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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/).

Distributed acoustic target tracking is an important application area of wireless sensor networks. In this paper we use algebraic geometry to formally model 2-dimensional acoustic target tracking and then prove its best degree of required sensing coverage. We present the necessary conditions for three sensing coverage to accurately compute the spatio-temporal information of a target object. Simulations show that 3-coverage accurately locates a target object only in 53% of cases. Using 4-coverage, we present two different methods that yield correct answers in almost all cases and have time and memory usage complexity of

Surveillance and monitoring of battlefields is one of the most important requirements in critical times. Sometimes it is also quite risky or impossible to have direct surveillance of a real field. That is why for such surveillance tasks remote acoustic target tracking is mostly used. Remote acoustic target tracking can be done by using wireless sensor networks (WSNs). A 2-dimensional acoustic target tracking of a target object by sensed spatio-temporal information obtained using sensor nodes is a 3-dimensional problem. It is often impossible to place wireless sensor nodes in predetermined positions, and this is why in most applications motes (i.e., sensor nodes) are spread randomly in a field from the air. For the sake of simplicity in this paper, we assume that sensor nodes are spread with a uniform distribution in a field and constitute a multi-hop wireless network. Sensor nodes use distributed processing for doing target tracking. In the study reported in this paper we further ignore the signal processing aspects of target tracking using sound sensing.

Quality of service (QoS) plays a critical role on the performability of WSNs that are used for remote target tracking. Attaining the best QoS from the application and end-user's viewpoint though requires the best possible adjustment of WSNs' parameters in support of QoS metrics. This task, namely the QoS management, is the responsibility of WSNs middleware [

In this paper we theoretically determine the minimum best sensing coverage of WSNs for 2-dimensional acoustic target tracking that guarantees high accuracy of results without occurrence of outliers. We also present the best possible processing and fusion method that can yield more accurate results with the least sensing coverage and less communication overhead. Since we study the acoustic target tracking from a theoretical point of view in this paper, we ignore the environmental factors like humidity and temperature. The contributions of our paper are applicable to other methods of target tracking like particle filtering [

The rest of paper is organized as follows. Section 2 presents related work in the area of acoustic target tracking. Section 3 presents the basics of acoustic target tracking and our basic geometric based proposed method, its simulation results and pitfalls. Section 4 discusses the acoustic target tracking from a theoretical perspective and formally represents the different bases of outliers in target tracking and proposes simple solutions for one type of outliers. Section 5 introduces two extended new methods and proves that they eliminate the source of second type of outliers. Section 6 evaluates our proposed methods by reporting the results of our simulations. Section 7 concludes the paper and puts forward some future works.

Basics of sensor nodes positioning for localization and location tracking are discussed in textbooks such as in [

Studies by Wang

Target tracking studies can be divided into two categories of single target tracking and multiple target tracking. The main aim in multiple target tracking is to separate multiple moving targets from each other [

Another method for target tracking is the Bayesian framework presented by Ekman

He

Using acoustic signal energy measurements of individual sensor nodes to estimate the locations of multiple acoustic sources is another approach for target tracking [

Using special types of Kalman filtering is another approach to overcome some of the problems of acoustic target tracking such as the problem of global time synchronization [

Beyond studying the various scenarios that are related to locating sensor nodes, some researchers have studied tracking objects with constant velocity with uncertain locations [

Clearly, there is a rich set of research on the subject of target tracking using WSNs. Several methods like Kalman filtering, statistical methods and numerical methods had been deployed for solving the target tracking problem, but analytical geometric methods had not been used for target tracking. We have opted to use analytical geometric methods in order to attain better tuning of differing QoS parameters that are required and set by end-users at the application-level of WSNs but need to be realized at the middleware-level of WSNs.

This section presents the basics of acoustic target tracking in Part 3.1. Part 3.2 proposes a new method based on algebraic geometry for solving simultaneous equations of acoustic target tracking that is originated from the basic model of acoustic target tracking, Part 3.3 presents a simulation model for validating our proposed method, and Part 3.4 shows the results of evaluation of the proposed method.

We assume that all motes (sensor nodes) have microphones for sensing sound waves. Localization and time synchronization of all motes are done with high accuracy. A target in an unknown location (_{o},y_{o}_{o}_{i},y_{i},t_{i}_{o}y_{o},t_{o}_{o}_{o}y_{o}

Based on the Δ_{o}y_{o},t_{o}

We rewrite simultaneous equations of target tracking and get

Let

The simultaneous equations of target tracking in

There are only two simultaneous equations in

We rewrite

We represent unknown variables

Factorizing

The inherent structure of the problem causes the delta of

For our simulations we used the VisualSense simulator [

We used three techniques in our simulations: (1) highly accurate time synchronization with 10^{-12} seconds precision, (2) target tracking by using the information of only three different sensing motes in each set of simultaneous equations, and (3) a simple formal majority voter. We used a variation of formal majority voter presented in [

The accuracy of the best time synchronization algorithms in real cases is in the order of 10^{-6} seconds [^{-12}) is not attainable in real cases; we can consider our simulations to be under ideal perfect time synchronization. ^{+3}. Simulation results showed that most of the times we had accurate spatio-temporal information of target tracking. But sometimes the network reported results that had big error values. Because these outliers in results were intolerable, we tried to detect the source of this most frequently happening outliers as is described in Section 4.

In Part 4.1 of this section we model and discuss the 2-dimensional acoustic target tracking problem as a geometric problem. In Part 4.2 we then present its dual that is easier to solve and enables us to formally present the sources of outliers in target tracking results. Part 4.3 mathematically describes how acoustic target tracking is done in our method. Part 4.4 introduces the sources of outliers in our method. Part 4.5 mathematically introduces the sources of some outliers in the 2-dimensional acoustic target tracking with three sensing coverage and proposes a simple time test method that can easily eliminate the occurrence of one type of outliers. In Part 4.6 we introduce the second sources of outliers. In Part 4.7 we summarize the simulations results and show the frequency of appearing each type of outliers in target tracking.

In 2-dimensional space and ideal environmental conditions we can assume that sound propagates in a circular form from a target's location with respect to time. If we consider the third dimension as time, the propagation of sound waves in 2-dimensional space with respect to time makes a vertical circular right cone.

_{o}y_{o},z_{o}

We need to solve simultaneous equations in

We can better represent the target tracking problem as a geometric problem if we rewrite

Let us assume time to be the third dimension that grows in upward direction. Now we can visualize sound propagation in 2-dimensional space with respect to time as a 3-dimensional right circular cone with a specific aperture angle. We have a cone equation and three known points on the surface of its up nappe and we want to determine the coordinate of its apex point that is (

Logically, the up nappe is of interest to target tracking, while degree two equations in simultaneous equations in

Point (_{i},y_{i}_{i}_{i},y_{i},t_{i}_{i}_{i}

We presented the target tracking problem in the form of a simple geometric problem in Part 4.1. In this part we present the dual of this geometric problem. Solving the dual of this problem is easier than solving the original problem and gives interesting results that will demonstrate the source of outliers in target tracking results.

The set of equations in simultaneous equations of _{i},y_{i},z_{i}

The intersection curve of two vertical right circular cones of sensing information of two different sensor nodes resides on a plane.

Sensing information of each mote

Aperture angles of sensing cones of all sensor nodes are equal and can be calculated using

All variables ^{3}) is:

The intersection of sensing cones of each pair of sensing nodes resides on a plane we call it _{ij}

Cones that are related to sensing nodes are assumed to be unbounded. Three cones of sensing nodes can have three different paired combinations and will have three intersection planes.

All planes passing from a common straight line form a

Let us assume that equations of two planes are as follows:

Three planes make a pencil if the third plane's equation satisfies the condition of

The intersection planes of each three sensing cones that are constructed from the sensed information of motes make a pencil.

Assume that our three sensing cones are _{ij}, π_{ik}, π_{jk}_{ij}_{ik}

To obtain the equation of the third intersection plane of a pencil, we substitute the equations of planes _{ij}_{ik}

If we set _{1} = −1 and _{2} = +1 in _{jk}

So the intersection planes of three sensing cones that are constructed from sensed information of motes make a pencil.

We call common line of a sheaf of planes as axis of pencil. We denote a pencil that is constructed from intersection planes of three sensing cones _{ijk}_{ijk}

The equation of two planes of a pencil is sufficient for computing axis of pencil. Furthermore, we saw that the equation of a third intersection plane _{jk}_{ij}_{ik}_{jk}

Two sensing cones in a pair intersect with each other on a degree two curve in 3-dimensional space.

We proved in Lemma 1 that the intersection of each pair of sensing cones is a plane. Except special cases, when a plane passes from the apex point of a cone or when a plane lies on the surface of a cone, the intersection of a plane and a cone can produce four different degree two curves [

We call a degree two curve that is generated from the intersection of two sensing cones as the _{ij}

Solving simultaneous equations of three different sensing nodes can generate incorrect answer.

As proved in Lemma 1, the intersection curve of two circular right vertical cones in a pair with equal aperture angles reside on a plane. In Lemma 2 we proved that the intersection planes of three sensing cones that are constructed from sensed information of motes make a pencil that intersect on a common line. This is obvious that a line can meet the target tracking cone of

_{2}_{1}

We can eliminate a mathematically correct but unfeasible answer most of the times. If the axis line of pencil crosses the up and the down nappes, then one of the answers belongs to the past and the other one belongs to the future. Sensors cannot hear the sound of a target that is going to be generated in the future. So the points that reside on the intersection point of up nappes in

Let _{ij}_{ij}_{ij}_{ij}_{ij}_{ik}

Let us now represent the outer product of these two normal vectors as vector _{1} in

_{1}:

_{1} that is parallel with the axis line of pencil. Vector _{1} can be represented in a simpler form as:

The axis of a vertical right cone of sound propagation of a target object is a normalized vector as follows:

The angle between two vectors in 3-dimensional space can be computed using their internal product [_{1} (representing the axis of pencil) and vector _{2} (representing the axis of cone) is shown by

The tangent of the angle between the axis of pencil and the axis of sound propagation cone is given by

Based on

When the axis of a pencil intersects with the down nappe in two different points we will have two answers that are mathematically correct, but the target object was located in a specific location and generated sound waves in a specific point of time and only one of these two answers is correct. In this case we cannot use a simple time test to eliminate the incorrect answer. The random selection of one of these two answers by mistake is the source of outliers in target tracking results especially when we have accurate localization and time synchronization. _{1}, _{2} and _{3} points and the target tracking results are shown with _{1} and _{2} points. The real position of target object is shown with point

We extensively studied the results of acoustic target tracking with 3-coverage in more than 400 simulations and summarized the results as they are shown in

28.88% of the times we obtained two correct answers, both of which resided on the down nappe, and we could not detect the feasible answer; we randomly chose one of them resulting in the selection of incorrect answers 14.44% of times. In 18.01% of the times that two points resided on the down nappe, one or both answers were inaccurate. This happened when sensing nodes resided very close to a line and error propagation was high. In the remaining parts of this paper we try to eliminate the errors which their source is that the axis line of sensing pencil crossing the down cones in two different points, wherein the correct one is not recognizable.

We proved that solving simultaneous equations of target tracking using information of three sensor nodes always can produce incorrect answers. One type of these outliers that is not under our control is directly related to the specific position of sensor nodes. In this case, we must not consider the results of target tracking because of the high error propagation in such special cases. In this section we present a solution that completely eliminates the generation of outliers in the target tracking results that are related to the weaknesses of our assumptions and methods. We prove that our proposed methods completely solve the problem. In Part 5.1 we propose our first method, called analytic four coverage tracking (AFCT), and prove its correctness, and in Part 5.2 we propose our second method, called redundant answers fusion (RAF), to eliminate outliers in the results.

In this part we introduce the AFCT method, which is the first extension to our basic proposed method that was introduced in Part 3.2. The AFCT method selects the correct spatio-temporal information of a target object when simultaneous equations yield two answers, none of which can be easily eliminated as incorrect with a simple time test. By using four sensing coverage we can solve this problem. Let us assume that a forth mote, in addition to three motes that had sensed the sound of a target, senses the same sound wave of a target object. The sensing information of the fourth mote creates a fourth cone in addition to the previous three cones of

The intersection planes of four sensing cones make a pencil.

In target tracking using the sensing information of three sensor nodes, the equation of two intersection planes is sufficient for calculating the axis of pencil for target tracking. _{12} and _{13}.

If we can find values for _{1} and _{12} that satisfy

The first three equations of simultaneous equations in

The 3-dimensional sensing information of three previous sensor nodes are shown in _{1}, _{2}, and _{3} points. Only one plane passes from three points that do not reside on a straight line in the 3-dimensional space. The probability that a random chosen point in the 3-dimensional space resides on a specific plane is nearly zero. Therefore, the probability that the sensing information of a fourth sensor in the form of cone makes intersection planes with three previous cones belonging to the pencil that three cones of three previous sensor nodes had made, is approximately zero.

We proved that the probability that the sensing information of the fourth sensor node belongs to the same pencil of three previous sensor nodes is zero. The information of four different sensing motes that do not reside on the same plane, make four different pencils. Because the coordinates of all four points originated from the same base, triple combination of four sensing motes' information satisfies the condition of a pencil. Four unbounded cones can have six different pairs of combinations with each other that make six intersection planes. Triple combination of sensing information of four sensing nodes makes four different pencils as it is shown in

All planes passing through a common point are known as

An interesting property of these four pencils is that they all pass through a unique common point most of the times. This point is the correct spatio-temporal information of the target object. As it is shown in _{123} axis line, which is the axis line of sheaf of planes _{12}, _{13} and _{23} made by the information of three sensing motes 1, 2 and 3. Using the sensing information of a fourth sensing mote helps to distinguish the correct answer. This figure shows that by using the information of a fourth sensing mote, the axis line of four pencils of planes passes through a common point, which is the correct spatio-temporal information of the target object.

The use of the sensing information of four sensor nodes in 2-dimensional acoustic target tracking results in a unique correct result.

We proved in Lemma 2 that the intersection planes of three sensing cones make a pencil. Now we extend that work and prove that the intersection planes of four sensing cones make a bundle of planes. We assume that we have four sensing nodes and their pair-wise intersection planes are _{ij}

Now we must prove that four different pencils intersect with each other on a common point. Let us assume a pencil _{ijk}_{ijl}_{ijl}_{ij}, π_{il}_{jl}_{ijk}_{ij}, π_{ik}_{jk}_{ijk}_{ijl}_{ij}_{il}_{jl}_{ijl}_{ik}_{jk}_{ijk}

The equations of intersection planes _{ij}, π_{jl}_{ik}

We can write the equations of planes in

If we assume

_{ij}_{ik}_{ij}, π_{ik}_{jk}_{jk}

A fusing mote gathers the reported information of all its neighboring motes as well as its own sensed information if it has the sensing capability. It then uses the information of the first three sensing motes to construct the simultaneous equations of

We can use three sensing coverage for accurate target tracking using appropriate local fusion methods. Fusion can be performed locally by the motes that reside on the routing path to the sink node, because local fusion greatly reduces the number of messages to be transmitted and increases the lifetime of the network [

Our proposed method for fusion by intermediate motes in the routing path to the sink node woks as follows. Each mote collects the 3-dimensional spatio-temporal results of reported target tracking information and categorizes the results based on the Euclidean distance similarity in the form of clusters. Then it selects an answer from a group that has more members. This is a special type of formal majority voter that is well suited to fusion by intermediate motes [

In this section we compare the simulation results of the AFCT and RAF methods with the basic method proposed in Part 3.2. We also discuss about the cause of peak errors in simulation results of these two extend methods and time and memory usage complexity of them.

^{−10}, implying that in 200 times of target tracking in 400 seconds of simulation run, no target tracking error was encountered. This method uses the sensing information of four sensor nodes in the required conditions. For this purpose, sometimes the sensing information of each sensor node was broadcasted in two hops distance to neighboring sensor nodes.

As ^{−10} and ignorable but it shows existence of small outliers. In Section 3.2 we explained our proposed method for solving simultaneous equations. The coefficient matrix of

For solving these simultaneous equations we must compute the inverse of this coefficient matrix. If the determinant of this matrix is zero, then this matrix is singular and is not invertible and the system of simultaneous equations of target tracking does not have any answer. This condition holds when the first row of matrix _{1}(_{1}, _{1}), _{2}(_{2}, _{2}) and _{3}(_{3}, _{3}) are located in a straight line. In computation of the inverse matrix ^{−1} we need to use the inverse of the determinant of matrix

^{−11} and no incorrect target tracking was encountered in 400 seconds of simulation run. In this method, each set of simultaneous equations was composed of sensing information of three sensor nodes. But for accurate target tracking we needed four sensing coverage. Four sensing coverage caused at least two different sets of simultaneous equations to be generated and using a proper fusion method, accurate spatio-temporal information of a target object to be computed.

AFCT and RAF methods compute the spatio-temporal information of target object without iteration. Time and memory usage complexity of both methods are

Three sensing coverage has been assumed in the literature [52,53] as a sufficient condition for 2-dimensional acoustic target tracking using WSNs. Using geometric representation for modeling, we theoretically proved that three sensing coverage generates outliers in target tracking results and that these outliers occur irrespective of the target tracking method that is used; they occur in target tracking using Kalman filtering, particle filtering and many other methods that assume three sensing coverage is sufficient for 2-dimensional acoustic target tracking.

To reduce the computational overhead of finding the spatio-temporal information of a target object by way of solving quadratic equations, we proposed an alternative method based on geometric algebra that uses linear equations instead of quadratic equations. Our proposed method solves target tracking equations with comparably lower computational overhead and it is applicable to WSNs whose sensor nodes have low processing power. Our simulation results for 2-dimensional target tracking with three sensing coverage in ideal cases, where time synchronization and localization errors were ignored, showed that the target tracking results contain outliers most of the times. Simulation results also showed that in more than half of the cases, three sensing coverage generated outliers in acoustic target tracking. We theoretically proved that four sensing coverage, rather than three sensing coverage, is the necessary condition for accurate 2-dimensional target tracking.

We proposed two other extended methods based on the basic proposed method to improve it, in order to accurately compute the spatio-temporal information of a target object in 2-dimensional space. Both of these methods were based on having four sensing coverage for accurate target tracking. We proved that the AFCT method, which was based on algebraic geometry, accurately computes the spatio-temporal information of a target object with 100% confidence under defined conditions. We also showed through simulation that target tracking errors can be eliminated. The RAF method used a customized fusion method and assumed that each set of simultaneous equations uses the sensing information of only three sensing nodes. But because of four sensing coverage we had at least two different sets of simultaneous equations that allowed us to exploit the capabilities of the second proposed fusion method. Simulation results showed that target tracking was carried out perfectly without any error.

The main contribution of our paper was in formally correcting the belief that

In real applications of acoustic target tracking, accuracy and precision of target tracking results are two important QoS metrics from view point of the application layer and end user. These metrics are closely related to the time synchronization and localization precision of sensor network. In this paper we discussed about one source of errors that greatly decreases the accuracy of target tracking results. It is difficult to recognize and prove the existence of this source of errors by considering localization and time synchronization errors. Therefore we carried out our studies reported in this paper by assuming very low time synchronization and localization errors.

In real applications, an end user may require different levels of QoS metrics based on the state of the system. If the end user wants to have control of the accuracy and precision of target tracking results, it is necessary to know the factors that influence the accuracy and precision of target tracking results and also know the relationship between influencing factors and the accuracy and precision of target tracking results. Knowing these, it will be possible for a middleware to guarantee the required level of accuracy and precision of target tracking errors by adjusting the time synchronization and localization precision parameters. To achieve this, the error propagation of our proposed method is currently under further study.

We would like to sincerely thank Professor Edward Lee, the founder of Ptolemy, for his support and guidance in helping us through simulation of our ideas reported in this paper. The authors would like to thank Iran Telecommunication Research Center (ITRC) for their partial financial support under contract number 8836 for the research whose results are partly reported in this paper.

Basic schema of target tracking in 2-dimensional space [

The square errors of target tracking using majority voter.

A vertical right circular double cone.

Sound propagation with respect to time in 2-dimensional space.

Intersection of three cones for tracking a target object.

Sensing cones, constructed pencil and axis of pencil.

A sheaf of planes and their intersection points with sensing cones at two points.

(a) Two planes of a pencil and its axis line. (b) Axis of a pencil and its degree with the axis of cone.

The intersection of axis line of a pencil with down nappe gives two answers where their sound propagation cones pass through three points of sensing information.

Statistics of different conditions in 2-dimensional target tracking using information of three sensor nodes.

Four points residing on a plane.

Triple combination of four different sensing motes' information yielding four different pencils that pass through a unique common point.

Intersection curves of four sensing cones intersected on a common point.

(a) Target tracking in a four sensing coverage area. (b) Connectivity graph of sensor nodes. (c) Connectivity graph of sensor nodes in another part of simulation field.

The square error of target tracking using AFCT method.

Target tracking error using RAF method.