Problem statement and assumptions

As it is customary in stochastic filtering problems, let us fix an underlying probability space (Ω, F (X ), P)on which random experiments will be defined, where Ω is the sample space, F(X )

CHAPTER 3. MULTI-TARGET TRACKING OF DEPENDENT TARGETS

is a σ-algebra on some (state) space X , characterizing a collection of probabilistic events, and P is a probability measure that associates probabilities to events. It is generally assumed that each target follows a state process {xt}_t≥0, xt ∈ X ⊆ R^dx, from which realizations can be evaluated at subsequent discrete-time steps t ∈ {t0, t1, . . . , tk, . . . |k ∈ N⁰}, and so we denote xk := xt_k hereafter. Likewise, an observation process, {yt}t≥0, yt ∈ Y ⊆ R^dy, is defined to describe measurements about target states, from which discrete-time outcomes, yk := ytk, are provided by at least one sensor. The description also establishes the measurable spaces of the state and observation processes respectively, (X , Ft(X )) and (Y, Ft(Y)), with σ-fields Ft(X ) , σ{x^t|t ≥ 0} and Ft(Y) , σ{y^t|t > 0} that compose the filtrations {Ft(X )}t≥0 and {Ft(Y)}t≥0 of the σ-algebra F. From now on, when talking about the state and observation processes, we refer to their discrete-time versions, {xk}_k∈N0 and {yk}_k∈N.

In this chapter we index objects by finite sets of distinct natural numbers as

In := {i1, . . . , in|i1, . . . , in∈ N, i16= · · · 6= in}

for n elements, and any such index set will be of the same form. In multiple target tracking, a number n of targets may possibly be in the scene. By indexing all targets in the scene by In,k := {i1, . . . , in}, each target state is described as x⁽ⁱ⁾_k such that i ∈ In,k. The set of all targets in the scene at time step k will be denoted as x^(1:n)k = x^I_k^n,k := {x⁽ⁱ⁾_k |i ∈ I_n,k}. In favor of simplicity, throughout this chapter the notation x^(1:n)k is used interchangeably with x^Ik^n,k to represent the finite set containing all objects x⁽ⁱ⁾_k indexed by In,k. Though we acknowledge that an index set In,kmight contain natural numbers (identifiers) not necessarily in the interval [1..n], in which case x^(1:n)_k would be an abuse of notation. Similarly, a number m of measurements about targets are collected, indexed by Im,k := {j1, . . . , j_m}, and so each measurement can be identified as y^(j)k such that j ∈ Im,k. The complete set of measurements is denoted as Y_k , y^(1:m)k := {y^(j)_k |j ∈ I_m,k} and we write the history of observations up to time step k as Y1:k := {Y1, . . . , Yk}. Multi-object state and observation spaces can be denoted as the Cartesian products Xⁿ = Q

i∈In,kX and ¯Y^m = Q

j∈Im,k

Y¯ respectively, where ¯Y , Y ∪ Yfa

and, in this context, the observation process is assumed to admit false alarms y^(j)k,fa ∈ Y_fa as realizations and missed detections may be present (m ≤ n).

The ultimate goal of any multiple-target tracker is to infer estimates about target states conditioned on the information provided by the observation history, that is

P_k[ϕ] = Eh

ϕ(x^(1:n)_k )|F_k( ¯Y^m)i

≡ ˆ

Xⁿ

ϕ(x^(1:n)_k )P_k(dx^(1:n)_k , Y_1:k), (3.1)

for a test function ϕ : Xⁿ → X⁰, measurable in Fk(Xⁿ) = σ{x^(1:n)_t |0 < t ≤ tk}, where P_k(·, Y_1:k)is a posterior multi-target probability measure, and x^(1:n)_k should be understood in terms of the joint event X^(1:n)k =Tn

i=1X⁽ⁱ⁾_k in the measurable space (Xⁿ, F_k(Xⁿ)). In practical terms, computing estimates as in (3.1) involves expressing the posterior multi-target measure explicitly, which constitutes the main scope of almost all multiple target trackers. In order to express the multi-target posterior measure, trackers require further assumptions, which are typically given as follows.

Assumption 3.1. Each target moves independently of one another, with motion modeled by a single-target Markov transition density pt(x⁽ⁱ⁾_k |x⁽ⁱ⁾_k−1). Each target state is assumed to follow a

148

state equation of the form

x⁽ⁱ⁾_k = f(x⁽ⁱ⁾_k−1) + g(ξ_k), i ∈ I_n,k−1, (3.2) where f : X → X is the state transition function, and g : X → X is a function of the state noise process {ξk}_k∈N0, ξk ∈ X , with independent realizations over time steps.

Assumption 3.2. Measurements generated from targets are independent on one another, with single-target likelihood function `j(x<sub>k</sub>) = p<sub>`</sub>(y_k^(j)|x_k). Each target state is assumed to produce at most one measurement, obtained according to an observation equation of the form

y_k⁽ⁱ⁾= h(x⁽ⁱ⁾_k ) + υk, i ∈ In,k, (3.3) where h : X → Y is the observation function, and {υk}_k∈N0, υk ∈ Y, is the observation noise process, whose realizations are independent over distinct measurement outcomes, and independent of the state noise process {ξk}_k∈N0.

Assumption 3.3. Each sensor detects each target with state x⁽ⁱ⁾_k with probability pd(x⁽ⁱ⁾_k ).

Hence, there may be misdetections with probability q_d(x⁽ⁱ⁾_k ) = 1 − p_d(x⁽ⁱ⁾_k ).

Assumption 3.4. False alarms (clutter) may affect the observation process, being indepen-dent of the target-originated measurements. The number of false alarms is assumed Poisson-distributed, i.e., m_fa ∼ Pois(m|λ_fa), where λ_fa is expected number of false alarms per time frame, and the position of each false alarm is assumed uniformly distributed in a surveillance region occupying a volume V , i.e. yfa ∼ U (∂V ) where ∂V is the boundary (surface) of the surveillance region. We define the expected volumetric density of false alarms per time frame according to λV,fa= λfa/V . At each time step k, the complete set of measurements is then given by

y_k^Im,k = y^I_k^n,k∪ y^I_fa^mfa,k

and sensors cannot distinguish target-generated measurements from false alarms.

In the Joint Probabilistic Data Association (JPDA) framework, estimating the joint pos-terior distribution Pk is structured by the additional assumption that target’s states are in-dependent conditioned on the observation information. Denoting p a probability density of Pk with respect to the Lebesgue measure, the posterior multi-target density under the JPDA framework would be of the product form, p(x^(1:n)_k |Y_1:k) =Qn

i=1p(x⁽ⁱ⁾_k |Y_1:k), where p(x⁽ⁱ⁾_k |Y_1:k) are the marginal probability densities of each target state. This form simplifies the multi-target density representation since it can be completely described by the individual target marginal densities. However, this simplicity comes at a cost: loss of ability to explain dependency between targets that may probabilistically share the same measurement in the observation path.

The uncertainty of whether a measurement may have originated from each of two (or more) targets arises when they are in sufficient proximity such that the measurement is likely to have come from either of them. In addition, such uncertainty is exacerbated if targets remain in mutual proximity for several time steps and, in that case, assuming target state independence would discard sensible information. With the intent of partially avoiding this information loss, Horridge & Maskell [94] proposed a filter where target dependency is maintained via a

CHAPTER 3. MULTI-TARGET TRACKING OF DEPENDENT TARGETS

targets. In this context, the filter proposed by Horridge & Maskell [94] makes two additional assumptions as follows.

Assumption 3.5. The marginal probability densities of each target state assume a mixture form according to

p(x⁽ⁱ⁾_k |Y_1:k) = X

c⁽ⁱ⁾_k ∈Γ⁽ⁱ⁾_k

p(x⁽ⁱ⁾_k |c⁽ⁱ⁾_k , Y_1:k)p(c⁽ⁱ⁾_k |Y_1:k), (3.4)

where c⁽ⁱ⁾_k is a discrete random variable taking values in a discrete countable set Γ⁽ⁱ⁾_k = {γ₁⁽ⁱ⁾, γ₂⁽ⁱ⁾, . . . , γ⁽ⁱ⁾_n

γ,k(i)},

with posterior probabilities expressed as p(c⁽ⁱ⁾_k |Y_1:k) = Pr{c⁽ⁱ⁾_k |Y_1:k}, and where n_γ,k(i) = |Γ⁽ⁱ⁾_k | is the event set cardinality for the component c⁽ⁱ⁾_k . The target states are assumed independent conditioned on the set of component variables c^(1:n)_k = {c⁽¹⁾_k , . . . , c⁽ⁿ⁾_k }, i.e.

p(x^(1:n)_k |c^(1:n)_k , Y_1:k) =

n

Y

i=1

p(x⁽ⁱ⁾_k |c^(1:n)_k , Y_1:k),

but the components themselves may be dependent on each other over all targets.

Assumption 3.6. The target dependency structure is described by a probabilistic tree, G_T = (VT, ET), where VT = In,k is the set of vertices and ET = {(u, v)|u, v ∈ VT, u 6= v} is a set of vertice pairs that explains edges in the tree. In this tree, each node represents a target i, proba-bilistically described by a (discrete) random variable, c⁽ⁱ⁾_k , and the edges represent probabilistic dependency between two targets. Adjacent nodes in the tree are related by discrete conditional distributions of the form p(c⁽ⁱ⁾_k |c^pa(i)_k , Y_1:k), where pa(i) stands for the parent of node i. In addition,

i ⊥⊥ an(i) \ pa(i) | pa(i), i ∈ I_n,k, (3.5) where an(i) stands for the set of all ancestors of node i, which means that p(c⁽ⁱ⁾_k |c^an(i)_k , Y_1:k) = p(c⁽ⁱ⁾_k |c^pa(i)_k , Y_1:k). The dependency structure over all targets can be expressed by the joint dis-tribution:

p(c^(1:n)_k |Y_1:k) =

n

Y

i=1

p(c⁽ⁱ⁾_k |c^pa(i)_k , Y_1:k). (3.6) We will increment those assumptions to: (i) model the uncertainty of target existence, (ii) maintain a set of unconfirmed tracks, (iii) model the possibility that new targets may appear in the scene. These assumptions will be used to develop a mechanism for track management, involving initiating, confirming, and deleting tracks. As a consequence, conditional probability densities of target states in Assumption 3.5 will also be conditioned on the target existence.

This is particularly useful for deciding whether a track (on a hypothetical target) should be confirmed or discarded, based on its evaluated probability of existence.

Assumption 3.7. Targets can exist or not, in a probabilistic sense, according to a binary existence random variable ek ∈ {0, 1} at time step k. The existence process, {ek}_k∈N, is assumed to follow a first-order Markov process. Let E⁽ⁱ⁾_k , {e⁽ⁱ⁾k = 1} be the event that target i exists, and E¯_k⁽ⁱ⁾, {e⁽ⁱ⁾k = 0} the complementary event. The posterior existence probability of each target is described by p(E⁽ⁱ⁾_k |Y_1:k). The Markov transition kernel of the existence process, denoted by

150

p(e⁽ⁱ⁾_k |e⁽ⁱ⁾_k−1), models possibilities that an existing target at time step k − 1 may die (disappear) or survive until the next time step. The probability of survival is denoted as ps= p(E_k⁽ⁱ⁾|E_k−1⁽ⁱ⁾ ).

Marginal probability densities of each target state will be conditioned on target existence and read as

p(x⁽ⁱ⁾_k |E_k⁽ⁱ⁾, Y_1:k) = X

c⁽ⁱ⁾_k ∈Γ⁽ⁱ⁾_k

p(x⁽ⁱ⁾_k |c⁽ⁱ⁾_k , E_k⁽ⁱ⁾, Y_1:k)p(c⁽ⁱ⁾_k |Y_1:k). (3.7)

Assumption 3.8. Targets that have never been detected, or have been detected but show prob-ability of existence within the interval p(E_k⁽ⁱ⁾|Y_1:k) ∈ (τdel, τconf), where τdelis a lower threshold for track deletion and τ_conf is an upper threshold for track confirmation, take part in set of nu,k unconfirmed targets, indexed by Iu,k = {i^u₁, . . . , i^u_nu,k}. For simplicity, each unconfirmed target has a single state component, distributed according to p(x⁽ⁱ⁾_k |E_k⁽ⁱ⁾, Y_1:k). While keeping track of unconfirmed targets, if p(E⁽ⁱ⁾_k |Y_1:k) > τ_conf for a target i ∈ I_u,k, then the corresponding track is confirmed, removed from the unconfirmed set, and incorporated in a set of nc,k con-firmed targets, indexed by I_c,k = {i^c₁, . . . , i^c_n

c,k}. After each filtering cycle, if any confirmed or unconfirmed target has p(E⁽ⁱ⁾_k |Y_1:k) < τdel, then its track is deleted.

Assumption 3.9. New targets may appear independently of the existing targets at time step k, being indexed by I_b,k = {i^b₁, . . . , i^b_n

b,k}. The number nb = n_b,k of new appearing targets is assumed to be Poisson-distributed according to nb ∼ Pois(n|λb) where λb = λV,bV is expected number of new targets per time frame, and λ_V,b is the expected volumetric density of new targets per time frame. Each new target i ∈ Ib,k is assumed to be spatially distributed according to pbirth(x⁽ⁱ⁾_k ). Each new target has existence probability given by pb, λ^b.

The filter that results from Assumptions 3.1–3.9 requires a solution for the data association problem. Later on we will explain how uncertainty of measurement-to-target associations will be taken into account, but for now it suffices to say that the posterior (marginal) association probabilities will be exactly computed by an algorithm reminiscent of Pearl’s algorithm [154]

in a Bayesian network¹, called Efficient Hypothesis Management (EHM) [135, 95, 94]. We will adopt the third variation of EHM as in [94], which handles multiple state (mixture) components per target. The resulting filter will be referred to as JPDA-EHM3 hereafter.