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Consensusbased distributed adaptive target tracking in camera networks using Integrated Probabilistic Data Association
EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 13 (2018)
Abstract
In this paper, a novel consensusbased adaptive algorithm for distributed target tracking in large scale camera networks is presented, aimed at situations characterized by limited sensing range, highlevel clutter, and possibly occulted targets. The concept of Integrated Probabilistic Data Association (IPDA) is introduced in the distributed adaptive tracker design so that the proposed algorithm, named IPDA Adaptive Consensus Filter (IPDAACF), incorporates probabilities of acquiring targetoriginated measurements, conditioned on either target perceivability or target existence. A distributed adaptation scheme represents the core element of the algorithm, allowing fast convergence under a large variety of operating conditions, emphasizing the influence of the nodes with the highest probability of obtaining targetoriginated measurements. A theoretical analysis of stability and reduction of noise influence allows getting an insight into the relationship between the local trackers and the global consensus scheme. A comparison with analogous existing methods done by extensive simulations shows that the proposed method achieves the best performance, in spite of lower communication and computation requirements.
1 Introduction
Recent rapid improvement in quality and resolution of imaging sensors and availability of lowcost smart cameras, together with the development of sensor network technology, have paved the way for creation of large scale camera networks. Examples of very successful applications are more and more numerous, especially in the fields of widearea surveillance, disaster response, and environmental monitoring. Detection and tracking of objects of interest is one of the fundamental functions of camera networks, see, e.g., [1] and the references therein.
In this context, distributed estimation schemes are becoming increasingly popular, due to their high precision and scalability to large number of sensors, with no need for centralized actions, e.g., [2–4]. In spite of the fact that the theoretical fundamentals of target tracking are basically the same for different types of sensors (radars, sonars, cameras), camera networks impose some unique and interesting challenges, arising from the fact that (1) most of the cameras are directional and have limited sensing range (LSR) or fieldofview (FoV) so that at each time instant there might be a significant share of sensors that do not observe the target (see Fig. 1 as an illustration) and (2) environment is highly cluttered and targets can be temporarily occulted. Majority of the existing distributed estimation algorithms of general type based on consensus cannot be directly applied in this context. The Kalmanconsensus filter (KCF) [5] has been applied to target tracking problems by sensor networks with LSR in [6]. The algorithm assumes internode exchange of local target state estimates, together with the socalled information vectors and matrices, which depend on local measurements and their covariances. The informationweighted consensus filter (ICF) has been found to outperform KCF under limited local observability scenarios [4]. It uses the fact that the local information contained in the nodes becomes correlated as the state estimates converge. The communication requirements incorporate, besides those of KCF, the local error covariance matrices. Both KCF and ICF have been applied to distributed tracking problems in cluttered environment [7, 8], incorporating elements of the Probability Data Association (PDA) methodology [9–11]. In [12], an algorithm for decentralized state estimation has been proposed, based on a combination of local estimators of Luenberger type and a flexible dynamic consensus strategy. It assumes exchange of only state estimates between neighboring nodes. Its adaptive modification has been proposed in [2] in the context of target tracking problems with sensors having limited sensing range, but without assuming presence of clutter. We shall refer to this algorithm as Adaptive Consensus Filter (ACF). It has been demonstrated that ACF largely outperforms KCF and slightly outperforms ICF, preserving, at the same time, significantly lower communication bandwidth requirements [3].
In this paper, we assume cluttered environment and possibly temporarily occulted targets and propose a new distributed adaptive tracking algorithm for camera networks, representing an extension of ACF from [2] based on Integrated Probabilistic Data Association (IPDA); this algorithm will be denoted further as IPDAACF. The concept of IPDA has been introduced by Musicki, Evans, and Stankovic in [13] and further extended in different directions, see, e.g., [9–11, 14, 15]. The notion of target perceivability [14, 15] has been utilized to construct two distinct consensusbased tracking algorithms, having the form of the basic Probabilistic Data Association (PDA) recursion [9–11]. The algorithms deal with two types of the socalled “ β”parameters, representing probabilities of getting a targetoriginated measurement conditioned either on target perceivability or on target existence; in this sense, we have the algorithms denoted as IPDAACF^{1} and IPDAACF^{2}, respectively. The core element of the proposed algorithm, making it competitive and even superior to the algorithms with higher communication requirements proposed in [7, 8], is the distributed adaptation scheme, developed as a substantial generalization of the adaptation strategy originally described in [2, 3] (see also [16]). The adaptation scheme, aiming at giving emphasis to the nodes with the highest probability of obtaining targetoriginated measurements, is based on the locally obtained “ β ”parameters as indicators of target observability in cluttered environment; these parameters are used to improve both tracking accuracy and convergence rate of achieving consensus among the nodes. Stability and noise immunity of the whole distributed tracking method are also theoretically analyzed in detail, by applying the methodology introduced in [2, 12], as well as the recent results concerning properties of the modified deterministic discrete Riccati Eq. [12, 17, 18]. The paper also introduces appropriate modifications of the original versions of KCF and ICF incorporating IPDA methodology [7, 8], necessary for a fair comparison with the proposed algorithm (IPDAKCF and IPDAICF). Tracking quality of the proposed algorithm has been analyzed in detail by simulation. The presented results give an insight into the main properties of the algorithm and show that it outperformsboth alternative algorithms, in spite of lower communication and computation requirements.
The outline of the paper is as follows. Section 2 is devoted to the problem definition and the basic notions related to the IPDA methodology. Section 3 contains a description of the new distributed tracking algorithm (IPDAACF), including the algorithm definition and presentation of the adaptation strategy. A theoretical analysis of stability of the proposed adaptive tracking algorithm, as well as of the reduction of noise influence is given in Section 4. Section 5 is devoted to the formulation of IPDAKCF and IPDAICF algorithms. Section 6 contains the results of simulation analysis, and Section 7 concluding remarks.
Notation. In the paper, we shall use t for discrete time and subscript i to indicate ith node (sensor) in a network; more specifically, we shall use the following notations:

x Target state vector

F State transition matrix

e Process noise vector

G Process noise covariance matrix

\(\mathcal {G}\) Directed graph reflecting the
network communication topology

\(\mathcal {N}_{i}\) Inneighborhood of ith node

\(\mathcal {J}_{i}\) Closed inneighborhood of ith node

z_{i,j}jth measurement

v_{ i } Measurement noise vector

H_{ i } Measurement noise covariance matrix

Z_{ i }(t) Set of all measurements at time t

\(Z_{i}^{t}\) Set of all measurements up to time t

\(O_{i}^{t}\) Event that the target is perceivable at time t

π_{ i }(tt−1) Predicted probability of target perceivability

π_{ i }(tt) Updated probability of target perceivability

\(P^{D}_{i}\) Probability of detection

\(P^{g}_{i}\) Probability of targetoriginated measurement
falling inside the gate (FoV)

\(\tilde {z}_{i,j}\) Residual of the jth measurement

S_{ i } Covariance of \(\tilde {z}_{i,j}\)

λ_{ i } Clutter spatial density

V_{ i } Gate volume

θ_{i,j} Event that the jth measurement
is target originated

θ_{i,0} Event that
the targetoriginated measurement
is not in the gate or nonexistent

\(\beta ^{[1]}_{i,j}\) Probability of θ_{i,j}, j=0,1,…,m_{ i },
conditioned on \(O_{i}^{t}\) and on \(Z_{i}^{t}\)

\(\beta ^{[2]}_{i,j}\) Probability of θ_{i,j}, j=0,1,…,m_{ i },
conditioned on the event that the target
is present and on \(Z_{i}^{t}\)

ξ_{ i }(tt−1) Predicted state estimate

ξ_{ i }(tt) Updated state estimate

P_{ i }(tt−1) Predicted estimation error
covariance matrix

P_{ i }(tt) Updated estimation error
covariance matrix

L_{ i } Kalman gain matrix

\(\mathcal {C}\), \(\mathcal {A}\) Single step consensus operators

ω_{ i } Scalar defining node importance

u_{ i } Information vector

U_{ i } Information matrix

y_{ i } Sum of information vectors
across neighbors

B_{ i } Sum of information matrices
across neighbors

∥·∥ Spectral norm
2 Problem definition; Integrated Probabilistic Data Association
Consider a target modeled as a dynamic stochastic linear timeinvariant discretetime system, given by
where \(t \in \mathcal {I}^{+}\), \(\mathcal {I}^{+}\) is the set of nonnegative integers, \(x \in \mathbb {R}^{n}\) is the target state vector, e zeromean white Gaussian noise with covariance matrix Q, and F and G constant matrices with appropriate dimensions (see, e.g., [19]).
Assume that we have Nintelligent sensors forming a network, formally represented by a directed graph\( \mathcal {G}=(\mathcal {N},\mathcal {E})\), where \(\mathcal {N}\) is the set of nodes and \(\mathcal {E}\) the set of arcs. Assume also that \(\mathcal {N}_{i}\) is the inneighborhood of the node i, containing the head nodes of the arcs entering the node i; define also \(\mathcal {J}_{i}=\mathcal {N}_{i} \cup \{i\}\). Each node (camera) is supposed to have, in general, limited sensing and communication ranges, which determine at each t the set of nodes that can observe the target and the sets of neighboring nodes exchanging messages (these sets may contain, in general, low percentage of the total number of nodes). We assume that real measurements are cluttered, i.e., in addition to data originating from the target, a set of measurements corresponds to no targets [9–11, 19]. At time t, node i gets m_{ i }(t) measurements, denoted as z_{i,j}(t), j=1,…,m_{ i }(t). Under the hypothesis that the measurement z_{i,j}(t) originates from the target, the sensing model for the ith node is given by
where \(z_{i,j}(t) \in \mathbb {R}^{p_{i}}\) is the measurement vector, H_{ i } a constant output matrix, and v_{ i } zeromean white Gaussian measurement noise with covariance matrix R_{ i }. Let \(Z_{i}(t) = \{z_{i,1}(t), z_{i,2}(t), \ldots, z_{i,m_{i}(t)}(t) \}\phantom {\dot {i}\!}\) denotes the set of all \(m_{i}(t)=m_{i}^{T}(t) + m_{i}^{f}(t)\) measurements obtained by node i at time t, where \(m_{i}^{T}(t)\) is the number of targetoriginated measurements, and \(m_{i}^{f}(t)\) the number of false clutteroriginated measurements. Let \(Z_{i}^{t}=\{Z_{i}(1), \ldots, Z_{i}(t) \} \).
The purpose of tracking is to estimate the state of a target in real time, based on a set of current measurements. For tracking in clutter, conventional state estimation techniques cannot be used because several measurements are available at every scan, and at most, one measurement can arise from the target. The elegant Probabilistic Data Association (PDA) methodology has been successfully applied in this case, including a variety of very successful applications to radars, sonars, and electrooptic systems, e.g., [9–11, 20–25]. This methodology is based on the implicit strong assumption that the target is always perceivable. The concept of target perceivability is a part of the general methodology of Integrated Probabilistic Data Association (IPDA) and represents a refinement of the concept of track existence (which does not address the possibility that a target cannot be detected) and of target observability (which presumes the presence of a target) [13–15]. According to [15], a target is perceivable if it is present and can be detected by the sensors used. If \(O_{i}^{t}\) and \(\bar {O}_{i}^{t}\) denote the events that the target is perceivable and not perceivable, respectively, at time t by node i, we shall follow [13] and assume that the target perceivability can be modeled by a first order Markov model, with \(p_{i}^{11}=P\left (O_{i}^{t}O_{i}^{t1}\right)\) and \(p_{i}^{21}=P\left (O_{i}^{t}\bar {O}_{i}^{t1}\right)\), see [14, 15]. If π_{ i }(tt−1) and π_{ i }(tt) are the predicted and updated probabilities of target perceivability for a given track, respectively, the total probability theorem gives
According to the basic references related to the IPDA methodology [13–15], the updated and predicted target perceivability probabilities are connected by
where
for m_{ i }(t)=m≠0, and
for m_{ i }(t)=0, \(P^{D}_{i}\) is the detection probability (assuming that the target is perceivable), \(P^{g}_{i}\) is the probability that a targetoriginated measurement falls inside the gate (FoV) of the sensor i,
for m_{ i }(t)≠0, and
for m_{ i }(t)=0, where \(\tilde {z}_{i,j}(t)=z_{i,j}(t)\hat {z}_{i,j}(t)\) is the residual of the jth measurement by node i (\(\hat {z}_{i,j}(t)\) denotes the optimal estimate of z_{i,j}(t)), and S_{ i }(t) its covariance,
\(\mu _{i}(t,m)=P \left \{m_{i}^{f}(t)=mZ_{i}^{t1} \right \}\), λ_{ i }(t) is the clutter spatial density and V_{ i }(t) the gate volume (under the classical Gaussian assumption for the target and the Poisson model for clutter [14, 15]).
Let θ_{i,j} be the event that the jth measurement of the ith node is target originated, and θ_{i,0} that the targetoriginated measurement is not in the gate (FoV) or nonexistent. Define
It is possible to show that the probabilities \(\beta ^{[1]}_{i,j}(t)\) are identical to the wellknown weights (“ β ”parameters) of the classical PDA filter [10, 11]:
for j=1,…,m_{ i }(t), where \(q_{i}=P^{D}_{i} P^{g}_{i}/\left (1P^{D}_{i} P^{g}_{i}\right)\). Probabilities \(\beta ^{[1]}_{i,j}(t)\) are used within the formulation of the target perceivabilitybased PDAF in [15].
In general, it can be of interest to estimate the state of a target that is presentno matter whether it can be detected or not. One may define the following probabilities [15]:
where T stands for the event that a target is present. Following [15], one can show that
for j=1,…,m_{ i }(t), where \( r_{i}(t)=P^{D}_{i} P^{g}_{i} \pi _{i}(tt1)/\left [1P^{D}_{i} P^{g}_{i} \pi _{i}(tt1)\right ]. \) The probabilities (8) represent a part of the target existencebased PDAF formulated in [15] and depend on the probabilities (6) and the perceivability probabilities (4). Recently, the concept of target existence probability has been used in the context of multitarget tracking in multistatic passive radar systems [26]. Notice that \(\beta ^{[2]}_{i,j}(t) \leq \beta ^{[1]}_{i,j}(t)\), j=1,…,m_{ i }(t) and that, consequently, \(\beta ^{[2]}_{i,0}(t) \geq \beta ^{[1]}_{i,0}(t)\) (see [14, 15] for a more complete discussion).
In the context of distributed target tracking by sensor networks considered in this paper, all the above statements hold locally, for each sensor. It will be seen below that the choice between \(\beta ^{[1]}_{i,j}(t)\) and \(\beta ^{[2]}_{i,j}(t)\) offers an additional possibility for adaptation to the current situation concerning the target, the entire sensor network, and its environment.
3 Adaptive consensus filter: IPDAACF
3.1 Tracking algorithm
In this subsection, we shall present the main IPDAACF algorithm, with its two versions (IPDAACF ^{l}, l=1,2), depending on the choice of the “ β ”parameters: for l=1, one assumes target perceivability (IPDAACF^{1}) and for l=2 target existence (IPDAACF^{2}). The algorithm has the following form:
i=1,…,N, where ξ_{ i } is an estimate of x generated by the ith node, \(\tilde {z}_{i}(t)= \sum _{j=1}^{m_{i}(t)} \beta ^{[l]}_{i,j}(t) \tilde {z}_{i,j}(t), \tilde {z}_{i,j}(t)=z_{i,j}(t)H_{i} \xi _{i}(tt1)\), l=1,2, \(\mathcal {C}(\cdot)\) is a single step consensus operator
where c_{ ij }(t), i,j=1,…,N, are (generally) time varying weights, such that N×N matrix C(t)=[c_{ ij }(t)] (consensus matrix) is rowstochastic for all t, and
is the local Kalman gain obtained from (1) and (2), using
The above algorithm follows structurally the consensusbased overlapping estimation algorithm proposed in [12]. Instead of the local estimators of Luenberger type from [12], we have here trackers of PDA type [10]. The algorithm contains two main parts:

The filtering part, in which the local measurements are processed

The prediction part, in which the agreement between the nodes is enforced through a convex combination of the estimates communicated by the neighboring nodes \(\mathcal {C}(\xi _{i}(tt))\).
Obviously, the algorithm requires only the exchange of state estimates (size n×1). The choice of l in \(\beta ^{[l]}_{i,j}(t)\) results from a predefined setting. For l=1, one obtains the estimation scheme resulting directly from the PDAF [10], while for l=2, the algorithm follows the existencebased PDAF from [15] and becomes more capable of coping with high level of clutter and possible target occlusions (typical for camera networks  see, e.g., [27] for a general discussion).
The choice of the consensus matrix C(t) can be based, in general, on different principles, including the wellknown standard Metropolis weights (see, e.g., [28] and the references therein) and optimization procedures providing the fastest convergence to consensus [29, 30]. However, in the above context (limited FoV, clutter, target occlusions), such approaches provide insufficient tracking accuracy of the proposed algorithm in realtime applications. One should bear in mind that we have, in the context of the proposed algorithm, the socalled “running consensus” [31, 32], consisting of an inseparable pair estimation algorithm  consensus algorithm, each possessing its own dynamics. It would be possible, in principle, to formulate the problem of defining the consensus matrix in a theoretically unified way, including both the estimation algorithm and the consensus scheme; however, such methodologies can hardly provide results applicable in real time. In order to provide a practically efficient solution, the next subsection contains a presentation of a distributed adaptation applicable in real time, aiming at providing adequate timevarying consensus weights in (10), using the locally available realizations of the βparameters, in accordance with the basic principles of the IPDA methodology.
3.2 Distributed adaptation strategy
Distributed adaptation strategy represents the core element of the IPDAACF algorithm. In the context of the scenario considered in this paper, we follow the basic line of thought from [2, 3, 12] and construct a novelimproved fully distributed adaptation procedure generating C(t) in (10), dynamically giving more importance to the nodes with higher probability of receiving targetoriginated measurements, enabling, at the same time, fast information flow through the network. In the sequel, we shall first give a short outline of the original adaptive scheme from [2], and then, we shall present the general adaptation strategy for the tracking algorithm presented above.
1) Basic adaptation scheme: no clutter
The distributed adaptive tracking algorithm ACF in [2] has been derived from [12]; it requires additional internode communication of \(\beta ^{[1]}_{i,0}(t) \in \{0,1 \}\) that are available locally (indicators of measurement availability at time t). Using this information, each node computes \(\mathcal {J}_{i}\) scalars \(\chi ^{i}_{j}(t)\), representing observation histories of the neighboring nodes. The nonnormalized consensus gains are given by \(c_{ij}^{\chi }(t) = c_{ij}^{\chi }(0) k^{\chi ^{i}_{j}(t)},\) for \(\chi ^{i}_{j}(t)\geq 0,\) and \( c_{ij}^{\chi }(t) = c_{ij}^{\chi }(\infty)\left (c_{ij}^{\chi }(\infty)c_{ij}^{\chi }(0)\right)k^{\chi ^{i}_{j}(t)},\) for \( \chi ^{i}_{j}(t) \leq 0\), where \(c_{ij}^{\chi }(0)\) are the initial values and \(c_{ij}^{\chi }(\infty)\) the desired final values; parameter k∈(0,1] determines the rate of change of \(c_{ij}^{\chi }(t)\). The normalized consensus gains, ensuring row stochasticity, are obtained from \(c_{ij}(t)=\frac {c_{ij}^{\chi }(t)}{\sum _{j\in {\mathcal {J}}_{i}} c_{ij}^{\chi }(t)}\), when \( j \not \in \mathcal {J}_{i}\), c_{ ij }(t)=0. It has been found that the resulting ACF algorithm from [2] outperforms KCF from [6].
2) General adaptation strategy
The adaptation procedure proposed in this paper essentially extends and modifies the basic adaptation scheme from [2] in two principal directions: (1) instead of binary indicators of target presence, we utilize real numbers \(\beta _{i,0}^{[l]}(t)\) defined by (6) or (8) since they reflect the uncertainty of obtaining a targetoriginated measurement; (2) in order to enable fast diffusion of the target state estimates throughout the network, the adaptation scheme incorporates its own, specially designed dynamics.
Formally, we introduce the node importance vector ω(t)=(ω_{1}(t),…,ω_{ N }(t))^{T}, generated by the following recursion:
which starts from ω(0)=(ω_{1}(0),…,ω_{ N }(0))^{T}=(0,…,0)^{T}, where γ(t)=(γ_{1}(t),…,γ_{ N }(t))^{T}, \(\gamma _{i}(t)=1\beta _{i,0}^{[l]}(t)\), A=(I+A_{adj})_{rs}, where A_{adj} represents the adjacency matrix of the underlying network graph, and Y=(X)_{rs} denotes, for N×N matrices X and Y, that Y is obtained from X in such a way that the elements of its rows are divided by the corresponding rowsums, while α>0 is a sufficiently small real number. Using the node importance vector, the elements of the consensus matrix C(t) are given by
for \(j \in \mathcal {J}_{i}\); c_{ ij }(t)=0 otherwise.
The algorithm requires that each node receives at each t real numbers ω_{ j }(t) from its neighbors. It is obvious that the recursion (13) is implementable in a distributed way, having in mind that locally
where ν_{ i } is the number of nonzero elements of ith row of A_{adj}.
The input vector γ(t+1) in (13) reflects the current networkwide perceivability, i.e., its elements represent the local probabilities of getting targetoriginated measurements (in the case of no clutter, γ_{ i }(t) is a binary random variable, equal to one when the target is observed, and to zero otherwise [2]). It can be considered that γ_{ i }(t) provides a direct measure of “quality” of the current estimate of the ith node, obtained after the local filtering phase (prior to the application of the consensus scheme). Namely, when a target is successfully being tracked and ith node receives measurement from it, \(\beta _{i,j}^{[l]}(t)\) connected to that measurement has high values (close to one), while all the other \(\beta _{i,j}^{[l]}(t)\) are generally small, as well as \(\beta _{i,0}^{[l]}(t)\) (close to zero). On the other hand, if the ith node does not receive measurement from the target, all \(\beta _{i,j}^{[l]}(t)\) will be generally close to zero and \(\beta _{i,0}^{[l]}(t)\) close to one. Obviously, the difference between the choices l=1 and l=2 lies in the fact that in the second case, the resulting tracking algorithm is less sensitive to the availability of measurements than in the case when l=1; this choice is well adapted to the high level of clutter and to the targets that are temporarily not perceivable (as stated above). Another choice for γ_{ i }(t) could be \(\gamma _{i}(t)=\max _{j=1, \ldots, m_{i}(t)}\beta _{i,j}^{[l]}(t)\), which yielded in our experiments the results comparable to those obtained by the above formulated algorithm.
The role of the first term at the right hand side of (13) containing αA is to enable fast decentralized diffusion of the local state estimates throughout the whole network. Under the standard assumption that the graph \(\mathcal {G}\) has a center node (a node from which all the nodes are reachable), matrix A has one simple eigenvalue at one, while the remaining ones are inside the unit circle. In this case, we have that A^{k} converges, when k→∞, to a constant matrix with equal rows composed of positive elements. This means, having in mind (13), that all the nodes will have asymptotically positive values ω_{ i }(∞), i=1,…,N. The multiplying constant α defines the memory length of the algorithm; obviously, for α<1, the recursion (13) is asymptotically stable. The chosen value of α should be small, in order to be able to efficiently exploit current measurements, and large enough, in order to sufficiently smooth out stochastic variations in the random sequence {γ(t)}.
In order to better clarify the basic effects of choosing α and A, consider w.l.o.g. the situation in which γ(t+1)=γ^{[1]}, where \(\gamma _{1}^{[1]} \approx 1\) and the remaining \(\gamma _{i}^{[1]} \approx 0\), i=2,…,N. The asymptotic response of (13) to such an input signal is given by
Having in mind (16) and (14), there are now two main possibilities: (1) the rows of C(∞) with c_{i1}≠0 contain, in fact, c_{i1}≈1, while the remaining elements are of the order of magnitude of α; (2) the rows of C(∞) with c_{i1}=0 contain the nonzero elements having different order of magnitude, depending on graph topology, irrespective of α. In the first case, the convex combination from (9) will give the highest emphasis to the neighbors observing the target with high probability, while in the second, node priorities result from the given graph. In the first case, we have prompt reaction for small α, while in the second, diffusion rate of the state estimates is maximized. In our experiments, we adopted α to be around 0.05.
Asymptotic properties of the proposed adaptation scheme capture the perceivability history in a more efficient way than the original scheme from [2]. For example, in the cases when two nodes have the same γ_{ i }, the first term at the right hand side of (15) gives more weight to the node that previously received information, coming more likely from the nodes that had observed the target (see the example below and the simulation results).
The proposed algorithm is summarized in Algorithm 1.
Example. In order to demonstrate clearly the positive effect of introducing specially designed dynamics into the original adaptive scheme from ACF, we shall consider the following simple illustrative example of a sensor network with the ring topology and N=6 nodes, where each node is connected to 2 neighboring nodes and only the first node observes the target; there are no false measurements. The adaptation parameters have been chosen as \(c_{ij}^{\chi }(\infty)=1\), \(c_{ij}^{\chi }(0)=0.01\), k=0.2 (original adaptive scheme from [2]), and α=0.03, \(\beta ^{[l]}_{i,0}(t)=0\) if the ith node observes the target at time instant t and \(\beta ^{[l]}_{i,0}(t)=1\) otherwise (the proposed general adaptation strategy). Figure 2 shows the time evolution of the resulting nonzero consensus weights connected to the original scheme (ACF_{1}) and the proposed algorithm (ACF_{2}). It can be seen that the main difference between the algorithms is in the consensus weights connected to the nodes that do not have measuring nodes in their neighborhoods. The original algorithm assigns equal consensus weights in the consensus matrix rows connected to these nodes (3rd, 4th, and 5th row in Fig. 2a), while the proposed general algorithm gives more weight to the nodes that are closer to the measuring node (3rd, 4th, and 5th row in Fig. 2b), enabling better information flow through the network which, in turn, allows faster convergence and better tracking.
Remark 1
The exposed methodology for distributed adaptive generation of the consensus weights in (10) is mainly motivated by the idea to exploit the locally available “ β ”parameters in order to assign higher importance to the nodes with higher probabilities of getting targetoriginated measurements; it has no other direct connection with the main state estimation algorithm based on the PDAF. Therefore, it is easy to conclude that the results of this section can be readily used in the context of any nonlinear estimation algorithm, provided an estimate of the probabilities contained in the “ β ”parameters is available. In this sense, it is straightforward to construct adaptive algorithms based, e.g., on the extended Kalman filter (like in [8]), on the unscented Kalman filter, and particle filters. It is also possible to apply the proposed methodology within the framework of the MLPDA method applicable in real time [20–25]. The adaptation strategy should again cope mainly with the limited FoVs (or sensing ranges).
4 Stability and reduction of noise influence
4.1 Stability
Stability and noise reduction of the proposed distributed tracking algorithm will be analyzed starting from the following global network model:
where X(t+1t)=(ξ_{1}(t+1t)^{T},…,ξ_{ N }(t+1t)^{T})^{T}, Φ(t)=[Φ_{ ij }(t)], i,j=1,…,N, in which \( \Phi _{ij}(t)=c_{ij}(t) F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right) L_{j}(t) H_{j}\right ]\) are m×m blocks, Ψ(t)=[Ψ_{ ij }(t)], in which Ψ_{ ij }(t)=c_{ ij }(t)FL_{ j }(t) are m×p blocks defined by (11), and \(Z(t)=\left (\sum _{k=1}^{m_{1}(t)} \beta ^{[l]}_{1,k}(t) \tilde {z}_{1,k}(t), \right. \left.\ldots, \sum _{k=1}^{m_{n}(t)} \beta ^{[l]}_{N,k}(t) \tilde {z}_{N,k}(t)\right)^{T}\).
To the best of the authors knowledge, no general methodology is directly applicable to the stability analysis of the model (17), not to mention many open questions related to the local recursions of PDA type [17]. We shall here make an attempt to apply the methodology proposed in [2,12] in relation with distributed estimation schemes with overlapping subsystems and to get an additional insight into the complex relationship between the local trackers and the adaptive consensus scheme. Using the basic results from [12,33–35], we define the norm of the blockmatrix Φ(t)=[Φ_{ ij }(t)] as
where ∥Y∥_{ τ } denotes the norm of a square matrix Y defined as \(\Y \_{\tau }=\left \D_{\tau }U^{T}Y U ~ D_{\tau }^{1}\right \_{\infty }\), in which U is an orthogonal matrix in the representation Y=UΔU^{T}, where Δ is an upper triangular matrix containing the eigenvalues of Y at the diagonal (according to the Schur’s theorem [35]) and D_{ τ }=diag{τ,τ^{2},τ^{3},…,τ^{n}}. It is essential to notice that it is possible to find, for any given ε>0, such a τ_{ ε }>0 that, for all τ>τ_{ ε }, ρ(Y)≤∥Y∥_{ τ }≤ρ(Y)+ε, where ρ(Y) is the spectral radius of a square matrix Y [12,35]. Without going into details related to specific properties of the adopted norm in the case of (17), notice that the main idea behind the adopted definition is to take \(\ \Phi (t)\^{*}_{\tau }\) as an estimate of ρ(Φ(t)) based on the local algorithm properties expressed by ∥Φ_{ ij }(t)∥_{ τ }.
Concentrating on Φ(t), we shall start from the following sufficient conditions for asymptotic stability of the linear timevarying system (17) [36,37]:

1)
Φ(t) is strictly Schur for all t

2)
For some constant μ∈[ 0,1)
$$\begin{array}{@{}rcl@{}} &\ \Delta(\Phi(t) \otimes \Phi(t)) \ \leq \frac{\mu}{(nN)^{\frac{1}{2}}} \sigma_{\text{min}}(\Phi(t) \otimes \Phi(t)  I)& \\ &\cdot \sigma_{\text{min}}(\Phi(t+1) \otimes \Phi(t+1)  I),& \end{array} $$(18)where Δ(Φ(t)⊗Φ(t))=Φ(t+1)⊗Φ(t+1)−Φ(t)⊗Φ(t), ⊗ denotes the Kronecker’s product and σ_{min}(·) the minimal singular value [37].
For τ large enough, \(\left \ F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right)L_{j}(t) H_{j}\right ] \right \_{\tau }\) becomes a good estimate of \(\rho \left (F\left [I \left (1 \beta ^{[l]}_{j,0}(t)\right)\!\!\right.\right. \left.\left.{\vphantom {\beta ^{[l]}_{j,0}}} \! L_{j}(t) H_{j}\right ] \right)\), i.e., there exists such a τ^{′}>0 that for all τ>τ^{′}, we have \(\left \ F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right)L_{j}(t) H_{j}\right ] \right \_{\tau } < \rho \left (F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right) L_{j}(t) H_{j}\right ] \right) + \varepsilon \), for any ε>0. Coming back to the system (17), we derive that
At this point, an insight into the local recursions (12) and (11) provides the following general conclusions:

1)
When P_{ i }(tt) diverges (see [17] for general observations concerning stability properties of PDAF), \(\rho \left (F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right) L_{j}(t) H_{j}\right ] \right)\) diverges and, consequently, \(\ \Phi (t) \^{*}_{\tau }\) defined by (19) diverges, as well;

2)
When P_{ i }(tt) remains bounded, and condition (18) holds for some μ∈[0,1), the corresponding L_{ i }(t) could be not stabilizing in the sense of ensuring the condition \(\rho \left (F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right) L_{j}(t) H_{j}\right ]\right) < 1\) for all i,j=1,…,N. However, in spite of this, the condition \( {\lim }_{\tau \to \infty } \ \Phi (t) \^{*}_{\tau } < 1\) in (19) may still be achieved, provided the coefficients \(c_{ij}(t) \left (\sum _{j=1}^{N} c_{ij}(t) =1\right)\) are chosen appropriately. The overall stabilizing property results from the network; from this point of view, it is obvious that the adaptation procedure in the proposed algorithm should be such that the nodes with higher detection probability (lower \(\beta ^{[l]}_{i,0}(t)\)) should be given higher priority (see the previous subsection for more details). This is exactly provided by the proposed adaptation scheme given in Section 3.2;

3)
When P_{ i }(tt) is stabilizing in the sense that \(\rho \left (F\left [I \left (1\beta ^{[l]}_{j,0}(t)\right) L_{j}(t) H_{j}\right ]\right) < 1\) for all i,j=1,…,N [38], \( {\lim }_{\tau \to \infty } \ \Phi (t) \^{*}_{\tau } < 1\) for admissible choice of the consensus weights.
Having in mind the stochastic timevarying nature of the basic recursions (12), the above general conclusions require verification in all particular cases; one should also bear in mind that, to the authors best knowledge, there is no rigorous stability analysis of the PDAF in the general case which would give a possibility to estimate ρ(Φ_{ ij }(t)) [17,39]. In order to provide additional clarifications to the above qualitative statements, we shall focus our attention on an approximation of (12), in which the random matrices P_{ i }(tt) and the probabilities \(\beta ^{[l]}_{i,0}(t)\) are replaced by their expected values (see, e.g., [17,39]). Without going here into details, we shall utilize the results from [17,18] and obtain that P_{ i }(tt) in (12) gives rise to the following modified deterministic discrete Riccati equation
where \(q_{2}^{(i)}\) is a constant \(\left (q_{2}^{(i)} \in (0,1]\right)\), while \(\bar {L}_{i}(t)\) and \(\bar {S}_{i}(t)\) follow analogously from the formulae for L_{ i }(t) and S_{ i }(t) in (12), i=1,…,N. Under the standard detectability condition of the pairs (F,H_{ i }) and positive definiteness of the noise covariance matrices R_{ i }, the important result from [18] leads to the conclusion that the recursion (20) converges (irrespective of the initial condition) to the unique positive definite solution of the following modified algebraic Riccati equation, resulting from (12) and (11):
for all values of \( q_{2}^{(i)} \in \left (\tilde {q}_{2}^{(i)},1\right ]\), where \(1\frac {1}{(\rho (F))^{2}} \leq \tilde {q}_{2}^{(i)} \leq 1\frac {1}{(M(F))^{2}}\) is the critical value of the characteristic parameter \( q_{2}^{(i)}\), \(M(F)=\prod _{l=1}^{n} \max \{1, \lambda _{l}(F) \}\). Let \(p_{1}^{(i)}=1\left (1q_{2}^{(i)}\right)^{\frac {1}{2}}\) and \(p_{2}^{(i)}=1+\left (1q_{2}^{(i)}\right)^{\frac {1}{2}}\); then, every positive definite solution of (21) renders the matrix
to be Schur for all \(p^{(i)} \in \left [p_{1}^{(i)}, p_{2}^{(i)}\right ]\) (the stabilizing solution of (21)) [18].
Assume next that the described approximations are adopted instead the original local PDAFs and that the recursions (20) converge to the unique positive definite matrices \(\bar {P}_{i}\) defined by (21). According to (9), asymptotic stability of the resulting approximation of the tracker model depends exclusively on the eigenvalues of \(\bar {F}_{i}(p^{(i)})\), where \(p^{(i)}=1\bar {\beta }^{[l]}_{i}=P_{D}^{(i)} P_{G}^{(i)}\), \(P_{D}^{(i)}\) representing the local detection probability and \(P_{G}^{(i)}\) the local probability that the correct measurement (if the target is detected) lies within the predefined gate [17]. Accordingly, we immediately conclude that for \(\bar {\beta }^{[l]}_{i} < \left (1q_{2}^{(i)}\right)^{\frac {1}{2}}\), the adopted approximation of the tracker model is asymptotically stable; however, for \(\bar {\beta }^{[l]}_{i} > \left (1q_{2}^{(i)}\right)^{\frac {1}{2}}\), this model may be not asymptotically stable, in spite of the fact that (21) has a positive definite solution [18]. Using (19), we obtain
As above, the condition \( {\lim }_{\tau \to \infty } \ \Phi (t) \^{*}_{\tau } < 1\) may be achieved for (17) not only when \( \bar {F}_{i}(1\bar {\beta }_{i,0})\) is Schur for all i but also in the situations when for some nodes, matrix \( \bar {F}_{i}(1\bar {\beta }_{i,0})\) is not Schur. In this sense, the overall stability of the estimator depends directly on the choice of c_{ ij }(t): the better the coefficients c_{ ij }(t) follow the essentially time varying detection probabilities, expressed through the importance of the nodes, the closer the whole system is to stability. A more detailed analysis could be connected to specific target and observation models, taking into account the corresponding structure of the solution of (21) and its dependence on the underlying parameters (notice, for example, that ρ(F)=1 for standard kinematic target models). Such an analysis would be beneficial for practice not only from the point of view of stability (which is expected to hold for a large variety of definitions of c_{ ij }(t)) but also from the point of view of tracking accuracy.
4.2 Reduction of noise influence
In order to provide an insight into the influence of measurement noise to the overall estimation error, we can write simply E(t+1t)=X(t+1t)−X(t+1), where X(t)=(x(t)^{T},…,x(t)^{T})^{T}, and obtain from (17) that the stochastic driving term in the relation for E(t+1t), appearing as a consequence of the measurement noise, is given by V_{ c }(t)=Ψ(t)v(t), where v(t)=(v_{1}(t)^{T},…,v_{ N }(t)^{T})^{T}. The effect of this driving term to the overall estimation error covariance can be immediately seen from the expression for P_{ c }(t)=E{V_{ c }(t)V_{ c }(t)^{T}}. Assuming the deterministic modified Riccati Eq. (20) and the state transition matrix (22), as well as uncorrelatedness of the local measurement noises, we obtain that the covariance of V_{ c }(t) at node i, conditioned by the consensus weights, is given by
For equal noise covariances R_{ j }=R, we obtain the following inequality \(\sum _{j=1}^{N} c_{ij}(t)^{2} \left (p^{(j)}\right)^{2} F L R L^{T} F^{T} \leq F L R L^{T}F^{T}\), having in mind row stochasticity of C(t). The last term represents the covariance of the driving term in the case when only one node has access to measurements (p^{(i)} =1). Evidently, the applied consensus scheme decreases the noise influence by averaging over the set of measuring nodes.
5 Alternative IPDAbased distributed trackers
5.1 Kalmanconsensus filter: IPDAKCF
We shall define information vectors and information matrices as \( u_{i}(t) = H_{i}^{T} R_{i}^{1}\sum _{j=1}^{m_{i}(t)}\beta _{ij}^{[l]}(t) z_{ij}(t)\) and \(U_{i}(t) = H_{i}^{T} R_{i}^{1}H_{i}\), respectively, as well as their sums across neighbors: \( y_{i}(t) = \sum _{j\in \mathcal {J}_{i}} u_{j}(t)\) and \(B_{i}(t) = \sum _{j\in \mathcal {J}_{i}} U_{j}(t)\) [6,7].
The socalled JPDA Kalmanconsensus filter (JPDAKCF) has been proposed in [7]. In the context of the present paper and the IPDA methodology, it is given in the following form:
where ε is a small positive scalar. The algorithm is derived by decomposing the global Kalman filter for the whole network and adding the consensus term at the filtering level [5]. Notice that the algorithm requires communication of information vectors (size n×1) and information matrices (size n×n) between the neighbors, in addition to the exchange of state estimates (size n×1).
5.2 Informationweighted consensus filter: IPDAICF
The tracking algorithm which takes into account crosscovariances between the state estimates across different nodes is the socalled informationweighted consensus filter (ICF) [4]. A multitarget version of this algorithm based on the PDA methodology is described in [8]. In the context of the present paper and the IPDA methodology, the single consensus step version of this algorithm is represented by:
where
by \(\mathcal {A}(x_{i}(t))\), where x_{ i }(t) is a local variable at the instant t, we denote the single step averaging consensus operation
The algorithm requires internode communication of one n×1 vector and two n×n matrices.
Notice that ICF was originally designed to be applied with multiple consensus steps between two consecutive time instants. This requirement represents a significant communication burden for the whole tracking scheme; in this paper, our focus is on the single consensus step version, compatible with the trackers formulated above.
Remark 2
To summarize, the communication requirements of the above considered algorithms are as follows (per node and per time step): IPDAACF: O(n) (more exactly, n+1), IPDAKCF: O(n^{2}) (more exactly, n^{2}+2n), and IPDAICF: O(n^{2}) (more exactly, 2n^{2}+n). Regarding the computational requirements, complexity of all the methods is, in general, O(n^{3}). However, the algorithms IPDAKCF and IPDAICF require computation of matrix inversions (size n×n, see (25) and (26), respectively), which is not the case with the IPDAACF algorithm. Therefore, in addition to favorable communication requirements (which are of crucial importance for the efficiency of large sensor networks), the proposed algorithm exhibits also the lowest computational demands.
6 Simulation results
We shall consider a network of N=15 cameras (nodes) aimed at distributed tracking of a target moving within a 500×500 space [4,8]. The cameras are randomly distributed in space with random orientations resulting in overlapping fieldofviews (FoVs), represented by equilateral triangles with 300 units height. The communication range is set to 200 units. The dynamics of a target is modeled using the constant speed model. The process covariance Q is set to diag(10,10,1,1) [4]. Target’s initial position is randomly selected within the given square area, and its initial speed is set to 2 units per time step, with random direction. The measurement noise covariances are set to 100I_{2}. The initial state estimates are randomly set around actual target’s initial state, with covariance 10Q. The initial error covariance matrices are set to diag(100,100,10,10) for all the nodes.
First, we shall examine scenarios with no clutter. An experiment has been performed with 250 Monte Carlo runs, over which estimation errors (average distance per node between the target position estimates and the actual position) and disagreements between the nodes (variance of the estimates) have been computed. Each run uses different target trajectory and network topology, designed in the way described above. In the chosen simulation setting, at each time iteration, 2.3 nodes on average receive measurements from the target, and one node communicates with 5.3 nodes for each network topology. Parameter ε in ICF has been set to \(1/\max _{i}(\mathcal {N}_{i}+1)\) at each run; parameter ε in KCF has been set to 0.07. ACF has been simulated in two described variants, with ACF_{1} parameters chosen as in the example from Section 3.2, and with α set to 0.05 for ACF_{2}. Figure 3 shows the resulting estimation errors (top) and disagreements (bottom). It is obvious that, owing to the introduced refined adaptation procedure, ACF outperforms both KCF and ICF, in spite of much lower communication requirements.
In the situations with clutter, \(P_{i}^{D}\) has been calculated individually for each node at each time instant by integrating the probability density function of the predicted estimate over the triangular area visible to the camera; \(P_{i}^{g}\) has been set to 0.999 for all the nodes and λ_{ i }(t)V_{ i }(t)=1/32 [8]. The target perceivability has been modeled using \(p_{i}^{11}=0.98\), \(p_{i}^{21}=0\), and π_{ i }(00)=0.2 for all the nodes. IPDAACF with the refined adaptation scheme has been simulated for two choices of l (IPDAACF ^{l}, with l=1 corresponding to the classical PDA filter, and l=2 to the targetexistencebased IPDA filter), together with the IPDAICF algorithm (both choices of l have produced very similar results so that only the case l=1 has been considered). The IPDAKCF algorithm does not give comparable performances. Figure 4 illustrates performance of IPDAACF^{2}, giving snapshots of the position estimates of all the nodes, together with the target trajectory for one simulation run. It shows that the proposed algorithm gives networkwide accurate target state estimates. The resulting average estimation errors (top) and disagreements (bottom) of different algorithms are shown in Fig. 5. The optimal centralized Kalman filter (CKF) has also been simulated. It can be seen that the proposed algorithm with the target existencebased IPDA filter again outperforms ICF. CKF obviously represents a reference scheme with the best tracking results; however, it assumes that all the measurements are available to all the nodes. It is far from being realistic to perform all the necessary communications before the next updating of the state estimates. However, we have also investigated achievable performance in the case of the flooding protocols realizable by the simulated networks (see, e.g., [40]), regardless of the communication cost. Obviously, depending on the adopted network diameter, different results can be obtained, represented typically as delayed estimates generated by CKF. For the networks simulated in the way described above, the results are much inferior to those obtained by the proposed algorithm (having in mind the intrinsic need for tracking moving targets); the average diameter in our experiments has been found to be 4.4. The results presented in Fig. 5 give an obvious illustration. The proposed algorithm has been found to be superior even in the case of networks with the average diameter equal to one.
To assess the performance of the algorithms in case of different communication topologies, the original communication networks have been pruned to their spanning trees, and random edges progressively added, so that the resulting networks have the average node degree δ ranging from 2 to 14. Parameter ε in IPDAICF has been set to 1/(δ+1). We can see in Fig. 6 that in terms of the tracking error, all the algorithms exhibit similar behavior, while IPDAACF^{2} outperforms IPDAICF in terms of the disagreement between the nodes. IPDAICF slightly outperforms IPDAACF^{2} in terms of the tracking error in the case of dense communication graphs.
In order to illustrate the idea underneath the consensus scheme design in the proposed adaptation algorithm, we have plotted in Fig. 7, for one simulation run, the actual target perceivabilities of the nodes (represented by binary signals equal to 1 when the target is visible and to 0 otherwise), together with the corresponding values of γ_{ i }(t) used in (15). It can be seen that, for each node, γ_{ i }(t) follows the binary perceivability signal, resulting in a consensus scheme which efficiently potentiates the nodes receiving measurements from the target.
As a part of the rationale behind the choice of l=2 in the design of the proposed algorithm, we have simulated a situation when a target is temporarily occulted in the circle with 50 units radius (Fig. 8). One target trajectory crossing the circle has been fixed, and 250 experimental runs have included random network topologies. Figure 9 shows the resulting average estimation errors (top) and disagreements (bottom) for different algorithms. We can see that choosing l=2 instead of l=1 yields substantially lower disagreement between the local estimates when the target is occulted since in these situations, the introduced target perceivability probability is small, resulting in the state estimates relying on prediction, rather than on the measurements generated by clutter.
7 Conclusion
In this paper, a new distributed adaptive consensusbased tracking algorithm has been proposed for camera networks in the case of limited FoVs, highlevel clutter and occulted targets, using the methodology of (IPDA) [13]. The algorithm is defined either in the perceivabilitybased or the existencebased form, representing distributed consensusbased versions of the classical PDA tracker from [10,11] and the existencebased tracker from [15], respectively. Special care is taken of the design of a distributed adaptation procedure, based on utilization of the “ β ”parameters as a measure of the probabilities of observing the target by particular nodes. A new distributed adaptation algorithm, based on the generation of node importance, is proposed for defining appropriate weights in the consensus matrix, providing high tracking precision and fast convergence of the estimates over the network. The paper contains a theoretical analysis of stability and noise rejection capabilities of the proposed algorithm. The applied methodology of analysis is based on the idea from [12] and on recent results related to the properties of the modified Riccati equations resulting from the PDA recursions [18]. It has been shown that the algorithm provides convergence even in the case when some nodes may be unstable, owing to a proper choice of the consensus weights. The proposed algorithm has been verified by extensive simulations. It has been shown that it outperforms similar algorithms known from the literature—IPDAKCF from [7] and IPDAICF from [8]—in spite of lower communication requirements; derivation of onestep consensusbased versions of these algorithms, consistent with the form of the proposed algorithm, is given in a separate section. The results related to occulted objects and highlevel clutter are of special interest. The proposed algorithm can be readily applied to all distributed target tracking problems (radars, sonars) under analogous assumptions. We do hope that it can become a simple and efficient tool for engineering practice.
Further efforts could be oriented in the direction of the development of complementary distributed track initiation, confirmation, and termination algorithms using IPDA methodology [13,15]. Generalization of the algorithm to the multitarget case in the sense of applying the basic methodology of JPDAF [10,41] in the distributed multiagent context represents a complex but straightforward task.
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Ali, K., Ilić, N., Stanković, M. et al. Consensusbased distributed adaptive target tracking in camera networks using Integrated Probabilistic Data Association. EURASIP J. Adv. Signal Process. 2018, 13 (2018). https://doi.org/10.1186/s136340180534z
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DOI: https://doi.org/10.1186/s136340180534z