wherehiis a complex-valued test function defined onXi, that is,hi:Xi→C. Then, the joint PGFL of thenprocesses is defined by
Ψ(h1, . . . , hn)≡
n
Y
i=1
Ψi(hi). (3.1)
The product form of the joint PGFL Ψ(h1, . . . , hn) holds if and only if the point processes corresponding to the test functionsh1, ..., hnare mutually independent (for a more general case that allows allows target correlation, the PGFL must be specified appropriately). In any event, realizations of the joint process are Cartesian products of finite point sets in the spacesXi.
The marginalization with respect to one point process (which is in the studied example from above used to model a single target) is done by setting the corresponding test–
functions equal to the identity function. Thus, the PGFL of theith marginal process is given by
Ψi(hi) = Ψ(. . . ,1, hi,1, . . .). (3.2) Realizations of the ith marginal process are finite point sets in the space Xi. For mutually independent processes, the PGFLs of the marginal processes are identical to the factors in the product (3.1).
Then processes can be superposed if the spaces Xi are identical, which is denoted byX and referred to as the ground space. The fact that there is only one target state space implies that there is only one test–function. The PGFL of the superposed process is given by thediagonal of the joint PGFL, that is,
ΨX(h)≡Ψ(h, . . . , h), (3.3)
where the test function of the superposed processh(·) is defined onX. Realizations of the superposed process are finite point sets inX.
The filters of Section 3.3 donotuse superposition, and therefore have as many (possi-bly different) target state spaces and test functions as there are targets. The pointillist filters of Section 3.4do use superposition, and thus all targets share the same state space and there is only one test function. Finally, the hybrid pointillist filters dis-cussed in 3.5 superpose some targets and not others. There, only the superposed targets share the same state space.
The initial reference time is denoted by t0. The measurement sample times are de-noted by tk,k∈N. It is assumed thattk−1< tk for allk. The recursive time index is suppressed throughout for ease of presentation.
Targets are assumed to be represented by states in the space denoted by X, where X ⊂Rdim(X). Measurements are points in a measurement spaceY ⊂Rdim(Y). The general models used for pointillist filters that do not superpose targets are dis-cussed in this section. Point target and extended target detection models are disdis-cussed in Section 3.2.2. Clutter modelling is described in Section 3.2.3.
3.2.1 Target Motion and Measurement Models
For each target a prior PDF is specified at the recursion start time t0 ≡tk−1. Six PDFs are needed:
• µ0(x0), the (prior) target PDF at timetk−1,
• p0(x|x0), the Markovian target motion (transition) model fromx0∈X at time tk−1 tox∈X attk,
• µ(x), the predicted target PDF at timetk,
• p(y|x), the likelihood function of a measurementy∈Y at timetk conditioned on target statex∈X attk,
• p(y), the PDF of a measurement at timetk conditioned on the sequence of all measurements up to and including timetk−1,
• p(x|y), the Bayes posterior PDF at timetkconditioned on measurements up to time and including timetk.
Three of these PDFs are determined by the others:
µ(x) = Z
X
µ0(x0)p0(x|x0)dx0 (3.4) p(y) =
Z
X
µ(x)p(y|x)dx (3.5)
p(x|y) =µ(x)p(y|x)
p(y) . (3.6)
The last expression is Bayes Theorem.
3.2.2 Target Detection Modeling
Different target detection models are proposed in this section. First, the standard assumption that a target generates at most one measurement is formulated and af-terwards the more general extended target model, which enables a target to generate
multiple measurements that are distributed according to a joint distribution in the measurement space is proposed. Finally, the general target–oriented measurement model is presented in terms of a PGFL.
3.2.2.1 Targets With At Most One Point Measurement
Missed target detections are modeled by assuming that a target that is known to be present at state x ∈ X at time tk is detected with probability PkD(x), where 0≤PkD(x)≤1. The probability of missing the target detection is then 1−PkD(x).
Suppressing the recursion time index, for allx∈X let
a(x)≡1−PkD(x) and b(x)≡PkD(x). (3.7) The corresponding PGF is defined by
GBMDM|x (z)≡a(x) +b(x)z, (3.8) wherez ∈C. Target detection probabilities may or may be not be the same for all targets, depending on the application.
3.2.2.2 Extended Target Model
An extended target is assumed to have a well defined statex∈X, e.g., an appropri-ately defined centroid. It is assumed that extended targets generate a random number M≥0 of independent and identically distributed (point) measurements in the space Y. The distribution of a target–originated measurement is taken to be the likelihood functionp(y|x). The number of measurements can also depend on the target state.
The conditional PGF of the RVM is defined by GM|x(z)≡
∞
X
m=0
Pr{M=m
x}zm, (3.9)
where Pr{M =m|x}denotes the probability thatmmeasurements are generated by the extended target with statex∈X. The target is said to be detected ifM ≥1.
ForM≥1, let
dm(x)≡Pr{M=m
target at statexis detected}, (3.10) so thatP∞
m=1dm(x) = 1. Using the probabilities (3.7) gives GM|x(z) =a(x) +b(x)
∞
X
m=1
dm(x)zm (3.11)
≡a(x) +b(x)GD|x(z), (3.12)
where GD|x(z) is the PGF of the number of measurements generated by a detected target atx. It reduces to the “at most one measurement per target” model forM≡1, that is, toGBMDM|x(z) in (3.8) whend1(x) = 1 andGD|x(z) =z.
3.2.2.3 Generalized Target–Oriented Measurement Model
A more general target–oriented measurement model is used to formulated the gener-alized PHD intensity filters from Sections 3.4.4 and 3.5.2. The genergener-alized PHD filter is originally derived in [CM12], the generalized intensity filter is presented first in (iFilter) [Deg14]. The general target–oriented measurement model used there cannot be formulated using a single PGF due to the fact that in this case measurements that correspond to one target can be correlated (measurements of different targets are assumed to be uncorrelated). Instead the target detection is incorporated into the PGFL of the target–oriented measurement process. The respective PGFL is defined by
ΨgenBMD(h, g)≡ Z
X
h(x)µ(x)
p0(∅|x)
+X
n≥1
1 n!
Z
Yn n
Y
i=1
g(yi)pn(y1, ..., yn|x)dy1...dyn
dx, (3.13)
where pn(y1, ..., yn|x) denotes the generalized symmetric likelihood function, which is defined on Yn. As mentioned in [CM12] the probability of detection is defined more generally than the standard detection model from Section 3.2.2.1. Therefore, p0(∅|x) denotes the probability of a missed detection andpn(y1, ..., yn|x) includes the probability of detecting (y1, ..., yn)T∈Yn, given a specific target statex∈X.
3.2.3 Clutter Modeling
The clutter process (also called the false alarm process) in the measurement space Y is assumed to be either a non–homogeneous time–dependent Poisson point process (PPP), a generalization called a cluster process or an arbitrary general point process.
For all cluster processes, including PPPs, given the number of points, the points are independently and identically distributed (i.i.d). In contrast, clutter models given by general point processes can be used to model correlated clutter measurements. The PGF of the probability distribution of the number of points plays a key role. PPPs are flexible, well understood, and widely used in diverse applications [Str10].
PPP clutter is considered first. Denote the intensity function of the clutter process at timetk byλk(y). The PPP is homogeneous ifλk(y)≡constant onY; otherwise,
it is non-homogeneous. Time dependence is suppressed in the notation, so thatλk(y) is writtenλ(y). The mean number of clutter points inY is
Λ≡ Z
Y
λ(y)dy. (3.14)
To assure that Λ is finite (and simplify the discussion), it is assumed that Y is bounded. Define the clutter PDFpΛ(y) to be the normalized intensity function,
pΛ(y)≡λ(y)/Λ. (3.15)
In this notation, the PGFL of the PPP clutter process is [Kar91]
ΨPPPC (g)≡exp
−Λ + Λ Z
Y
g(y)pΛ(y)dy
, (3.16)
where the test–functiong:Y →Cis bounded by one, non–negative and Lebesgue–
integrable.
The PGFL of clutter when modeled as a cluster process is defined by ΨClusterC (g)≡GC
Z
Y
g(y)q(y)dy
, (3.17)
whereq(y) is the PDF of a clutter pointy∈Y andGC(z) is the PGF of the number of clutter points, namely,
GC(z)≡
∞
X
c=0
Pr{C=c}zc, (3.18)
where Pr{C=c}denotes the probability thatcclutter measurements are generated.
Note that (3.16) is a special case of (3.17) forGC(z) = exp(−Λ + Λz).
The PGFL of an arbitrary general clutter process is denoted by ΨgenC (g) in the fol-lowing.