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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

Recent advances in integrated electronic devices motivated the use of Wireless Sensor Networks (WSNs) in many applications including domain surveillance and mobile target tracking, where a number of sensors are scattered within a sensitive region to detect the presence of intruders and forward related events to some analysis center(s). Obviously, sensor deployment should guarantee an optimal event detection rate and should reduce coverage holes. Most of the coverage control approaches proposed in the literature deal with two-dimensional zones and do not develop strategies to handle coverage in three-dimensional domains, which is becoming a requirement for many applications including water monitoring, indoor surveillance, and projectile tracking. This paper proposes efficient techniques to detect coverage holes in a 3D domain using a finite set of sensors, repair the holes, and track hostile targets. To this end, we use the concepts of Voronoi tessellation, Vietoris complex, and retract by deformation. We show in particular that, through a set of iterative transformations of the Vietoris complex corresponding to the deployed sensors, the number of coverage holes can be computed with a low complexity. Mobility strategies are also proposed to repair holes by moving appropriately sensors towards the uncovered zones. The tracking objective is to set a non-uniform WSN coverage within the monitored domain to allow detecting the target(s) by the set of sensors. We show, in particular, how the proposed algorithms adapt to cope with obstacles. Simulation experiments are carried out to analyze the efficiency of the proposed models. To our knowledge, repairing and tracking is addressed for the first time in 3D spaces with different sensor coverage schemes.

One among the main WSN issues that should be addressed while dealing with target tracking and monitoring applications, in 3D environments with obstacles, is area coverage. This is because a sensor can detect the occurrence of events or the presence of hostile targets only if they are within its sensing range. Coverage reflects how well a zone is monitored or a system is tracked by sensors. Therefore, the WSN detection performance depends on how well the wireless sensors observe the physical space under control.

Several metrics have been provided in the literature to measure the quality of coverage. Among these metrics, one can mention the following: (a) the number of coverage holes; (b) the proportion of uncovered area with respect to the area under monitoring; and (c) the so called Average Linear Uncovered Length (ALUL), which has been developed in 2D zones to estimate the average distance a mobile target can traverse before being detected by one sensor [

Obstacles in monitored 3D domains may complicate seriously the role of the monitoring sensors, increase their power consumption, and limit the coverage efficiency of the process providing coverage control [

This paper proposes a coverage assessment approach amenable to implement advanced target tracking functionalities. First, it provides a technique based on the concept of retraction by deformation applied to a special space, called the Rips complex, associated with the deployment of a set of sensors to develop a low complexity algorithm for locating coverage holes. Second, it constructs a collaborative mechanism to repair coverage holes, assuming that the sensors have mobility capabilities. Third, the paper builds on higher-order Voronoi diagrams to define an efficient scheme to coordinate tracking activities of single and multiple targets. To the best of our knowledge, this is the first time where retraction by deformation and higher-order Voronoi tessellations are used for hole assessment and target tracking in 3D domains with obstacles using sensors. The major contributions of this paper are as follows:

The definition proposed to distributively reduce the Rips complex associated to the sensors is general, in the sense that it applies to a large variety of sensor, detection techniques, monitored domains, and obstacles.

The proposed cooperative coverage repairing approach considerably reduces the uncovered areas and provides efficient handling of obstacles with respect to existing methods. The detection and localization of holes is done with low complexity.

We show that the higher-order Voronoi tessellations we utilize are useful for performing multiple tasks including activity scheduling and coordination. In addition, we show that local coverage information, when gathered using the Voronoi diagram, can be used to implement coverage preserving mobility models.

The remaining part of this paper is organized as follows: Section 2 describes the state of the art of coverage control in various areas in general and in 3D spaces in particular. Section 3 surveys the definition of the mathematical objects needed for coverage and tracking control, the Vietoris complex and the Voronoi diagram and discusses the retraction by deformation. Section 4 discusses different schemes based on the Vietoris complex to detect and count the coverage holes in 3D domains, locate these holes, and repair them. It also defines a special procedure to reduce the complexity of the Vietoris complexes without modifying their topological properties. Section 5 sets up models for coverage assessment, sensor mobility, and target tracking. Section 6 analyzes the complexity of the algorithms constructed in this paper and sets some extensions of our results to more general types of sensors. Section 7 develops simulation experiments to evaluate the performance of a monitoring system implementing our techniques. Section 8 concludes this paper.

Studies on coverage, holes, and boundary detection have been addressed using three main categories of techniques: geometric methods, statistical/probabilistic methods, and topological methods.

Studies using probabilistic approaches usually make assumptions on the probability distribution of the sensor deployment. Fekete

A number of literature has addressed the static or “blanket” coverage. Dynamic or “sweeping” coverage [

More recently, the robotics community has explored how networked sensors and robots can interact and augment each other: (see e.g., [

Since coverage verification is inherently a geometric problem, many research done in this area are based on computational geometry, and more precisely on the Voronoi Tessellation (and its dual, Delauney Triangulation). Motivated from the early success of the application of geometric techniques to cope with coverage problems (

The most important drawback of these approaches is that they are too computationally expensive to be implemented in real-time contexts. Another severe limitation is the impact of localization uncertainty on the performance of these approaches. These claims are well-documented in ([

Meguerdichian

Huang

Ghrist

The objective of this section is to provide a mathematical model for accurately gauging the coverage degree of a monitored domain in the 3D space ^{3} and repairing the coverage holes. This model uses the Vietoris Complex [

The following assumptions will be used in the next subsections: Let ^{3} with non-empty boundary ∂^{3} × ^{3} → ^{+} denoting the Euclidean distance. We denote by ^{3} to monitor _{p}, y_{p}, z_{p}^{3}. Let us notice, finally, that the sensors can be deployed inside

Let us assume that the sensors in ^{3} perpendicular to segment [^{3} containing the

We also denote by _{M}_{M}

The Voronoi cell generated by

The Voronoi cell of a sensor is convex and contractible. The common boundary of two Voronoi cells _{S}_{S}^{3} by the Voronoi cells associated to all sensors. Thus, every cell of the subdivision contains the nearest neighbors defined in

In particular, the Voronoi diagram ^{D}^{3}.

Since in this paper, we are rather interested in partitioning a domain

One can easily see that the order 1 Voronoi diagram
^{D}S^{D}

We consider a set of points _{1},..., _{n}_{i}_{1≤i≤n} will be simply referring indifferently to as sensor nodes and points. We suppose that each sensor is capable of covering a disk of radius _{c}_{vi,rc} = {^{3} : ||_{i}_{c}

A _{0}, _{1}, .., _{k}_{i} ≠ v_{j}

Now let us discuss the definition of the Vietoris-Rips complex. This complex captures the features related to connectivity and coverage of WSNs.

_{ε}

A subset of

The reader, however, may wonder whether such topological structure can be computed in practice by tiny motes equipped with radio devices and limited storage capabilities. To answer this question, we propose a simple mechanism allowing a fully distributed construction of the Vietoris-rips complex. Through a 3-step broadcast of connectivity information, each sensor node can be aware of what simplices it belongs to, and what other simplices its neighbors belong to. To this end, we assume that every sensor node has a unique identifier (typically a layer-2 address) and has enough space to maintain a table of identifiers. The protocol performs as follows:

Initialization: Every sensor _{i}

Edge construction: Sensor _{i}_{i}

Simplicial iteration: On receiving the information from its neighbors, sensor _{i}_{i}

An informal explanation of the construction algorithm is as follows. Simplices of higher dimension are constructed iteratively. In the first iteration, the 2-simplices are constructed by applying the following rule:

Let

Let _{0} ∈ _{0} is a map _{0} = _{0} can be defined by stating that loops _{1} and _{2} are equivalent if they are homotopic with respect to _{0}; meaning that there exists a homotopy _{1} and _{2} such that

We denote the equivalence class of a loop _{0} by [_{0} is denoted by _{1}(_{0}) and is called the fundamental group. It can be equipped with a multiplication defined by [_{1}] ★ [_{2}] = [_{1}._{2}], for all loops [_{1}] and [_{2}] based at _{0}, where _{1}._{2} is the loop obtained by attaching _{1} to _{2}. A second group of homotopy, denoted by _{2}(_{0}) can be defined as the set of homotopy equivalence classes of applications ^{2} _{0}. It is an Abelian group, [

On the other hand, a map

A deformation retraction of a space

In other words, The subset

Finally, let _{i}_{k}_{k}_{+1}. Thus, it can be shown easily that, _{0} and _{n}

Let _{1}, .., _{k}_{1} = {_{2}, .., _{k}_{1} − ∂_{1}) be the part of the boundary of _{1}. Then

Let _{k}_{k}_{k}_{+1} and _{k}_{+1} is obtained from _{k}_{k}_{0} has no simplex with external face that is retractible.

In this section, we propose a novel distributed technique to count the coverage holes of WSN using the retraction theory of spaces. In particular, we show that the Vietoris-rips complex associated with the WSN can be reduced to a simpler space that is tightly related to the number of holes.

In the following, let ^{3} be a compact domain in the 3D space ℝ^{3} and ∂_{1}, .., _{n}_{b}

We assume, in this subsection, a complete absence of localization capabilities and metric information, in the sense that the sensors in the network can determine neither distance nor direction. Under these assumptions, we are interested in designing distributed algorithms for coverage assessment and hole detection.

To this end, we need first to introduce a special procedure, called

Let _{ε}_{0}, ..., _{3}} be a 3-simplex in _{ε}_{0}, _{1}, _{2}} does not belong to another 3-simplex in _{ε}_{ε}_{1} and _{1} be the set of points _{ε}_{0}_{1}_{2}} and the subset of _{1} generated by the other two faces, respectively. Then its is easy to construct a map _{1} : _{1} _{1} such that:

Map _{1} can be easily extended to a map
_{ε}_{1}) ∪ _{1}.

Repeating the map _{ε}

_{ε}

_{ε}

_{ε}

_{ε}_{ε}_{i}_{i}_{+1}, we can deduce that
_{ε}

The second statement of the theorem can be deduced from the following features:

a holes is a path connected component that is surrounded by the delimiting space (
_{ε}

Retracting a 3-simplex in _{ε}

The retraction process does not create holes since it operates on the simplices that have free faces.

To count and locate holes, we set up a 3-step algorithm. In the first step, we construct the external boundary of _{ε}

Let us assume that the boundary ∂

Every sensor node detecting a boundary component of

The information related to boundary detection, when received by sensors should be put together to form the external boundary of _{ε}

The nodes broadcast information related the external boundary of _{ε}

Counting the coverage holes can be set up by an algorithm that repeats iteratively the following major procedures:

Boundary retraction: Let _{n}_{ε}_{n−1} be one of its external faces, then _{n}_{n}_{n−1}), if _{n−1}, if

Boundary deflation: When all the simplices on the boundary of have been retracted, a pre-selected node in

It is worth noticing that, when a deflation of a 2-simplex on the boundary _{ε}

One can conclude, therefore, that any time a deflation is operated, a hole can be located by simply constructing its boundary using the nearest nodes to that hole.

Let us here assume that the 3D domain ^{3}/3) the volume of the area covered by a sensor and by _{0} =

Repairing holes aims at extending the coverage by eliminating the holes, or at least by shrinking considerably their size. An algorithm can be defined to this purpose. It can be built based on the following general rules:

A node detecting the external boundary ∂

A node on the external boundary of _{ε}

When two neighbor nodes on the external boundary of _{ε}_{1}, and one of them is not seeing the boundary of

A node seeing the boundary should inform its neighbors so that they can move accordingly.

When the distance between a sensor _{2}, then

A node on the external boundary, finding itself unable to move informs, its successor to move towards its direction.

In this section, we use 3D Voronoi diagrams to optimize sensor coverage and target tracking performance. We first propose a strategy to measure the uncovered zones of the monitored region. Then, we develop two mobility models that provide target tracking using order k Voronoi diagrams and optimize the coverage ratio of a zone using Voronoi cells. Finally, we extend these models to multiple target tracking. We assume in this section that the sensors have spherical coverage. The vector-guided case can be addressed using similar techniques.

Assume that a location _{1}, _{2}), for 0 ≤ _{1} ≤ 2_{2} ≤

The Average Linear Uncovered Length (ALUL), denoted by

More generally, when

the ALUL metric was developed to deal with a static deployment, which is not the case of our study. When a mobility model is implemented, the topology of the WSN is no longer static. To overcome this, we extend this notion so as to support sensor node mobility. The ALUL should also vary according to time and should use a function, denoted by the ℒ(_{m}

Due to sensor node mobility, the ALUL, over time, in a point

Finally, _{m}_{m}

From the performance evaluation perspective, two important points should be highlighted:

_{m}

_{m}

In this section, we show how the Voronoi cells can be used to implement target tracking using a sensor mobility model. In fact, we define two mobility models:

The first model is called k-mobility model. Sensor nodes in this model move toward the regions where the hostile target is supposed to be and collaborate to keep the target controlled by k sensors all the time, To this end, the order k Voronoi diagrams are used and maintained all the time.

The second model is called simplified model. It relies on estimating the uncovered zones within the Voronoi cells, using the ALUL metrics and moving sensor nodes toward the “uncovered zones”.

While the first model is triggered by the occurrence of targets, the second model aims at adapting the covered area so that the targets can be detected with higher probabilities. Obviously, the k-mobility model is more energy-consuming than the second since it encompasses the prediction of the target position and requires tracking using k sensor nodes. Therefore, we suppose that the second model can be used when energy resources become scarce. The performance of both models will be assessed in Section 7. Moreover, one can notice that the prediction function we are using is tightly related to the coverage of the zones where the targets are expected and that the mobility models assume that nearest sensor nodes can move to these zones while reducing the coverage of other zones where targets are not expected. In fact, the greater is the number of target detection signals, the better is the prediction precision to command sensor movements.

In the following, we distinguish two cases: (a) a target crossing a

The mobility algorithm is triggered upon the detection of a target presence. Every detecting sensor sends its detection signal to the relevant intermediate sensor (called IS). The latter collects all detection signals, verifies their integrity, deduces the current zone where the target might be, estimates the positions of the target in the next of time slot, and commands k sensors to move to monitor the new zone to ensure tracking continuity.

Typically, the selected zone of target presence is taken among other zones (when more then k sensors detect the target presence). These zones are ordered according to the probability of presence of the target. The zone selected is the one presenting the highest probability among those which are

The mobility algorithm is defined through five steps:

Assume that _{i}_{i}_{t,i}_{i}, z_{t,i}_{i}_{t,i}_{i}, θ_{t,i}_{t,i}_{t,i}_{t,i}

In the case where detection signals are sent to different intermediate nodes; the intermediate coordinate to gather all signals (or at least

IS constructs:

The zone of target presence

The most likely target presence zone ^{τ}^{τ}

Then, IS computes the order k Voronoi cell ^{S}^{τ}

IS estimates the zone ^{τ,+}(_{t}_{t}

IS selects ^{τ,+}(

When a criteria is applied for the selection of k sensors to cover the new position, some of the selected sensors (say

In this case, only

IS computes the most likely zone of target presence let _{t}_{1}, ..., _{k′}

For each _{i}

For each _{i}_{i}

IS classifies the _{i}_{i}_{j}

IS selects the nearest _{i}_{j≤k′}_{j}_{i}_{i}_{i}, p_{i}, p_{i}, q_{i}_{i}_{i}_{i}

To enhance coverage while keeping more mobility freedom, we implement a group mobility model in which ground sensors move in groups in order to preserve the

We propose hereafter a mobility model which is based on the use of simple Voronoi diagram to identify and reduce coverage holes.

This model can serve to implement a mobility strategy where a sensor node looks for one or more neighbors that are at least 2_{f}

The following result extends this strategy to the case where the monitored region is required to be _{i}^{3}, denotes the boundary of

For the sake of parsimony, we do not provide proofs for these corollaries in this paper.

_{i} in S, if |N_{i}^{D}^{D}_{i}_{i} in S, if |X_{i}^{D}^{D}_{i}

This lemma shows how simple Voronoi diagrams can be used to detect the coverage holes based on the distance between the sensor node and the edges of its Voronoi cell. It is based on the concept that the Voronoi tessellation is a partition of the points belonging to the monitored area according to their proximity to the sensor nodes. In other terms, if a point is not detected by the sensor node located at the generator of the Voronoi cell it belongs to, it cannot be detected by any other sensor node. If a sensor detects that the distance to one among the edges of its Voronoi edges is more than its coverage range, it has to move towards this edge to cover the corresponding hole. The uncovered can therefore be gradually reduced using this distributed strategy. However, a sensor node can detect that more than one of its Voronoi neighbors do not fulfill the condition of the lemma, it will therefore move towards the most distant neighbor.

The major advantage of this strategy, with respect to the advanced strategy, is that it relies on simple Voronoi diagrams to deal with

A more accurate comparison between the two models will be carried out in the simulation section.

The two tracking models presented in the above can be extended to the tracking of multiple targets. To describe the extension let us assume, for the sake of clarity, only two targets are detected by sensors in _{t}_{t}_{′} be the reported positions.

The extension of the simplified model considers two cases:

Only one node has detected the presence of the two targets: In that case, the sensor keeps monitoring one of the targets and invites the nearest neighbor to the second target to monitor the second and provides it with relevant information it collects.

More than one node have detected the targets: In that case, two sensor among those that have detected the targets are selected to keep monitoring the targets independently.

On the other hand, the k-mobility model extends in following way: if d nodes detect the targets, these sensors are divided into two subsets, each in charge of monitoring one target, then the subsets are extended so that any of them contains k sensors.

In this section, we analyze the complexity of the different algorithms we have developed in the previous sections for the detect and locate holes or to repair coverage holes. Our approach to estimate the complexity can be based on the following metrics:

The number of messages exchanged between the sensors during the execution of the algorithm.

The number of additions and deletions of simplices to the Vietoris complex.

The number of sensor movements made during the execution of an algorithm.

Some other operations can be added for a more accurate estimation of complexity. These metrics may include, for example, the number of storing operations made at the node level to update the related data structures. The messages exchanged during the execution of an algorithm can be of different types. In particular, they can be sent to a neighbor to tell it to change its status from internal (to the Vietoris complex) to external (

For the sake of clarity, we will focus on the complexity on the detection and counting of coverage hole. In this case, let

The number of messages sent during the execution of the algorithm should be lower or equal to the number of messages exchanged if all the polyhedra (external and internal) have been retracted first and that after deflation all the facets have been retracted. In that case, one can state that the number _{1} of messages sent is given by:

Now let us assume, without loss of generality, that the deployment of sensors (initial and current) guarantees that every node in the Rips complex of

The results presented in the previous sections can be extended in two dimensions: the type of the sensors and the occurrence of obstacles in the domain under monitoring.

It is worth noticing that the additions made to the developed algorithms do not modify significantly the complexity of the algorithms. In particular, the complexity of the hole counter remains linear (as shown in the simulation discussed in the following section).

In this section, we carry out a set of experiments to prove the efficiency of the proposed techniques. We first address the coverage hole problem by evaluating the performance of the higher-order Voronoi-based strategy for coverage optimization. To this purpose, we define a metric representing the ratio of uncovered area with respect to the total area of the monitored region. Second, We assess the target tracking approach by estimating the maximum linear distance that can be made by a hostile target without being detected. Finally, we evaluate the complexity of our coverage control and mobility techniques. We use the number of transmitted messages as a main criterion to estimate this complexity since data transmission consumes much more power than computational steps in WSNs.

The first experiment aims at evaluating the hole reduction strategy based on three-dimensional Voronoi tessellations with spherical coverage. We define the following metric to evaluate the performance of hole reduction.

A similar experiment is conducted for vector guided sensors, assuming that at every step of the iteration, the mobility is provided along with an orientation of the vector to achieve better coverage.

The approach performs better with spherical sensors for the first iterations. Indeed, the normalized uncovered proportion reaches 70%, with spherical sensors, after 10 iterations, while it stays under 10% for semi spherical sensors.

The approach performs the same for both types of sensors after 30 iterations.

This can be explained by the fact that the density of sensors is the same for both types and, therefore, it takes more mobility for semi spherical sensors the coverage holes.

The Average Linear Uncovered Length (ALUL), denoted by ℒ(_{m}

From the performance evaluation perspective, _{m}

In order to visually illustrate the performance of coverage reduction models, we use the local node density distribution that gives the number of sensors that cover every point of the monitored region. ^{2}-size monitored zone, the coverage degree considerably varies according proximity to the mobile target. In fact, after 5 mobility steps, the local sensor density is less than 1 in regions that are far from the target location (which is (70,30)) and reaches 2.7 in points that are close to this target.

More interestingly,

To confirm these results, we used the ALUL_{m}

We notice that the proposed mobility models, denoted by Advanced Voronoi-based Mobility Model (AVBMM) and Distributed Voronoi-based Mobility Model (DVBMM), clearly outperform the existing models. They also return a better performance than the Density-Preserving Mobility Model. This is because the latter model, despite its ability to guarantee a nearly uniform node density within the monitored area, does not take into account the presence of hostile targets in the zone of interest.

In this subsection, we evaluate the communication overhead resulting from the proposed retract-based coverage control approach. To this end, we only consider the complexity of the detection and localization steps in our algorithm and do not address the complexity of the repair step, since the repair step complexity is mainly dependent on the first deployment. However, one can easily deduce that if the deployment guarantees that holes size do not exceed a threshold, then the linear complexity can be verified.

We considered that the dimensions of the monitored zone are 10 m × 10 m × 3 m. We varied the number of nodes deployed within this zone and we measured the number of messages required to setup our coverage control protocol. We first supposed that all sensor nodes have a spherical coverage of range 0.5 m. ^{2} to 5 sensors per ^{2}. The major remark is that this number is nearly linear with respect to the number of sensors per area unit.

Moreover, we considered the case where sensors have semi-spherical coverage (with the same range).

The number of exchanged messages is independent of the density. It is close to 4 for the spherical sensors and 10 for semi spherical sensors. This fact may appear strange; however, one can notice that when a deployment is performed, the detection and location will only search for holes surrounded by the Vietoris space. The latter is reduced when the density is low.

The number of messages exchanged by the semi spherical sensors for detection and localization is 2.5 times higher than the number observed for spherical. Two reasons can be mentioned for this. First, the area covered by a semi spherical sensors is half the area covered by spherical sensors. Second, the guiding vectors is randomly oriented.

This paper developed a low complexity approach to detect and localize sensing holes in 3D spaces. It also constructed efficient algorithms to repair holes and track (multiple targets). Our approach has built on two concepts, the Vietoris complex and the Voronoi diagram, and demonstrated that the technique called retraction by deformation achieves low complexity algorithms for the detection of coverage holes in WSNs.

Our approach can be easily extended to more general sensors, for which the Vietoris complex and the Voronoi diagram can be defined. Such sensors can be called conical sensors or vector guided sensors and can represent camera sensors.

Evolution of the uncovered area proportion according to time.

Local node density distribution after 5 mobility iterations.

Illustration of 3 mobility iterations for a context where two targets are considered.

Evolution of the ALUL_{m}

Illustration of complexity.