutter model in the generalized PHD filter. The definition of the false–measurement model for elements fromXφ, that is, the clutter model, of the generalized iFilter can be incorporated into the generalized iFilter by the definition of the likelihood functionp|·|(·|x) onXφforx∈X.
6.3.1.3 Computational Complexity and Probability Of Detection
The update equations (6.55) and (6.62) of both generalized PHD Intensity filters are numerically highly complex due to the sum over all partitions of the set of re-ceived measurements. The number of partitions is growing exponentially with the number m of received measurements and is given by the Bell number Bm. The exponential growth of the Bell number is visualized in Figure 6.11 and it is ob-vious that for an application of (6.55) and (6.62) approximations are inevitable.
In [RBGD15], [Gra12], [GLO12], [GO12a], [GO13], [GO12b] and [SC10] clustering approaches, which are essentially based on the spatial relation of measurements, are used to reduce the number of partitions. These approximations are possible, if mea-surements that are generated by the same target are spatially related in the measure-ment space. However, in BML a target generates multiple measuremeasure-ments per sensor scan which are not spatially related in the measurement space. Therefore, the par-titions in equation (6.55) need to be reduced without using any information about the spatial distribution of measurements. Instead, other criteria have to be identified to successfully approximate the update equations (6.55) and (6.62) with a feasible number (in terms of the numerical complexity) of partitions.
Furthermore, a modeling of the probability of detection via a simple Bernoulli trial (as it is done for the standard target–oriented measurement model from Section 3.2.2.1) is inappropriate if a target generates multiple measurements per sensor scan, since multiple measurement partitions might be processed which are subsets of each other.
Therefore, the definition of the probability of detection in (6.55) and (6.62) (as part of the likelihood function) has to be modeled in a generalized fashion (as already mentioned in [CM12]).
Set Size
0 2 4 6 8 10 12 14
Number of Partitions
100 101 102 103 104 105 106 107 108 109
1 1 1 1 1 1
1 2
1 1 1 1 1
2 4
1 1 1 1 1
5 10
2 1 1 1 2
15 26
7 2 1 1 7
52 76
22 7 2 1 22
203 232
57 23 8
1 57 877
764
162 65 30
2 162 4140
2620 694
163 94
10 694 21147
9496 3424
499 256
47 3424 115975
35696 14884
2179 628 177 14884 678570
140152 60424
10693
1882 562 60424 4213597
568504 293294
49336
9010
1574 293294 27644437
2390480 1768768
205349 51052
4005 190899322
Bell Number Nmin = 2, N
max = 2 Nmin = 4, N
max = 5 Nmin = 5, N
max = 7 Nmin = 6, N
max = 8 Nmin = 8, N
max = 10
Figure 6.11:Comparison of the Bell number and the number of partitions due to approxi-mation (6.85) c2014 IEEE.
reduction of the number of partitions by the application of clustering methods is not applicable if measurements that belong to a specific target are not spatially related in the measurement space. To this end, two novel approaches are presented in the following section, which approximate the update equation of generalized PHD inten-sity filters by reducing the number of investigated partitions without assuming an underlying spatial distribution of the measurements which belong to a specific target.
Finally, a generalized definition of the probability of detection is presented.
6.3.2.1 Incorporation of a Priori Information
The first proposed approximation of Equation (6.55) considers available a priori in-formation about the number of generated measurements per target and sensor scan.
The idea is to restrict the possible number of generated measurements, that is to as-sume that a target generates at leastNmin∈Nand at mostNmax∈Nmeasurements per sensor scan. This idea is straightforward and closely related to the approach for reducing partition hypothesis presented in [Alg10, Section 7.2.3]. Even though it might seem obvious how a restriction of the number of measurements per target will influence the update equation of the generalized PHD filter and iFilter, a detailed derivation is carried out in the following for the generalized PHD filter to demon-strate how a priori information and specific assumptions can be incorporated via a mathematically correct approach into an existing pointillist filter from Chapter 3.
To derive the respective PHD update equation the general higher order chain rule, presented in [CM12] is used to carry out the functional differentiation. Note that this
Analytical Step can also by done by the application of secular functions. Let g: (Y,B(Y))→(R,B(R)) (6.64) and
h: (X,B(X))→(R,B(R)) (6.65) be bounded (by one), non–negative and Lebesgue–integrable test–functions, where B(·) denotes the Borel–σalgebra of the respective space. First, the PGFL of the joint state is given analogously to Equation (3.64) by
ΨGenPHD(h, g) = ΨgenC (g)GN ΨgenBMD(h, g)
= (exp◦f)(h, g), (6.66) where
f(h, g)≡λ
Z
Y
c(z)g(z)dz−1
+µ
Z
X
s(x)h(x)Ψobs(g|x)dx−1
and the approximated PGFL of the likelihood function Ψobs(g|·) : X → R, which incorporates the a priori knowledge on the number of measurements per target, is defined by
Ψobs(g|x)≡p0(∅|x) +
Nmax
X
n=Nmin
1 n!
Z
Yn n
Y
j=1
g(zj)pn(z1, ..., zn|x)dz1...dzn. (6.67) It holds by definition that
ΨgenBMD(h, g) = Z
X
h(x)µ(x)Ψobs(g|x)dx (6.68) Note that the for changing the target–oriented measurement model, only Ψobs(g|·) has to be adapted. Applying the general higher order chain rule to determine the functional derivative of (6.66) with respect to impulses yields
∂mΨGenPHD
∂δz1· · ·∂δzm(h, g) = ∂m(exp◦f)
∂δz1· · ·∂δzm(h, g) = X
π∈Π(1:m)
∂|π|exp(f(h, g))
∂ξπ1· · ·∂ξπ|π|
= X
π∈Π(1:m)
exp(f(h, g))
|π|
Y
j=1
ξπj[g, h], (6.69) where
ξω(h, g)≡ ∂|ω|f(h, g)
∂ω1· · ·∂ω|ω|
=µ Z
X
s(x)h(x)∂|ω|Ψobs(g|x)
∂ω1· · ·∂ω|ω|
dx (6.70)
and the functional derivatives are taken with respect to the function g. For the evaluation of (6.70) the functional derivative of definition (6.67) has to be considered.
Therefore, letωbe an arbitrary element of a partition from Π(1:m). Then, the Gˆateaux derivative of the functional is given by
∂|ω|Ψobs(g)
∂ω1· · ·∂ω|ω|
=
Nmax
X
n=Nmin
1
n!·n·(n−1)·...·(n− |ω|+ 1)
× Z
Yn−|ω|
n−|ω|
Y
j=1
g(zj0)pn(i(ω), z10, ..., z(n−|ω|)0|x)dz10...dz(n−|ω|)0 (6.71)
if|ω|< Nmin. If|ω| ∈ {Nmin, ..., Nmax−1}it is given by
∂|ω|Ψobs(g)
∂ω1· · ·∂ω|ω|
=p|ω|(i(ω)|x) +
Nmax
X
n=Nmin
1
n!·n·(n−1)·...·(n− |ω|+ 1)
× Z
Yn−|ω|
n−|ω|
Y
j=1
g(zj0)pn(i(ω), z10, ..., z(n−|ω|)0|x)dz10...dz(n−|ω|)0 (6.72)
and if|ω|=Nmaxit is equal to
∂|ω|Ψobs(g)
∂ω1· · ·∂ω|ω|
=p|ω|(i(ω)|x). (6.73)
If|ω|> Nmaxthe derivative is
∂|ω|Ψobs(g)
∂ω1· · ·∂ω|ω|
= 0. (6.74)
Thus,
∂|ω|Ψobs(g)
∂ω1· · ·∂ω|ω|
=1A(ω)p|ω|(i(ω)|x) =
p|ω|(i(ω)|x),if|ω| ∈ {Nmin, ..., Nmax} 0, otherwise,
(6.75) (6.76) where
A≡
a:|a| ∈ {Nmin, ..., Nmax} . (6.77) In the following, the short–hand notation from (6.75) is used. Given the functional derivative of the PGFL of the joint state with respect to impulses the update equation
of the corresponding PHD filter can be determined. It is given by µX|Y(x|z1, ..., zm) =
∂mΨGenPHD(0,1)
∂δz1· · ·∂δzm −1
∂m+1ΨGenPHD(0,1)
∂δz1· · ·∂δzm∂δx
!
(6.78)
=
X
π∈Π(1:m)
|π|
Y
j=1
ξπj(0,1)
−1
X
π∈Π(1:m)
∂Bπ(0,1)
∂δx
, (6.79) where
Bπ(h, g)≡f(h, g)·
|π|
Y
j=1
ξπj(h, g) (6.80)
and
∂Bπ(h, g)
∂δx =µs(x)Ψobs[g|x]
|π|
Y
j=1
ξπj(h, g)
+
|π|
X
j=1
µs(x) ∂|πj|Ψobs(g|x)
∂πj,1· · ·∂πj,|πj| π
Y
k=1,k6=j
ξπk(h, g) (6.81) The evaluation of (6.79) yields the update equation of the approximated generalized PHD filter with Poisson clutter. It is given by
µX|Y(x|z1, ..., zm) =
µs(x)
p0(∅|x) + P
π∈Π(1:m)
|π|
P
j=1
1A(πj)p|πj|(i(πj)|x)
|π|
Q
k=1,k6=j
ηπ,kPHD
P
π∈Π(1:m)
|π|
Q
j=1
ηPHDπ,j
, (6.82)
where
ηPHDπ,j ≡ξπj(0,1) = 1{a:|a|=1}(πj)λc(i(πj,1)) +µ Z
X
s(x)1A(πj)p|πj|(i(πj)|x)dx.
(6.83) The update equation of the generalized iFilter is derived analogously. It is given by (6.82), but withηPHDπ,j replaced by
ηπ,jiFilter≡1{a:|a|=1}(πj)µs(φ)p|πj|(i(πj)|φ) +µ Z
X
s(x)p|πj|(i(πj)|x)dx. (6.84)
Due to the fact that some summands of Equation (6.82) are zero, computational effort can easily be saved. A summand of the sum over all partitions in (6.82) is zero if for the respective partitionπ∈Π(1:m)holds
∃j∈ {1, . . . ,|π|}:|πj| 6∈ {1, Nmin, ..., Nmax}, (6.85) since then either 1{a:|a|=1}(πj) = 0 or 1A(πj) = 0. Therefore, the computational effort can be reduced by rejecting the partitions which fulfill condition (6.85). After rejecting the partitions, Equation (6.55) can be evaluated, since except for the ap-pearance of 1A(·) = 0 it is identical to Equation (6.82).
Note that partitions are not rejected, if they have a subset which is of cardinality one. This is independent of the choice ofNminandNmaxand holds if a Poisson clut-ter model is chosen (for the generalized PHD) or if clutclut-ter scatclut-terers are allowed to generate only single measurements (generalized iFilter)). For the generalized iFilter this has to be modeled via the likelihood function onXφ. However, more enhanced clutter models could be included. For example, in a BML–scenario the context in-formation, which is available due to a ray tracer, does not consider cars and other road users. Therefore, typical clutter sources in a BML–scenario can be road users, which reflect the signal emitted by the mobile station and act as new point sources of the reflected electromagnetic wave(s). Thus, multipaths which are received due to the same clutter source are not independent and hence clutter models which enable multiple measurements per clutter source could enhance the proposed data fusion algorithms. Obviously, condition (6.85) then needs to be adapted.
6.3.2.2 Evaluation of Significant Summands
In practical applications, the likelihood function is close to zero or might even be represented by zero for unlikely events due to the numerical resolution of the computer.
Therefore, another practical approach for reducing the number of partitions which have to be considered in equation (6.82) is to evaluate only the terms for which the likelihood function value is above a specific significance–threshold. To this end, a criterion based on the cardinality of the partition elements is developed to determine these partitions. Let π ∈ Π(1:m) be an arbitrary partition which does not fulfill criterion (6.85) andx∈X be an arbitrary target state. Then, if
∃j∈ {1, . . . ,|π|}:|πj|>1 andp|πj|(i(πj)|x)≤τ (6.86) is fulfilled
|π|
X
j=1
1A(πj)p|πj|(i(πj)|x)
|π|
Y
k=1,k6=j
ηPHD/iFilter
π,k ≈0 (6.87)
approximately holds, whereτ >0 is a chosen suitable small threshold for the signif-icance of a partition. Note that|πj|>1 in (6.86) has to be fulfilled due to the first
summand inηPHD/iFilter
π,k , respectively, since otherwise it might happen that only the jth summand of (6.87) is approximately zero, while the other summands are signifi-cantly larger than zero. Hence, condition (6.86) can be used to reduce the number of the considered partitions. Ifτ = 0 in (6.86), “≈” can be replaced by “=” in (6.87).
Note that for the application of this condition the likelihood function has to be eval-uated for all possible subsets and all target states. The number of all possible subsets is given by the binomial series, e.g., for a set ofmmeasurements,
NSubsets=
m
X
k=0
m k
!
= 2m (6.88)
subsets have to be evaluated. However, depending onNminandNmaxthe application of condition (6.85) already reduces the number of subsets which have to be consid-ered significantly, that is for m >0 and 1 < Nmin ≤Nmax ≤m an application of condition (6.85) reduces the number of subsets of the measurement set, which have to be considered to
NSubsets= 2(Nmax+1)−Nmin+1. (6.89)