Consensusbased distributed adaptive target tracking in camera networks using Integrated Probabilistic Data Association
 Khaled Obaid Al Ali^{1, 2},
 Nemanja Ilić^{3, 2}Email authorView ORCID ID profile,
 Miloš S. Stanković^{4, 2} and
 Srdjan S. Stanković^{5, 2}
https://doi.org/10.1186/s136340180534z
© The Author(s) 2018
Received: 4 October 2017
Accepted: 7 February 2018
Published: 22 February 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.
Keywords
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 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.

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
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]).
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 \(Z_{i}^{t}=\{Z_{i}(1), \ldots, Z_{i}(t) \} \).
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].
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

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.
for \(j \in \mathcal {J}_{i}\); c_{ ij }(t)=0 otherwise.
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)}.
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.
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
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 \(Z(t)=\left (\sum _{k=1}^{m_{1}(t)} \beta ^{[l]}_{1,k}(t) \tilde {z}_{1,k}(t), \right. \left.\ldots, \sum _{k=1}^{m_{n}(t)} \beta ^{[l]}_{N,k}(t) \tilde {z}_{N,k}(t)\right)^{T}\).
 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].
 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.
to be Schur for all \(p^{(i)} \in \left [p_{1}^{(i)}, p_{2}^{(i)}\right ]\) (the stabilizing solution of (21)) [18].
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
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].
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 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.
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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