• level 1: Object refinement,

• level 2: Situation refinement,

• level 3: Threat refinement,

• level 4: Process refinement.

• Level 0—sub-object assessment: the pre-detection activities such as pixel or signal processing, spatial or temporal registration is present. Level 0 deals with the estimation and prediction of signal/object observable states on the basis of pixel/signal level data association and characterization.

• Level 1—object assessment: is concerned with estimation and prediction of target locations, behavior or identity. In this level, which is sometimes referred to as multi-sensor data fusion or multi-sensor integration, data is combined to assign dynamic features (e.g., velocity) as well as static (e.g., identity) to objects, hence adding semantic labels to data. This level includes techniques for data association and management of objects (including creation and deletion of hypothesized objects, and state updates of the same). Level 1 addresses the following functions: data alignment, data/object correlation, object positional/kinematic/attribute estimation, object identity estimation.

• Level 2—situation assessment: investigates the relations among entities such as force structure and communication roles. This level involves aggregation of level 1 entities into high-level, more abstract entities, and relations between entities. An entity in this level might be a pattern of connected objects of level 1 entities. Input data are assessed with respect to the environment, relationship among level 1 entities, and entity patterns in space and time. Level 2 addresses the following functions: object aggregation, contextual interpretation/fusion, event/activity aggregation, multi-perspective assessment.

• Level 3—impact assessment: outlines sets of possible courses of action and the effect on the current situation. The impact assessment, which is sometimes called significance estimation or threat refinement, estimates and predicts the combined effects of system control plans and the entities of level 2 (possibly including estimated or predicted plans of other environment agents) on system objectives. Level 3 addresses the following functions: estimate/aggregate force capabilities, predict enemy intent, identify threat opportunities, estimate implications, multi perspective assessment.

• Level 4—process refinement: is an element of Resource Management used to close the loop by re-tasking resources to support the objectives of the mission. Process refinement evaluates the performance of the data fusion process during its operation and encompasses everything that refines it, e.g., acquisition of more relevant data, selection of more suitable fusion algorithms, optimization of resource usage with respect to, for instance, electrical power consumption. Process refinement is sometimes called process adaption to emphasize that it is dynamic and should be able to evolve with respect both its internal properties and the surrounding environment. The function of this level is in some literature handled by a so called meta-manager or meta-controller. It is also rewarding to compare level 4 fusion to the concept of covert attention in biological vision which involves, e.g., sifting through an abundance of visual information and selecting properties to extract. Level 4 addresses the following functions: evaluation (real-time control/long term improvement), fusion control, source requirements, mission management.

• Signals Intelligence (SIGINT),

• Imagery Intelligence (IMINT),

• Measurement and Signature Intelligence (MASINT),

• Human Intelligence (HUMINT),

• Open-Source Intelligence (OSINT),

• Geospatial Intelligence (GEOINT).

• production of accurate and continuous tracks (e.g., objects, persons, single vehicles, group objects),

• system reaction rates (e.g., track extraction, detection of target maneuvers, track monitoring),

• sustainment of reconnaissance capabilities in case of either system or network failures (e.g., graceful degradation),

• system robustness against jamming and deception,

• compensation of degradation effects (e.g., sensor misalignment, limited sensor resolution),

• robustness against sub-optimal real-time realizations of sensor data fusion algorithms,

• processing of eventually delayed sensor data (e.g., out-of sequence measurements).

2.22.5.1 Sensor configuration

Sensor fusion networks can be categorized according to the type of sensor configuration. Durrant-Whyte distinguishes three types of sensor configuration as schematized in Figure 22.8 [57,58].

Competitive sensor data fusion: Sensors are configured competitive if each sensor delivers independent measurements of the same property. Sensor data represent the same attribute, and the fusion is to reduce uncertainty and resolve conflicts. Competitive sensor configuration is also called a redundant configuration. Sensors S1 and S2 in Figure 22.8 represent a competitive configuration, where both sensors redundantly observe the same property of an object in the environment space.

Complementary sensor data fusion: A sensor configuration is called complementary if the sensors do not directly depend on each other, but can be combined to give a more complete image of the phenomenon under observation. Fusion of the sensor data provides an overall and complete model. Examples for a complementary configuration is the employment of multiple cameras each observing disjoint parts of a room, or using multiple spectrum signatures to identify a land cover type, or using different waveform to identify an aircraft type. Sensor S2 and S3 in Figure 22.8 represent a complementary configuration, since each sensor observes a different part of the environment space.

In both competitive and complementary sensor configurations, there is an improvement of the accuracy of the target characteristics estimation consequent to the data fusion. In their seminal work H. Cramer and C.R. Rao found how to compute the best theoretical accuracy that can be achieved by an estimator. The lower bound of accuracy, i.e., the mean square error of any unbiased estimator, is given by the inverse of the so-called Fisher Information Matrix (FIM). The computation of the CRLB (Cramer-Rao Lower Bound) applies to problems involving the maximum likelihood estimation of unknown constant parameters from noisy measurements [59]. The best achievable improvement of target location and track accuracy can be quantified by the reduction of the CRLB consequent to the track fusion. In [60] this computation is reported in case of fusion of data from two sensors with an ideal unitary detection probability. In [61,62] the same computation has been proposed in case of detection probability less than one and false alarm probability higher than zero.

Cooperative sensor data fusion: A cooperative sensor network uses the information provided by two independent sensors to derive information that would not be available from the single sensors. An example for a cooperative sensor configuration is stereoscopic vision: by combining two-dimensional images from two cameras at slightly different viewpoints a three-dimensional image of the observed scene is derived. Cooperative sensor fusion is the most difficult to design, because the resulting data are sensitive to inaccuracies in all individual participating sensors. Thus, in contrast to competitive fusion, cooperative sensor fusion generally decreases accuracy and reliability. Sensor S4 and S5 in Figure 22.8 represent a cooperative configuration. Both sensors observe the same object, but the measurements are used to form an emerging view on object C that could not have been derived from the measurements of S4 or S5 alone.

These three categories of sensor configuration are not mutually exclusive. Many applications implement aspects of more than one of the three types. An example for such a hybrid architecture is the application of multiple cameras that monitor a given area. In regions covered by two or more cameras the sensor configuration can be competitive or cooperative. For regions observed by only one camera the sensor configuration is complementary.

2.22.5.2 Classical approach to surveillance

Sensor networks have countless applications, for example, we mention the sensor networks used in computer science and telecommunications, in biology, where they can be used to monitor the behavior of animal species such as birds or fishes, and in habitat monitoring, where they can be used to provide real-time rainfall and water level information used to evaluate the possibility of flooding. In the field of Homeland Protection one of the main task to be assigned to a sensor network is the surveillance with its most general significance. Automatic surveillance is a process of monitoring the behavior of selected objects (targets and/or anomalies) inside a specific area by means of sensors. A target generally consists of an object (e.g., a tank close to a land border or a rubber approaching to the coast) whose presence and characteristics can be detected and estimated by the sensor; an anomaly consists in a non usual behavior (e.g., a jeep moving off-road, the increasing of the radioactivity level within an area) that can be revealed by the sensor. Sensors typically provide the following functions:

To perform the previous functions, the sensors can be organized on the bases of several approaches. The classical approach to surveillance of wide areas is based on the use of a single or few sensors with long range capabilities. The signal received by the single sensor is processed by means of suitable digital signal processing subsystems. In this case the sensors are costly, with adequate computation and communication capabilities. Sensors are normally located in properly selected sites, to mitigate terrain masking problems; nevertheless, they provide different performance depending on the location of target inside the surveillance area. Typical sensors are radars (ground-based, air-borne, ship-borne or space-based), infrared or TV cameras, seismic, acoustical, radioactive sensors. Usually in this kind of networks, as represented in Figure 22.9, the information traffic goes from the sensor nodes to a single sink node called information fusion center that performs the target localization and tracking. According to the information received from the sensors the fusion center monitors the area where the sensors are deployed and decides, on the basis of the state estimates and their accuracy (e.g., a covariance matrix for a Kalman filter or a particle cloud for a particle filter) the actions to take.

In [63] an example of high-performance radar netted for Homeland Security application with a centralized data fusion process is described. The same classical approach is presented in [64] where this kind of sensor network is employed for natural resource management and bird air strike hazard (BASH) applications.

However if an intruder reaches and neutralizes the fusion center, the communication between the network nodes are interrupted and the whole network is exposed to the risk of becoming useless as a network even if the individual sensors may still be all working.

2.22.5.3 Collaborative signal and information processing (CSIP)

Nowadays, a novel approach to the automatic surveillance has been adopted; it is based on the use of many sensors with short range capabilities, low costs, and limited computation and communication capabilities. In case of a huge number of sensors, the use of information fusion centers is unpractical and their functioning is based on the information exchange between “near-by” sensors. The sensors can be distributed in fixed positions of the territory, but they could also be deployed adaptively to the change of the scenario. There are several approaches: they can be randomly distributed inside the surveillance area and if the number of sensors is high, the performance of the surveillance system can be considered independent of the location of the targets; then the signal received by each sensor is processed using the computational capabilities of a sub-portion of the sensor system and employed to re-organize dynamically the network. Sensors may be agile in a variety of ways, e.g., the ability to reposition, point an antenna, choose sensing mode, or waveform. Notice that the number of potential tasking of the network grows exponentially with the number of sensors. The goal of sensor management in a large network is to choose actions for individual sensors dynamically so as to maximize overall network utility. This process is called Collaborative Signal and Information Processing (CSIP) [65]. One of the central issues for CSIP to address is energy-constrained dynamic sensor collaboration: how to dynamically determine who should sense, what needs to be sensed, and who the information must be passed onto. This kind of processing system allows a limitation in the consumption of power. Applying a surveillance strategy which accounts for the target tracking accuracy and the sensor random location, only a limited number of sensors are awake and follow/anticipate the target movement; thus, the network self-organizes to detect and track the target, allowing an efficient performance from the energetic point of view with limited sensor prime power and with a reduced number of sensors working in the whole network. For example in [66], instead of requesting data from all the sensors, the fusion center iteratively selects sensors for the target localization: first a small number of anchor sensors send their data to the fusion center to obtain a coarse location estimate, then, at each step a few non-anchor sensors are activated to send their data to the fusion center to refine the location estimate iteratively. Moreover the possibility to actively probe certain nodes allows to disambiguate multiple interpretations of an event.

In [67] the techniques of information-driven dynamic sensor collaboration is introduced. In this case an information utility measurement is defined as the statistical entropy and it is exploited to evaluate the benefits in employing part of the network that consequently is re-organized. Other cost/utility functions can be employed as criteria to dynamically re-organize the sensor network as described in [68,69].

Several analytical efforts have been done to evaluate the performance of such networks in terms of tracking accuracy. As usual the CRLB has been taken as reference of the best achievable accuracy; in particular a new concept of conditional PCRLB (Posterior Cramer Rao Lower Bound) is proposed and derived in [70]. This quantity is dependent on the actual observation data up to the current time, and is implicitly dependent on the underlying system state. Therefore, it is adaptive to the particular realization of the underlying system state and provides a more accurate and effective online indication of the estimation performance than the unconditional PCRLB. In [71,72] the PCRLB is proposed as a criterion to dynamically select a subset of sensors over time within the network to optimize the tracking performance in terms of mean square error. In [73] the same criterion is proposed as a framework for the systematic management of multiple sensors in presence of clutter.

2.22.5.4 Self-Organizing Sensor Networks

Self-organization can be defined as the spontaneous set-up of a globally coherent pattern out of local interactions among initially independent components. Sensors are randomly spread out over a two dimensional surveillance area. In a self-organized system, its elements affect only close elements; distant parts of the system are basically unaffected. The control is distributed, i.e., all the elements contribute to the fulfillment of the task. The system is relatively insensitive to perturbations or errors, and have a strong capacity to restore itself. Initially independent components form a coherent whole able to efficiently fulfill a particular function [74]. Flocks of birds, shoals of fish, swarms of bees are examples of self-organizing systems; they move together in an elegantly synchronized manner without a leader which coordinates them and decides their movement. It has been shown that flocks of birds self-organize into V-formations when they need to travel long distances to save energy, by taking advantage of the upwash generated by the neighboring birds. Cattivelli and Sayed [75] propose a model for the upwash generated by a flying bird, and shows that a flock of birds is able to self-organize into a V-formation as if every bird processes spatial and network information by means of an adaptive diffusive process. This result has interesting implications. First, a simple diffusion algorithm is able to account for self-organization of birds. Second, according to the model, that birds can self-organize on the basis of the upwash generated by the other birds. Third, some information is necessarily shared among birds to reach the optimal flight formation. The paper also proposes a modification to the algorithm that allows birds to organize, starting from a V-formation, into a U-formation, leading to an equalization effect, where every bird in the flock observes approximately the same upwash. The same algorithm based on birds flight is extended in [76] to the problem of distributed detection, where a set of sensors/nodes is required to decide between two hypotheses on the basis of the collected measurements. Each node makes individual real-time decisions and communicates only with its immediate neighbors, in order that any fusion center is not necessary. The proposed distributed detection algorithms are based on diffusion strategies described in [77–79] and their performance is evaluated by means of classical probabilities of detection and false alarms.

These diffusion detection schemes are attractive in the context of wireless and sensor networks thanks to their intrinsic adaptability, scalability, improved robustness to node and link failure as compared to centralized schemes, and their potential to save energy and communication resources.

2.22.5.5 Self-synchronization mechanism applied to sensor network

Several studies have shown how a simple self-synchronization mechanism, borrowed from biological systems, can form the basic tool for achieving globally optimal distribution decisions in a wireless sensor network with no need for a fusion center. Self-synchronization is a phenomenon first observed between pendulum clocks (hooked to the same wooden beam) by Christian Huygens in 1658. Since then, self-synchronization has been observed in a myriad of natural phenomena, from flashing fireflies in South East Asia to singing crickets, from cardiac peacemaker or neuron cells to menstrual cycles of women living in strict contact with each other [80]. The goal of these studies is to find a strategy of interaction among the sensors/nodes that could allow them to reach globally optimal decisions in terms of a “consensus” value in a totally decentralized manner. Distributed consensus algorithms are indeed techniques largely studied in distributed computing [81,82]. The approaches suggested in [83,84] give a form of consensus achieved through self-synchronization that may result critical in wide-area networks, where propagation delays might induce an ambiguity problem. This problem is overcome in [85–87] where also a model of the network and of the sensors is proposed. Each of the N nodes composing the network is equipped with four basic components: (1) a transducer that senses the physical parameter of interest (e.g., temperature, concentration of contaminants, radiation, etc.); (2) a local processing unit that provides a function of the measurements; (3) a dynamical system, initialized with the local measurements, whose state evolves as a function of its own measurement and of the state of nearby sensors; (4) a radio interface that makes possible the interactions among the sensors. The criterion to reach a consensus value is the asymptotical convergence toward a common value of all the derivatives of the state, for any set of initial conditions and for any set of bounded. This condition makes the convergence to the final consensus independent of the network graph topology. However the topology has an impact on several aspects: the overall energy necessary to achieve the consensus and the convergence time. In general there exists a trade-off between the local power transmitted by a each sensor and the converge time depending on the algebraic connectivity of the network graph, as shown in [88]. In the practical applications these aspects cannot be neglected; for instance, the design of a network should account for the precision to achieve, and the time to get the consensus value at the given precision, versus such constraints as the energy limitations of the sensors. A global overview of the problem is given in [89].

2.22.5.6 From theory to real application problems

Moving from the functional model to a working implementation in a real environment involves a number of design considerations: including what information sources to use and what fusion architecture to employ, communication protocols, etc.

Admittedly, the fusion of data is decoupled from the actual number of information sources and, hence, does not require necessarily multiple sensors: the fusion, in fact, may be performed also on a temporal sequence of data that was generated by a single information source (e.g., a fusion algorithm may be applied to a sequence of images produced by a single camera sensor). However, employing a number of sensors provides many advantages as well explained in the previous Sections. Unsurprisingly, there are also difficulties associated with the use of multiple sensors.

A missed sensor registration may cause a failure in the correct association between signals or features of different measurements. This problem and the similar data association problem are very important and apply also to single sensor data processing. To perform data registration, the relative locations of the sensors, the relationship between their coordinate systems, and any timing errors need to be known, or estimated, and accounted for otherwise a mismatch between the compiled picture and the truth may result. An overstated confidence in the accuracy of the fused output, and inconsistencies between track databases, such as multiple tracks that correspond to a single target may appear. A missed registration can result from location and orientation errors of the sensor relative to the supporting platform, or of the platform relative to the Earth, such as a bearing measurement with an incorrect North alignment. Errors may be present in data time stamping, and numerical errors may occur in transforming data from one coordinate system to another. Automatic sensor registration can correct for these problems by estimating the bias in the measurements along with the kinematics of the target. However, the errors in sensor registration need to be known and accounted for [90]. In [91] a maximum likelihood (EML) algorithm for registration is presented using a recursive two-step optimization that involves a modified Gauss-Newton procedure to ensure fast convergence. In [92] a novel joint sensor association, registration, and fusion is performed exploiting the expectation–maximization algorithm incorporated with the linear Kalman filter (KF) to give simultaneous state and parameter estimates. The same approach can be followed also with non linear filtering techniques as the Extended KF (EKF) and the Unscented KF (UKF) as proposed in [93], where also the performance is evaluated by means of the PCRLB.

Next to the spatial sensor registration also the temporal alignment cannot be neglected. For instance, a critical aspect of a sensor network is its vulnerability to temporary node sleeping, due to duty-cycling for battery recharge, permanent failures, or even intentional attacks.

Other realistic problems, such as conflicting information and noise model assumptions, may enable the use of some fusion techniques. Noisy input data sometimes yield conflicting observations, a problem that has to be addressed and which does not arise in single sensor data processing. The administration of multiple sensors have to be coordinated and information must be shared between them.

2.22.5.7 Rethinking mathematical algorithms for net-centric approaches

Most of the optimization algorithms have been developed in a centralized framework, i.e., they have been conceived to perform centralized data fusion process. In the last years the trend is to employ network centric approaches, and the mathematical optimization algorithms must be able to support this approach. In the following an example of the adaptation of a “centralized-conceived” algorithm to the new trend is presented.

Consider the following minimization problem to solve:

(22.1)

where are real positive values and the function represents an ellipsoid function, whose axes do not coincide with the reference frame axes if . The problem of Eq. (22.1) can be solved by the steepest descent method in a centralized fusion process frame, hence it will be named “centralized steepest descent.” The centralized steepest descent method when used to solve minimization problems is an iterative procedure that, beginning from an initial guess, updates at every iteration the current approximation of the solution of the function to minimize with a step in the direction of the gradient of the own function. In a network centric approach it may be solved by the application of the Jacobi method² usually employed for the iterative solution of linear system equation.

Consider three agents (namely agent 1, 2, and 3) controlling the three variables . In the centralized data fusion process, represented in Figure 22.10a, the communication between the three agents is completely performed at the same instant of time; in the network centric case this does not happen. Consider the model of Figure 22.10b with the following communication scheme:

moreover the communications among agents is not instantaneous, but they succeeds in time.

The method of the centralized steepest descent applied to the function , given a starting point , is based on the following iterations:

(22.2)

where , and represents the step employed in the steepest descent method.

A network centric steepest descent method can be derived by the communication scheme represented in Figure 22.10b and described below. Given the starting point , the following iterations can be done:

(22.3)

where and represents the step employed in the steepest descent method. Figure 22.11 shows the comparison of the two methods for the previous model. Note that the three agents in the net-centric approach are those looking at the function to be minimized along the x, y, and z axes respectively. The black square and the red diamond in the curves represent respectively the starting point of the iteration and the final position. The black solid line shows the trajectory described by the variables obtained by the application of the centralized steepest descent method; the red solid line shows the behavior of the variables obtained by the net-centric steepest descent method. Note that the red line approaches the minimum by moving along the x, y, and z axes separately. The ellipsoids of Figure 22.11 represent the iso-level surfaces of the objective function. Notice that the telecommunication network modeled for the net-centric steepest descent determines the usual Jacobi iteration employed for the solution of linear systems associated to minimization problems [94–96]. In the following Section 2.22.6.1 this approach is applied to reach the optimal deployment of a sensor network.

2.22.6.1 A cooperative sensor network: optimal deployment and functioning

This section presents a mathematical model for the deployment of a sensor network, for the creation of consensus values from the noisy data measured and a statistical methodology to detect local anomalies in these data. A local anomaly in the data is associated to the presence of an intruder. The model of sensor network presented here is characterized by the absence of a fusion center. In other words the deployment, the construction of the consensus values, and the detection of local anomalies in the data are the result of local interactions between sensors. Nevertheless the local interactions will lead to global solution of the considered problem. This is an example of model of a network centric sensor network. The sensors are assumed to be identical and they measure a quantity pertinent to the properties of the area to survey able to reveal the presence of an intruder. In the proposed study case the sensors are able to measure the temperature of the territory in the position or in the “area” where they are located; in absence of anomalies there is a uniform temperature on the territory where the sensors are deployed. The sensor measures are noisy and can be considered synchronous. This measurement process is repeated periodically in time with a given frequency. From these measures a “consensus” temperature is deduced, pertinently to the territory where the sensors are deployed and an estimate of the magnitude of the noise contained in the data. Finally using these consensus values as reference values local anomalies are detected by the individual sensors. In the following we give some analytical details of the consensus method [97].

Let be a bounded connected polygonal domain in two dimensional real Euclidean space R². The domain represents the territory where the sensor network must be deployed; in our case the downtown part of the Italian city of Urbino, shown in Figure 22.12. Let denote the Euclidean norm of in R². Consider N sensors , located respectively, in the points , assumed to be distinct. To the sensor network deployed in the points corresponds a graph whose nodes are the sensors location and whose edges join the sensors able to communicate between themselves. This graph is assumed to be connected and can be imagined as laid on the territory. The assumption that the graph is connected is equivalent to assuming that the sensors constitute a network. For , a polygonal region is associated to each sensor ; this region is defined by the condition that the points belonging to are closest to the sensor , that is they are closest to , than to any other of the remaining sensors located in . It follows:

(22.4)

When for a given the minimizer of the function is not unique we attribute to , where i is the smallest index between the indices that are minimizers of the function f.

The collection of subsets defined in Eq. (22.4) and further specified by the condition above is a partition of and it is a Voronoi partition of associated to the Voronoi centers , as represented in Figure 22.13 [98], where the sets are the Voronoi cells. The sensor is located in , with , and monitors the sub-region of . Note that there is a Voronoi partition of associated to each choice of the Voronoi centers , that, completed with the graph that defines the communication between the sensors, constitute a deployment of the sensors on the territory .

After the definition of a Voronoi partition of , we want to determine the optimal one with respect to a pre-specified criterion, that in this study case is the fact that the Voronoi centers should coincide (as much as possible) with the centers of mass of the corresponding Voronoi cells . This property translates in mathematical terms the request that the sensors are well distributed on the territory. That is what is called optimal Voronoi partition, i.e., the Voronoi partition associated to the Voronoi centers whose coordinates are the solution of the following problem:

(22.5)

subject to the constraints:

(22.6)

where is the center of mass of the Voronoi cell . Moreover we require:

(22.7)

That is the Voronoi centers and the centers of mass of the Voronoi cells coincide. Note that in general depends on and that the function is in general a non linear function of . The solution of the problem expressed in Eqs. (22.5)–(22.7) after having specified the communications between the sensors is the optimal deployment, represented in Figures 22.13b and 22.14. When the condition of Eq. (22.7) cannot be satisfied, we may accept the available solution of Eqs. (22.5) and (22.6) as location of the Voronoi centers corresponding to the optimal deployment. Note that in general the solution of problem expressed in Eqs. (22.5)–(22.7) is not unique and it can be solved by the application of the steepest descent concept, revised in a network centric frame as shown conceptually in Section 2.22.5.7 [94]. This method can be used to solve the problem of Eq. (22.5) with an iterative procedure, that beginning from an initial guess, updates at every iteration the current approximation of the solution with a step in the direction of the gradient of the function . Moreover the steepest descent method must be adapted to the presence of the constraints of Eq. (22.6), of the condition of Eq. (22.7) and to the requirement that its implementation must lead to a network centric solution of the deployment problem. For sake of brevity, how to impose Eq. (22.6) will not be discussed here, however a treatment of constraints in the continuous analog of the steepest descent algorithm can be found in [99]. Note also that the solutions of Eqs. (22.5) and (22.6) that are of interest are usually interior points of the constraints (6). That is the constraint issue usually is not relevant in the solution of Eqs. (22.5) and (22.6). Similarly we will not pay attention to condition of Eq. (22.7). In fact with respect to Eq. (22.7), we will simply verify if the solution of the optimization problem determined by the steepest descent method satisfies Eq. (22.7). Let us concentrate our attention on the issue of building a network centric implementation of the continuous analog of the steepest descent method to solve Eq. (22.5). Assume that the sensor knows only the position of its neighbor sensors, that is of the sensors that belong to a disk with center and radius . Later we will show how to choose r. The solution of the optimization problem of Eq. (22.5) is found approximating the solution of the system of differential equations:

(22.8)

where denotes a real parameter, with the solution of the “network centric” system differential equations:

(22.9)

with

(22.10)

where

(22.11)

and being the center of mass of the Voronoi cell obtained computing the Voronoi partition of associated to the Voronoi centers . Assume that is large enough to guarantee that is neighbor of when the distance between and is zero, . Note that with this assumption we have . In Eqs. (22.8) and (22.9) the dot denotes the differentiation with respect to . We observe that Eq. (22.8) is known as the steepest descent differential equation. The continuous analog of the steepest descent method consists in obtaining the solution of the optimization problem of Eq. (22.5) computing the asymptotic value as goes to infinity of a solution of Eq. (22.8) equipped when with a suitable initial condition. This asymptotic value hopefully is a point that solves Eq. (22.5) and satisfies Eqs. (22.6) and (22.7).

Note that the function depends only on , that is can be computed in the location using only information available in . Approximating the gradient of F with the appropriate pieces of the gradients of the function , and using Eq. (22.9) instead than Eq. (22.8) we can find an approximation of the solution of Eq. (22.5) integrating numerically the initial value problem for Eq. (22.9). Note that the solution for the ith differential equation of Eq. (22.9) is computed in the location . This approximation of the solution of Eq. (22.5) is obtained using only local information so that it is “network centric.” When the asymptotic value as goes to infinity, the solution of Eq. (22.9) coincides with an asymptotic value of a solution of Eq. (22.8), solving numerically Eq. (22.9), we can obtain in a network centric manner a solution of Eq. (22.5). The choices of the optimal Voronoi partition and of the steepest descent method to determine it, are only one of the many other legitimate choices. In Figure 22.13 the polygonal region shown represents , for N = 20 and for , denoting with the full circle the position of the center of mass of the subset and with the empty circle the position of the sensors . The Figure 22.13a shows the Voronoi partition of the domain , associated to 20 Voronoi centers and the corresponding centers of mass of the associated Voronoi cells . Note that in Figure 22.13a we have . The Figure 22.13b shows an optimal Voronoi partition. Note that in Figure 22.13b we have . The Voronoi partition shown in Figure 22.13b satisfies Eqs. (22.5)–(22.7). The centers of Figure 22.13b have been obtained integrating numerically, using the explicit Euler method in Eq. (22.9), equipped with the initial condition given by the centers shown in Figure 22.13a. In Figure 22.14 we show the graph associated to the optimal Voronoi partition of shown in Figure 22.13b. The graph is obtained joining with branches the Voronoi centers that are (distinct) neighbors. In Figures 22.13 and 22.14 we have chosen , where k is a parameter that can be changed during the optimization procedure used to solve Eqs. (22.5)–(22.7).

Remind that we have assumed that the graph G associated to the optimal deployment is connected (see Figures 22.14 and 22.13b). Moreover we remind that, since there is not a fusion center, each node of the graph G does not know the positions of all the remaining nodes of the graph, in fact it knows only the positions of its neighbor nodes. Let L be the Laplacian matrix associated to G [100]. The matrix L is a symmetric positive semi-definite matrix. Let , be a real N dimensional vector depending on the real parameter . The superscript means transposed. We consider the system of ordinary differential equations:

(22.12)

equipped with the initial conditions:

(22.13)

where denotes the usual matrix vector multiplication, is a known initial condition and the dot denotes differentiation with respect to . Since G is connected we have:

(22.14)

where , is the solution of Eqs. (22.12) and (22.13). This result follows easily from the spectral properties of L [100]. Note also that the right hand side of Eq. (22.14) is the “average” of the initial condition . Note that Eq. (22.12) can be interpreted as the “heat equation” on the graph G, that the problem of Eqs. (22.12) and (22.13) can be seen as an initial value problem for the heat equation on G and that Eq. (22.14) can be understood as the approach to an asymptotic equilibrium “temperature” in an “heat transfer” problem. We assume that during the monitoring phase the sensor measures a physical quantity, such as, for example, the temperature, of the region where it is located. The sensors are identical, the measures made by the sensors are synchronous, repeated periodically in time and of course they are noisy. Moreover they are assumed to be independent. A first set of measures is taken by the sensors at time and is collected in the vector , where is the measure done by the sensor . The set of measure will be used to obtain the “consensus” value of the quantity monitored in at time . We choose:

(22.15)

Remind that the sensor located in knows and communicates with the sensors located in . In order to provide to the sensor , the consensus value in a network centric manner we proceed as follow: we choose in Eq. (22.13) and we integrate numerically the initial value problem of Eqs. (22.12) and (22.13) using the explicit Euler method to obtain a numerical approximation of . Note that the ith differential equation of Eq. (22.12) is integrated in the location , and that using the explicit Euler method this can be done using only information available in the location . Note that the analytic solution of Eqs. (22.12) and (22.13) is not “network centric” but its approximation with the explicit Euler method is “network centric.” In the former case to achieve the solution each node should know the whole graph, i.e., all the nodes. The ith node is not able to achieve the solution exploiting only the information in its posses: in this sense the solution is not “network centric.” Otherwise, exploiting the Euler approximation of the exponential of a matrix, the whole knowledge of the graph is not necessary: in this sense a “network centric” solution is achieved.

Once obtained we consider the following vector:

(22.16)

Then we choose in Eq. (22.13) and we integrate Eqs. (22.12) and (22.13) with the explicit Euler method as done above. In this way we obtain asymptotically a numerical approximation of where:

(22.17)

This approximation of is provided to each sensor in a network centric manner. Note that is an estimate of the magnitude of the noise contained in the data; in fact is the “sample” variance of the measures made by the sensor at time . The approximation of and obtained integrating numerically Eqs. (22.12) and (22.13) are the consensus values. These values are “global” values (that is they depend on all the measures made by the sensor network at time and have been provided to each sensor in a network centric manner (that is using only “local” interactions between sensors).

The sensor repeats periodically in time the measure of the quantity of interest and after a given time interval has as its disposal a set of measures that can be compared with the consensus values and to detect (local) anomalies. Let us assume that the set of measures made by the sensor is a sample taken from a set of independent identically distributed Gaussian random variables of mean and variance . In these hypotheses the Student t-test and the Chi-square test [101] are the elementary statistical tools that must be used to compare and (that are unknown) to and . The result of this comparison is the detection of local anomalies. A (statistical) significance is associated to the detected anomalies. The statistical tests used are based on the assumption that the measures come from a set of independent identically distributed Gaussian random variables. Note that the estimators and can be used in more general circumstances.

2.22.6.2 Modeling and performance analysis of a network of chemical sensors with dynamic collaboration

Typically the challenge in the deployment of an operational wireless sensor network (WSN) resides in establishing the balance between its operational requirements (e.g., minimal detection threshold, the size of surveillance region, detection time, the rate of false negatives, etc.) and the available resources (e.g., energy supply, number of sensors, communication range, fixed detection threshold of individual sensors, limited budget for the cost of hardware, maintenance, etc.) [102]. The issue of resource constraints is particularly important for a network of chemical sensors, because modern chemical sensors are equipped with air-sampling units (fans), which turn on when the sensor is active. Operating a fan requires a significant amount of energy as well as a frequent replacement of some consumable items (i.e., cartridges, filters). This leads to the critical requirement in the design of a WSN to reduce the active (air-sampling) time of its individual sensors.

One attractive way to achieve the described balance between the requirements and the constraints of WSN is to exploit the idea of dynamic sensor collaboration (DSC) [103,104]. The DSC implies that a sensor in the network should be invoked (or activated) only when the network will gain information by its activation [104]. For each individual sensor this information gain can be evaluated against other performance criteria of the sensor system, such as the detection delay or the detection threshold, to find an optimal solution in given circumstances. However, the DSC-based algorithms involve continuous estimation of the state of each sensor in the network and usually require extensive computer simulations [103,104]. These simulations may become unpractical as the number of sensors in the network increases. Furthermore, the simulations can provide the numerical values for optimal network parameters only for a specific scenario.

This motivates the development of another simple and analytic approach to the problem of network analysis and design. The main idea is to phenomenologically employ the so-called bio-inspired (epidemiology, population dynamics) or physics inspired (percolation and graph theory) models of DSC in the sensor network in order to describe the dynamics of collaboration as a single entity [105–110]. From a formal point of view, the equations of bio inspired models of DSC are the ones of the “mean-field” theory, meaning that instead of working with dynamic equations for each individual sensor we use only a small number of equations for the “averaged” sensor state (i.e., passive, active, faulty, etc.), regardless of the actual number of sensors in the system.

The analytic approach can lead to the valuable insights into the performance of the proposed sensor network system by providing simple analytical expressions to calculate the vital network parameters, such as the detection threshold, robustness, responsiveness and stability and their functional relationships.

The fluctuations in concentration C of the pollutant are modeled by the probability density function (pdf) with the mean as a parameter [111]:

(22.18)

Here the value can be chosen to make it compliant with the theory of tracer dispersion in Kolmogorov turbulence [111], but it may vary with meteorological conditions. The parameter , which models the tracer intermittency in the turbulent flow, can be in the range [0, 1], with corresponding to the non-intermittent case. In general it also depends on the sensor position within a chemical plume, thus near the plume centroid and may drop to near the plume edge. For , the pdf of Eq. (22.18) has a delta impulse in zero, meaning that the measured concentration in the presence of intermittency can be zero on some occasions. It can be easily shown that the pdf of Eq. (22.18) integrates to unity, so it is appropriately normalized.

Depending on the values of parameters , Eq. (22.18) allows simulation of pollutant distributions with the different correlation structure (e.g., intermittent and strongly non-Gaussian) corresponding to the rich variety of possible regimes of turbulent mixing occurring in the ambient environment; Figure 22.15 shows two examples of the same WSN operating in two different correlation structures of the chemical tracer.

We adopt a binary model of a chemical sensor, with reading V specified as:

(22.19)

where is the threshold (an internal characteristic of the sensor). It can be shown [112] that the probability of detection of an individual sensor embedded in the environmental model described by Eq. (22.18) is given by:

(22.20)

where

(22.21)

is the cumulative distribution function corresponding to pdf of Eq. (22.18), see [113].

Suppose that N chemical sensors are uniformly distributed over the surveillance domain of area S and adopt the following network protocol for dynamic collaboration. Each sensor can be only in one of the two states: active and passive. The sensor can be activated only by a message it receives from another sensor. Once activated, the sensor remains in the active state during an interval of time ; then it “dies out” (becomes passive). While being in the active state, the sensor senses the environment and if the chemical tracer is detected, it broadcasts a (single) message. The broadcast capability of the sensor is characterized by its communication range . This network with the described dynamic collaboration can be modeled using the epidemic SIS model (susceptible-infected-susceptible) [114]:

(22.22)

where denote the number of active and passive sensors, respectively. The nonlinear terms on the right hand side of Eq. (22.22) are responsible for the interaction between the sensors; parameter is a measure of this interaction. The number of sensor is assumed constant, hence we have an additional equation: . Since the parameter alpha describes the intensity of social interaction in a community [114] we can propose that:

(22.23)

where m is the number of contacts made by the activated (“infected”) sensor during its infectious period (i.e., the number of sensors that received the wake-up message from an alerting sensor). In our case . Then we have:

(22.24)

where G is a calibration constant. In order to simplify notation we will further assume that G is absorbed in the definition of . Equation (22.22) combined with can be reduced to one equation for :

(22.25)

where . By simple change of variables , this equation can be reduced to the standard logistic equation [115,116]:

(22.26)

The solution of the logistic equation is well-known:

(22.27)

where . Observe that the WSN will be able to detect the presence of a pollutant only if , because then as independent of . In this case, after a certain transition interval, the WSN will reach a new steady state with:

(22.28)

From (22.27) and using the expression for b stated above, the activation time (transition interval) is given by:

(22.29)

From Eq. (22.29) it follows that the key requirement for the network to be operational is that , that is:

(22.30)

where is a well-known parameter in epidemiology, referred to as the basic reproductive number [114]. Observe that is independent of ; however, according to Eq. (22.29) the response time of the WSN is strongly dependent on .

It remains to specify q, the number of sensors that should initially be active for the described WSN with dynamic collaboration to be effective. The initial condition is simply , that is on average . Eqs. (22.28)–(22.30) are important analytic results. For a given level of mean pollutant concentration and meteorological conditions (, these expressions provide a simple yet rigorous way to estimate how a change in network and sensor parameters (i.e., will affect the network performance (i.e., .

The examples of agent-based simulation of “information epidemic” in WSN, which satisfies the threshold condition of Eq. (22.30) is presented in Figure 22.16. We can observe that by change of the configuration parameters of WSN we can vary the activation time and the saturation limit of the detection system. Further development of the theoretical framework presented in this section can be found in [117–120].

2.22.6.3 Detection and localization of radioactive point sources with experimental verification

Recently there has been an increased interest in detection and localization of radioactive material [121–125]. Radioactive waste material is relatively easy to obtain with numerous accidents involving its loss or theft reported. The danger is that a terrorist group may acquire some radiological material and use it to build a dirty-bomb. The dirty bomb would consist of waste by products from nuclear reactors wrapped in conventional explosives, which upon detonation would expel deadly radioactive particles into the environment. The ability to rapidly detect and localize radioactive sources is important in order to disable and isolate the potential threat in emergency situations.

This section is concerned with radiological materials that emit gamma rays. The probability that a gamma radiation detector registers counts (N being the set of natural numbers including zero) in seconds, from a source that emits on average counts per second is [126]:

(22.31)

where is the mean and variance of the Poisson distribution. The measurements of radiation field are assumed to be made using a network of low-cost Geiger-Müller (GM) counters as sensors. In general, the problem of detection and localization of point sources or radioactive sources can be solved using either controllable or uncontrollable sensors. Controllable sensors can move and vary the radiation exposure time [127,128]. In this Section we will focus on uncontrollable sensors, placed at known locations with constant and known exposure times.

Assume that sources (r is unknown) are present in the area of interest. Furthermore, the assumption is that the area is flat without obstacles (“open field”). Each source is parameterized by its 2D location and its equivalent strength (a single parameter which takes into account the activity of the source, the value of gamma energy per integration and scaling factors involved, see [129]). Thus the parameter vector of source i is , while the total parameter vector is a stacked vector: . Suppose a network of GM counters is deployed in the field of interest. Let GM counter , located at , reports its count every seconds. Assuming that each GM counter has a uniform directional response and that attenuation of gamma radiation due to air can be neglected, the joint density of the measurement vector , conditional on the parameter vector and the knowledge that r sources are present, can be modeled as [129]:

(22.32)

Here is the mean radiation count at sensor j:

(22.33)

with

(22.34)

being the distance between the source i and sensor j, and the average count due to the background radiation (assumed known). The problem for the network of GM counters is to estimate the number of sources r and the parameter vector for each source . In this section we will present the experimental results obtained using real data and a Bayesian estimation algorithm combined with the minimum description length (MDL) for source number estimation.

A radiological field trial was conducted on a large, flat, and open area without any obstacles at the Puckapunyal airfield site in Victoria, Australia. The measurements were collected using the DSTOs³ Low Cost Advanced Airborne Radiological Survey (LCAARS) survey system which consists of an AN/PDR-77 radiation survey meter equipped with an RS232 interface module, a gamma probe and software written in Visual Basic running on a laptop computer. The gamma probe contains two GM tubes to cover both low and high ranges of dose rates. It was capable of measuring gamma radiation dose rates from background to 9.99 Sv/h⁴ without saturating [130] with a fairly flat response [131]. Three radiation sources were used in the field trial: source 1 was a cesium sources with , source 2 was also a cesium source with , and source 3 was a cobalt source with . The aerial image of the experimental site with the location of sources and the local Cartesian coordinate system is shown in Figure 22.17. Four data sets were collected during the field trails in the presence of r sources, with respectively [132]. Data sets with sources contains 50 count measurements in each measurement point.

Estimation of parameter vector , under the assumption that r is known, was carried out using the Bayesian importance sampling technique known as the progressive correction [125,133]. This technique assumes that prior distribution of , denoted , is available. The information contained in the measurement vector is combined with the prior to give the posterior pdf: . The minimum mean squared error estimate of is then the posterior expectation:

(22.35)

The problem is that the posterior pdf and hence the posterior expectation of Eq. (22.35) cannot be found analytically for the described problem. Instead, an approximation of Eq. (22.35) is computed via the importance sampling: it involves drawing samples of the parameter vector from an importance density and approximating the integral by a weighted sum of the samples. This is carried out in a few stages, each stage drawing samples from a “target distribution” which is gradually approaching the true posterior. The “target distribution” at stage is constructed as:

(22.36)

where with and . An adaptive scheme for the computation of S and factors is given in [125,133]. Assume that a random sample from is available and one wants to generate the samples or particles from . The progressive correction algorithm steps are then as follows [125]:

The procedure is repeated for every stage until . The initial set of particles is drawn from the prior density . The final estimate in Eq. (22.35) is approximated as

(22.37)

The number of sources was estimated using the MDL algorithm [59], which will choose that will maximize the following quantity:

(22.38)

where is the estimate obtained under the assumption that r sources are present and

(22.39)

is the Fisher Information Matrix. It can be shown that

(22.40)

with

(22.41)

The inverse of the FIM gives us the CRLB, which represents the theoretical lower bound for estimation error covariance [135]. Figure 22.18 shows the output of the progressive correction algorithm for data set 3 (with three sources present) after (a) and (b) stages of processing. The red stars indicate the locations of three sources. The green line shows the initial polygon A for the location of sources. The prior density for sampling the initial set of particles for source is:

(22.42)

where stands for uniform distribution over the polygon A and is the gamma distribution with parameters and . From Figure 22.18 we observe how the progressive correction algorithm localiszs the three sources fairly accurately.

As we mentioned earlier, 50 count measurements have been collected by each sensor. This allows us to find the root mean square (rms) estimation error using each snapshot of measurement data from all sensors. Table 22.2 shows the resulting rms errors versus the theoretical CRLB.

The theoretical CRLB was computed using the idealized measurement model as stated by Eqs. (22.32)–(22.34). Considering that this measurement model was very crude with a number of factors neglected (e.g., uniform directional response, neglected air attenuation, perfect knowledge of sensor locations, known and constant average background radiation, etc.), the agreement between the theoretical bound and the RMS estimation errors in Table 22.2 is remarkable. The experimental results in this table effectively verify the measurement model as well as the estimation algorithm.

Results for estimation of r are shown in Table 22.3. The table lists the number of runs (out of 50) that resulted in . It can be observed that the number of sources is estimated correctly in the majority of cases.

More results of experimental data processing can be found in [131,132]. In a recent study [136] it was found that by using all 50 snapshots of measurement data for estimation by progressive correction, results in a posterior pdf which is very narrow but does not include the true source positions. This indicates that the measurement model is not perfect, which is not surprising considering that it is based on many approximations. In situations where the measurement likelihood is not exact, it is necessary to introduce a degree of caution to make the estimation more robust. In the framework of progressive correction this can be achieved by . In this way the measurement likelihood is effectively approximated by a fuzzy membership function which has a theoretical justification in random set theory [137, Chapter 7].

If one wants to relax the assumption that radioactive sources are point sources, the problem becomes the one of radiation field estimation. This is an inverse problem, difficult to solve in general. By modeling the radiation field by a Gaussian mixture, however, the problem becomes tractable and some recent results are reported in [138].

2.22.7.1 Sensor deployment

Sensor deployment is a critical issue for intelligence collection in an uncertain dynamic environment. It concerns with making decisions about when, where, and how many sensing resources need to be deployed in reaction to the state of the environment and its changes.

Sensor placement needs special attention in sensor deployment. It consists of positioning multiple sensors simultaneously in optimal or near optimal locations to support surveillance tasks when necessary. Typically it is desired to locate sensors within a particular region determined by tactical situations to optimize a certain criterion usually expressed in terms of global detection probability, quality of tracks, etc. This problem can be formulated as one of constrained optimization of a set of parameters. It is subject to constraints due to the following factors:

In simple cases, decisions on sensor placement are to be made with respect to a well-prescribed and stationary environment. An example of a stationary problem is the placing of radars to minimize the terrain screening effect in detection of an aircraft approaching a fixed site. Another example is the arrangement of a network of intelligence gathering assets in a specified region to target another well-defined area. In the above scenarios, mathematical or physical models such as terrain models, propagation models, etc. are commonly available and they are used as the basis for evaluation of sensor placement decisions. Paper [139] presents a study for finding a solution to the placement of territorial resources for multi-purpose telecommunication services considering also the restrictions imposed by the orography of the territory itself. To solve this problem genetic algorithms⁵ are used to identify sites to place the resources for the optimal coverage of a given area. The used algorithm has demonstrated to be able to find optimal solutions in a variety of considered situations.

More challenging are those situations in which the environment is dynamic and sensors must repeatedly be repositioned to be able to refine and update the state estimation of moving targets in real time. Typical situations where reactive sensor placement is required are, for instance, submarine tracking by means of passive sonobuoys in an anti-submarine warfare scenario; locating moving transmitters using ESM (Electronic Support Measures) receivers; tracking of tanks on land by dropping passive acoustic sensors.

2.22.7.2 Sensor behavior assignment

The basic purpose of sensor management is to adapt sensor behavior to dynamic environments. By sensor behavior assignment is meant efficient determination and planning of sensor functions and usage according to changing situation awareness or mission requirements. Two crucial points are involved. Firstly the decisions about the set of observation tasks (referred to as system-level tasks) that the sensor system is supposed to accomplish currently or in the near future, on the basis of the current/predicted situation as well as the given mission goal. Secondly the planning and scheduling of actions of the deployed sensors to best accomplish the proposed observation tasks and their objectives.

Owing to limited sensing resources, it is prevalent in real applications that available sensors are not able to serve all desired tasks and achieve all their associated objectives simultaneously. Therefore a reasonable compromise between conflicting demands is sought. Intuitively, more urgent or important tasks should be given higher priority in their competition for resources. Thus a scheme is required to prioritize observation tasks. Information about task priority can be very useful in scheduling of sensor actions and for negotiation between sensors in a decentralized paradigm.

To focus on this class of problems, let us consider a scenario including a number of targets as well as multiple sensors, which are capable of focusing on different objects with different modes for target tracking and/or classification. The first step for the sensor management system should be to utilize evidences gathered to decide objects of interest and to prioritize which objects to look at in the time following. Subsequently, in the second step, different sensors together with their modes are allocated across the interesting objects to achieve best situation awareness. In fact, owing to the constraints on sensors and computational resources, it is in general not possible to measure all targets of interest with all sensors in a single time interval. Also, improvement of the accuracy on one object may lead to degradation of performance on another object. What is required is a suitable compromise among different targets.

2.22.7.3 Sensor coordination in a decentralized sensor network

As stated in the previous Sections, there are two general ways to integrate a set of sensors into a sensor network. One is the centra1ized paradigm, where all actions of all sensors are decided by a central mechanism. The other alternative is to treat sensors in the network as distributed intelligent agents with some degree of autonomy. In such a decentralized architecture, bi-directional communication between sensors is enabled, so that communication bottlenecks possibly existing in a centralized network can be avoided. A major research objective of decentralized sensor management is to establish cooperative behavior between sensors with no or little external supervision. In a decentralized sensor network scenario a local view perceived from a sensor can be shared by some members of the sensor community. Intuitively, a local picture from one sensor can be used to direct the attention of other sensors or transfer tasks such as target tracking from one sensor to another. An interesting question is how participating sensors can autonomously coordinate their movements and sensing actions, on grounds of shared information, to develop an optimal global awareness of the environment with parsimonious consumption of time and resources.

As for homogeneous sensor network, the CSIP approach can be exploited [141,142]: the network consists of different kinds of sensors, randomly distributed inside the surveillance area and if the number of sensors is high, the performance of the surveillance system can be considered independent of the location of the targets. Each sensor has a different functioning level. A first level sensor, with small sensing and communication capabilities may provide only detection information; a second level sensor may provide detection and localization information, with medium sensing and communication capabilities. Finally a third level sensor may provide tracking information and may be able to perform target recognition and classification. Usually the number of low level sensors exceeds the number of higher level sensors and only close sensors exchange data.

In [143] the network consists of two types of sensors: simple and complex as represented in Figure 22.19a. The simple ones have only the capability of sensing their coverage area with a reduced computation capabilities and they transmit data to complex sensors. The information they provide may be encoded, for example, by a “1” if sensor detects something crossing its coverage area and by a “0” otherwise. Complex sensors, instead, have computation capabilities; they are able to locate the target by applying sophisticated algorithms (e.g., in [143] the maximum likelihood estimation algorithm is applied). The topology simulated in [143], constituted by 80 simple sensors and 20 complex sensors, is represented in Figure 22.19b: the sensors are indicated by circles; the complex sensors are connected by the solid lines, simple and complex sensor by dashed lines. Figure 22.20 shows the number of active sensors during the target tracking: the theoretical value and the simulated value are compared. It is evident that in a self-organizing configuration the number of active sensors is optimized with the consequent advantage of saving of power.

An adaptive self-configuring system consists of a collection of independent randomly located sensors that, carrying ahead local interactions, estimate the position of the target without a centralized control unit that coordinates their communication. It is fault tolerant and adapts to changing conditions. Furthermore, it is able to self-configuring, i.e., there is not an external entity that configures the network. Finally, the task is performed efficiently, i.e., it guarantees both a reasonably long network life and good target tracking performances. From local interactions, sensors form an efficient system that follows the target, i.e., local communication leads to a self-organizing network that exploits the features of the theories of random graphs and of self-organizing systems. The most natural way to approach random network topology is by means of the theory of random graphs [144,145]. The theory of random graphs allows, for instance, to compute an upper bound to the estimated number of active sensors at each time step.

2.22.7.4 A mathematical issue for multi-sensor networks

When the fusion of heterogeneous signals is performed, there is a formal problem to solve. The signal received by the different sensors may be statistically dependent because of the complex intermodal interactions; usually the statistical dependence is either ignored or not adequately considered. Usually the multiple hypotheses testing theory is based on the statistical independence of the received signals, in our case this condition is not maintained, therefore techniques as the “copula probability theory” may be useful.

In probability theory and statistics, a copula can be used to describe the dependence between random variables [146]. The cumulative distribution function of a random vector can be written in terms of marginal distribution functions and a copula. The marginal distribution functions describe the marginal distribution of each component of the random vector and the copula describes the dependence structure between the components. Copulas are popular in statistical applications as they allow one to easily model and estimate the distribution of random vectors by estimating marginal distributions and copula separately. The Sklar’s theorem ensures that the joint cumulative distribution function (cdf) of random variables are joined by a copula function to the respective marginal distributions as [147]:

(22.43)

Further, if the marginals are continuous, is unique. By the differentiation of the joint cdf, the joint pdf is obtained:

(22.44)

The copula density , function of the N marginals from the N sensors, represents a correction term of the independent product of densities of Eq. (22.44).

Processing heterogeneous data set is not straightforward as they may not be commensurate. In addition, the signals may also exhibit statistical dependence due to overlapping fields of view. In [148] the authors propose a copula-based solution to incorporate statistical dependence between disparate sources of information. The important problem of identifying the best copula for binary classification problems is also addressed and a copula based test-statistic, able to decouples marginals and dependency information, is developed.

2.22.8.1 Multi-scale approach

The size of the region, the nature of the border and the complexity of the scenario require the provision of different pictures of the region with different field of view at different resolution and time scales, suggesting a multi-sensor/multi-scale approach integrated in a hierarchical architecture of the whole system. Typically a global field of view of the whole region is necessary at the higher Command and Control (C2) level to capture the overall situation. A higher level of resolution and refresh rate is necessary at the lower and local level to analyze and control in depth each single zone of a region. Therefore the surveillance segment may be structured according to a multilayer architecture where layers realize different trade-offs in terms of field of view and granularity and refresh time. The surveillance segment comprises several types of sensors, each one characterized by different achievable resolution, field of view, and revisiting time. A pictorial sketch of the surveillance architecture is depicted in Figure 22.21 for a notional country: sensors on board of satellites are expected to provide a global coverage of the monitored area at medium resolution with a low refresh rate, typically in the order of several hours or days; a higher resolution data and a higher refresh rate, in the order of seconds or tens of seconds, is provided by ground sensors on limited areas; airborne sensors (e.g., Unmanned Air Vehicle, UAV) will provide data on remote areas with good resolution data and short deployment time.

All data collected by the sensors are exploited by the fusion engine, highlighted in the figure. It is responsible to track and classify relevant entities present in the scenario and to provide a high quality representation of the situation. Also the data fusion process supports this multi-scale approach performing a distributed and network-centric processing at the various levels of the architecture, in accordance with available communication bandwidth and latency.

2.22.8.2 The electronic fence

The surveillance of critical perimeters is one of the most important issues in Homeland defense and Homeland Protection systems. The ground surveillance needs are relevant to border protection applications, but include also local area protection, such as critical infrastructure, military/civilian posts.

During the last 10 years special attention has been focused on the realization of so-called “electronic fence” for perimeter/border control and several developments have been carried out to demonstrate the efficiency of such systems. However several problems occurred when the electronic fences became operational, showing lacks in the practical use by the operators (i.e., high number of false alarms, loss of/slow communication links) together with the problem of the high funding required for the whole system. One example is described in [149], that requires now a total different approach for the surveillance of a wide national border ().

In the following an overview of the problems and solutions related to the implementation of an electronic fence is presented. The major components are:

Depending on the geographical deployment of the protection system, the data are then exchanged with C2 centers, both at local level and wide area (i.e., national) level. In Figure 22.22 an example of an electronic fence architecture is depicted. In this case a wide area to be controlled, such as a border of a nation, has been considered; the subnets are geographically distributed along the boundaries. The architecture has the advantage to be modular and scalable and it can be organized with different level C2 centers (local, regional, national), depending also on the size of the considered boundaries. Each subnet is able to ensure the data exchange among the sensors. An overview of the sensors that can be employed in an electronic fence is presented.

2.22.9.1 Modeling

To describe the dynamics of an epidemic outbreak we employ the generalized SIR (Susceptible, Infectious and Recovered) epidemic model with stochastic fluctuations [195–197]. According to this model, the population of a community can be divided into three interacting groups: susceptible, infectious and recovered. Let the number of susceptible, infectious and recovered be denoted by S, I, and R, respectively, so that S + I + R = P, where P is the total population size. The dynamic model of epidemic progression in time can be then expressed by two stochastic differential equations subject to the “conservation” law for the population:

(22.46)

where , and are two uncorrelated white Gaussian noise processes, both with zero mean and unit variance. The terms and are introduced into Eq. (22.46) to capture the demographic noise (random variations in the contact rate and in the recovery time ) [197,198]. Parameter in Eq. (22.46) is the population mixing parameter, which for a homogeneous population equals 1. In the presence of an epidemic, however, may vary as people change their daily habits to reduce the risk of infection (e.g., panic, self-isolation). In general, model parameters can be assumed to be partially known as interval values. In order to insure , standard deviations need to satisfy [199]:

(22.47)

Assuming that non-medical syndromic data are available for estimation and forecasting of the epidemic, we adopt a measurement model verified by [185,186], where a power law relationship holds for the odds-ratio between the observable syndrome and the (normalized) number of infected people i:

(22.48)

The power law exponent in Eq. (22.48) is in general syndrome specific. Since at the initial stages of an epidemic (which is of main interest for early detection and forecasting) we have: and , Eq. (22.48) can be reduced to a simple power-law model:

(22.49)

where is a constant and is introduced to model the random nature of measurement noise. It is assumed that is uncorrelated to other syndromes and dynamic noises . Since (e.g., number of Google searches), the noise term associated with syndrome j should be modeled by a random variable that provides strictly non-negative realizations. For this purpose we adopt the log-normal distribution, that is , with and being the standard Gaussian distribution.

Parameters typically are not known, but with a representative data set of observations the model of Eq. (22.49) can be easily calibrated (see for example the results of the linear regression fits in [186]). The data fit reported in [183] suggests that may be close to unity, although it is difficult to precisely specify its value because of significant scattering of data points). To cater for this uncertainty, we assume that can take any value in an interval, around . Unfortunately [185,186] do not report any specific values of fitting parameters, so we use in this study some heuristic values for in our simulations.

The problem now is to estimate the (normalized) number of infected i, and susceptible s at time t, using syndromic observations of Eq. (22.49), collected up to time t. Let x denote the state vector to be estimated; it includes i and s, but also the imprecisely known epidemic model parameters and . The formal Bayesian solution is given in the form of the posterior pdf , where is the state vector at time t and denotes all observations up to time t. Using the posterior , one can predict the progress of the epidemic using the dynamic model of Eq. (22.46).

2.22.9.2 Sequential Bayesian solution

For the purpose of computer implementation, first we need a discrete-time approximation of dynamic model of Eq. (22.46). The state vector is adopted as: , where ^T is the matrix transpose. Using Euler’s method with small integration interval , the nonlinear differential equations in Eq. (22.46) can be approximated as

(22.50)

where is the discrete-time index and

(22.51)

is the transition function; here denotes the ith component of vector . Discrete-time process noise in Eq. (22.50) is assumed to be zero-mean white Gaussian with diagonal covariance matrix , which according to Eq. (22.47) can be expressed as .

The optimal Bayes filter is typically presented in two steps, prediction and update. Suppose the posterior pdf at time is given by . Then the prediction step computes the pdf predicted to time as [194]:

(22.52)

where is the transitional density. According to Eq. (22.50), we can write . The prediction step is carried out many times with tiny sampling intervals until observation about syndrome j becomes available at . The predicted pdf at is denoted .

In the standard Bayesian estimation framework, the predicted pdf is updated using measurement by multiplication with the measurement likelihood function [200]. According to Eq. (22.49), the likelihood function in this case is , where . The standard Bayesian approach, however, cannot be applied because defined in this way is not a function: is effectively an infinite set (an interval) and therefore is one-to-many mapping.

An elegant solution to the imprecise measurement transformation is available in the framework of random set theory [137]. In this approach is modeled by a random set and the likelihood function represents the probability:, and is referred to as the generalized likelihood. More details and a theoretical justification of this approach can be found in [201]. The Bayes update using syndromic measurement is now defined as [137]:

(22.53)

For the measurement model Eq. (22.49) with additive Gaussian noise, the generalized likelihood has an analytic expression [201]:

(22.54)

where define the limits of the set and is the cumulative log-normal distribution. The recursions of the Bayes filter start with an initial pdf (at time , denoted , which is assumed known.

The proposed Bayesian estimator cannot be solved in the closed form. Instead we developed an approximate solution based on the particle filter (PF) [134,202]. The PF approximates the posterior pdf by a weighted random samples; details can be found in [134,202]. The only difference here is that importance weight computation is based on the generalized likelihood function.

2.22.9.3 Numerical results

Epidemic forecasting will be demonstrated using an experimental data set obtained using a large-scale agent based simulation model [203,204] of a virtual town of P = 5000 inhabitants, created in accordance with the Australian Census Bureau data. The agent based model is rather complex (takes a long time to run) and incorporates a typical age/gender breakdown, family-household-workplace habits, including the realistic day-to-day people contacts for a disease spread. The blue line in Figure 22.32 shows the number of people of this town infected by a fictitious disease, reported once per day during a period of 154 days (only first 120 days shown). The dashed red line represents the adopted SIR model fit, using the entire batch of 154 data points and integration interval days, with no process noise, i.e., in Eq. (22.50). The estimated model parameters are: . These estimates were obtained using the importance sampling technique of progressive correction [202]. Figure 22.32 serves to verify that the adopted non-homogeneous mixing SIR model, although very simple and fast to run, is remarkably accurate in explaining the data obtained from a very complex simulation system.

The true number of infected people in forecasting simulations is chosen to be the output of the agent based population model, shown by the solid blue line in Figure 22.32. The measurements are generated synthetically in accordance with Eq. (22.49) and discussions above, using the following parameters: monitored syndromes. Independent measurements concerning all Nz = 4 syndromes are assumed available on a daily basis during the first 25 days. The problem is to perform the estimation sequentially as the measurements become available until the day number 25, and at that point of time to forecast the number of infected people as a function of time.

The initial pdf for the state vector was chosen as with , where and denote the truncated Gaussian distribution, restricted to , and uniform distribution, respectively. The imprecise measurement parameter is adopted as , while its true value The number of particles is set to 10,000.

Figure 22.33 shows the histograms of particle filter estimated values of , and , after processing 25 days of syndromic data (i.e., in total 100 measurements). The histograms in this figure reveals that the uncertainty in parameters and has been substantially reduced after processing the data (compared with the initial and ). The uncertainty in , on the other hand, has not been reduced, indicating that this parameter cannot be estimated from syndromic data. While this is unfortunate, it does not appear to be a serious problem in forecasting the epidemic mainly because the prior on in practice is fairly tight ( ). This is confirmed in Figure 22.34 which shows a sample of 100 overlaid predicted epidemic curves (gray lines) based on the estimate of obtained after 25 days. Figure 22.34 indicates that the forecast of the peak of the epidemic is fairly accurate, while the forecast of the size of the peak is more uncertain. Most importantly, however, the true epidemic curve (solid red line) appears to be always enveloped by the prediction curves. More experimental results can be found in [199].