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Existing 3-dimensional acoustic target tracking methods that use wired/wireless networked sensor nodes to track targets based on four sensing coverage do not always compute the feasible spatio-temporal information of target objects. To investigate this discrepancy in a formal setting, we propose a geometric model of the target tracking problem alongside its equivalent geometric dual model that is easier to solve. We then study and prove some properties of dual model by exploiting its relationship with algebra. Based on these properties, we propose a four coverage axis line method based on four sensing coverage and prove that four sensing coverage always yields two dual correct answers; usually one of them is infeasible. By showing that the feasible answer can be only sometimes identified by using a simple time test method such as the one proposed by ourselves, we prove that four sensing coverage fails to always yield the feasible spatio-temporal information of a target object. We further prove that five sensing coverage always gives the feasible position of a target object under certain conditions that are discussed in this paper. We propose three extensions to four coverage axis line method, namely, five coverage extent point method, five coverage extended axis lines method, and five coverage redundant axis lines method. Computation and time complexities of all four proposed methods are equal in the worst cases as well as on average being equal to

3-dimensional acoustic target tracking is an extension of acoustic target tracking in 2-dimensional space in view of the fact that in real life applications we mostly deal with 3-dimensional target tracking. In target tracking temporal information of a target object in addition to its 3-dimensional spatial information must be computed. Target tracking in 3-dimensional space is a 4-dimensional problem and we consider time as the fourth dimension. The sensing information of each sensor node about a target object in its sensing coverage forms an equation. Simultaneous equations of target tracking are quadratic and non-linear therefore solving those using numerical methods is difficult, complex, and requires high computational resources that are generally constrained in the sensor nodes of wireless sensor networks (WSNs). In this paper we use geometry and algebra and exploit their relationship to model and prove some basic facts about 3-dimensional acoustic target tracking and propose new methods for computing the spatio-temporal information of a target object. Our proposed methods use linear simultaneous equations instead of quadratic equations and need less computational resources, making them more amenable to target tracking applications with timing constraints. We prove that four sensing coverage of a target object cannot always yield the correct spatio-temporal information of a target object and that this proven fact is quite independent of the kind of used target tracking method.

In this paper we show that simultaneous equations of four sensing coverage for a target object yield two different answers. Most of the times we can eliminate the infeasible answer by performing a simple proposed time test. We introduce a four coverage axis line method which works based on the four sensing coverage. We prove that the sensing information of five sensor nodes about a target object accurately determine the correct spatio-temporal information of the target object. To overcome this weakness, we propose to increase the sensing coverage to five sensing nodes and present three new methods as extensions to the four coverage axis line method. Five coverage extent point method uses the sensing information of five sensor nodes and by solving a set of four linear equations accurately determines the spatio-temporal information of a target object. Five coverage extended axis line method is based on the four coverage axis line method and if the sensing information of four sensor nodes about a target object does not satisfy conditions to remove the infeasible answer out of two answers of the set of its simultaneous linear equations, the sensing information of a fifth sensing node on the same target object is used to determine the correct spatio-temporal information of the target object. Our last proposed method called five coverage redundant axis line method is based on the five sensing coverage but uses the set of simultaneous equations of four sensing coverage in at least two different sets. This method deploys a customized version of formal majority voter among sensory nodes to compute the spatio-temporal information of a target object. The contributions of our paper are applicable to other methods of target tracking like Bayesian filtering, Kalman filtering [

The rest of this paper is organized as follows. Section 2 presents related work in the area of 3-dimensional acoustic target tracking. Section 3 presents the basics of 3-dimensional acoustic target tracking and geometric representation of problem and its equivalent dual geometric problem that can be solved easier than the main problem. Section 4 discusses theoretically the special properties of dual representations of 3-dimensional acoustic target localization and introduces the theoretical basis for a method based on the four sensing coverage. Section 5 introduces a four coverage axis line method based on the theoretic background of Section 4 and analyses the simulation results of its application to a real life problem. Section 6 theoretically proves that five sensing coverage always guarantees to yield the correct answer. It also presents three extended methods based on the four coverage axis line method in Section 5, alongside the necessary theorems and proofs and simulation results in support of these methods. Section 7 concludes the paper and suggests some future work.

A considerable part of the literature on WSNs discusses the issues of sensor node localization and location tracking [

Brooks

Ekman

Using WSNs for real-time target tracking with guaranteed deadlines had been studied by He

Barsanti

Combining geometry and algebra to represent the spatio-temporal information of target objects is a new idea. We had studied the geometric modeling of 2-dimensional acoustic target tracking using wireless sensor nodes and proved that three sensing coverage can only sometimes determine the correct spatio-temporal information of a target object in 2-dimensional acoustic target tracking. We proved that four sensing coverage is the best sensing coverage for 2-dimensional target tracking and proposed new geometric methods with low computational overhead [

In this paper we have ignored the signal processing aspects of acoustic target tracking and assumed that the sound waves of a target object is detected and differentiated from other environmental sounds by an appropriate signal processing method. We have also ignored the environmental phenomena that may affect the broadcasting of sound waves in 3-dimensional space and the reflections of sound waves when they meet the ground surface. Furthermore, we have assumed that every sensor node is equipped with a microphone for sensing sound waves, and localization and time synchronization of all sensor nodes are done with high accuracy.

When a target object in an unknown location (_{0}, _{0}, _{0}) generates sound waves at time _{0}, its sound waves broadcast in a spherical form in 3-dimensional space and reaches to each sensor node in the field after some time delay that directly depends on the Euclidian distance of sensor nodes from the target object. _{i}_{i}_{i}_{i}_{i}_{0}, _{0}, _{0}, _{0}) that represent the spatio-temporal information of target object in a distributed way. This information implies that the target object has generated a sound at time _{0} in position (_{0}, _{0}, _{0}) that has been detected by motes after some delay. To compute these four unknown variables, we need the sensing information of at least four sensor nodes to create simultaneous equations that contain at least four equations.

Let us assume that sound waves propagate with constant speed of v = 344.0 m/s. Based on simple formulas of physics for displacement with constant velocity Δ

All

The general equation of a right spherical double hypercone (spherical cone) in 4-dimensional space whose apex point has (_{0}, _{0}, _{0}, _{0}) coordinates is as follows [

Capital letters

If the _{0}, _{0}, _{0}, _{0}) point. Shape grows in spherical form centered at the apex point with the ratio of

When time increases (up nappe) or decreases (down nappe) relative to _{0}, the 3-dimensional shape of

By looking at simultaneous equations in _{i}_{i}_{i}_{i}_{i}_{i} on the surface of the up nappe.

We represented the 3-dimensional acoustic target localization problem as a geometric problem in previous section and now we want to present an equivalent dual representation of this problem from another viewpoint. Answers of these two dual interpretations are the same because both of them use the same set of

Each equation in simultaneous equations of _{i}_{i}_{i}_{i}_{i}

_{i}^{4}) is a 3-dimensional subspace of ℝ^{4} whose equation is as follows [

^{4}.

_{ij}

_{i}_{j}_{ij}_{ij}

To clarify the results of lemmas and theorems, we use an example 3-dimensional target localization problem. We assume that a target object at location _{1} (800, 400, 100, 0.75029), _{2} (122, 400, 150, 1.7479), _{3} (400, 800, 200, 1.1161), and _{4} (500, 200, 125, 1.3173). _{1} and _{2} alongside their intersection hyperplanes π_{12} in three different points in time.

Based on Lemma 1, we can now compute the intersection surface of two sensing hypercones more easily by computing the intersection of each sensing hypercone with their common intersection hyperplane. To introduce intersection surface of two sensing hypercones, we need to define hyperconic sections.

The ^{3}). Circle, ellipse, parabola, and hyperbola are four different quadratic curves that can be produced from the intersection of a cone and a plane [

The intersection of each pair of sensing hypercones is a hyperconic section that is mostly in the form of a hyperboloid of two sheets or an elliptic paraboloid. All sensing hypercones that we deal with in target localization in this paper are right circular hypercones with equal aperture angles whose axis are parallel. That is why the intersection hyperplanes of sensing hypercones do not have big angles with the axis line of hypercones, implying that their intersection surface will not be in the form of some of the hyperconic sections like sphere and ellipsoid. The intersection surface of two hypercones in

We use the relation of linear algebra with geometry to study the properties of intersection hyperplanes in ℝ^{4}. We can represent the equations of intersection hyperplanes of each pair of sensing hypercones in the form of

Three sensing hypercones can have three different paired combinations and thus have three intersection hyperplanes. In this part we study the geometric properties of three sensing information.

^{3} that pass from a common straight line form a

Now we extend the definition of pencil to ℝ^{4}

^{4} that pass through a common plane form a _{i}_{j}_{k}_{ijk}

If a third hyperplane’s equation satisfies the following condition:
_{1},_{2} ∈ ℝ, … then three hyperplanes make a pencil.

_{ij}_{ik}_{jk}_{ij}_{ik}

If we substitute the equations of hyperplanes _{ij}_{ik}

If we put _{1} = −1 and _{2} = +1 in

_{jk}_{jk}_{ij}_{ik}

_{ij}_{ik}_{jk}_{jk}_{ij}_{ik}

Our points of interest that give the spatio-temporal information of a target object are located on the intersection points of 4-dimensional degree two intersection surfaces of each pair of sensing hypercones. Computing the intersection of three sensing hypercones of target localization is a difficult work and needs heavy computation. Theorem 2 states that the intersection of three sensing hypercones resides on the axis plane of the pencil that is constructed from the intersection hyperplanes of sensing information. Computing the axis plane is easy and finding the intersection of sensing hypercones with this plane is easier than computing the intersection of three sensing hypercones.

In this part we demonstrate the simulations results of Theorem 2 about properties of sensing information of three sensor nodes 1, 2 and 3 that were introduced in Part 4.2.

In 4-dimensional space, the intersection surface of each pair of sensing hypercones reside on a hyperplane and three pair of three sensing hypercones meet each other on a curve that resides on the intersection plane of the pencil that these sensing hypercones make.

_{ijk}

If we consider all figures in

^{3} can meet each other in a common point and make a

Now we extend the definition of bundle of planes to ℝ^{4}.

^{4} that pass through a common line form a _{i}_{j}_{k}_{l}_{ijkl}

Each pair of triple combinations of four sensing hypercones will have two common hypercones. Let us assume two pencils _{ijk}_{ijl}_{ijl}_{ij}_{il}_{jl}_{ijk}_{ij}_{ik}_{jk}_{ijk}_{ijl}_{ij}_{il}_{jl}_{ijl}_{ik}_{jk}_{ijk}_{il}_{jl}_{ik}

We now prove that the equations of these three hyperplanes are linearly dependent. We write the equations of hyperplanes in

By setting

We can write

_{ij}_{ik}

We have proved in Part 4.4 that hyperplanes _{ij}_{ik}_{jk}_{jk}_{ij}_{jl}_{ik}_{jk}

We can present our proof in other words. Let us assume four sensing hypercones of sensing nodes _{ij}_{ik}_{il}_{jk}_{jl}_{kl}_{ij}_{ik}_{il}

Therefore, from the equations of the six intersection hyperplanes of four sensing hypercones, only three of them are linearly independent. Dimension of solution of these simultaneous equations of four intersection hypercones is one and shows that they will intersect in a common line.

Based on Theorem 3, we can find the axis line of a bundle of hyperplanes that are formed by the intersection hyperplanes of four sensing nodes; the axis line intersects with all sensing hypercones only in two common points. We can compute the intersection of the axis line with one of the sensing hypercones for computing the spatio-temporal information of a target object. This computation has lower overhead in comparison to computing the intersections of four sensing hypercones.

In this part we use the sensing information of the example introduced in Part 4.2 for demonstrating the properties of four sensing hypercones proven in Theorem 3. In Theorem 2 and Part 4.5 we proved and showed that each triple combination of sensing nodes’ information makes a pencil on whose axis plane resides the spatio-temporal information of a target object. Theorem 3 proved that four axis planes of intersection hyperplanes’ pencil meet each other in a common line and all intersection hyperplanes make a bundle of hyperplanes.

_{1} and _{2}. Both of these points lie on the axis line of the intersection hyperplanes. Our target localization equations are degree two; therefore we have

FCAL method produces two different answers both of which are mathematically correct but only one of them is the feasible 4-dimensional spatio-temporal information of the target object and the other answer is infeasible. We declare a simple method called the

_{1} is before the times of the sound sensing by all four sensor nodes and the time of the other answer, _{2} is after the time of the sound sensing of at least one of the four sensing nodes, answer _{1} is related to past time and answer _{2} is related to future time and is the infeasible answer. An example of case I was shown in

_{1} and _{2} is before the reported times of sound sensing by all sensor nodes, both answers are related to the past and time test cannot help the FCAL method to detect the feasible spatio-temporal information of a target object.

_{1} and _{2} are the same and both of them are the feasible spatio-temporal information of a target object. The time test method is successful in cases I and III but it cannot detect the correct answer in case II.

We proved that

Based on Theorem 3 we propose a simple combined algebraic and geometry based method, we called it Four Coverage Axis Line (FCAL) method in Part 4.7. Using properties of Theorem 3 we do not use heavy computations for computing the intersection of four sensing hypercones; instead in the first step we compute the axis line of four sensing hypercones and then we compute its intersection with one of the sensing hypercones. FCAL converts degree two systems of four simultaneous equations to a simple degree one system of three simultaneous equations and greatly decreases the computation overhead.

The simultaneous equations in

If we simplify the above equations we get the equations of intersection hyperplanes as follows:

_{1} as follows:

We represent unknown variables

Factorizing

Matrix _{1}, _{2}, _{3}, and _{4} are located such that no three of them are located on a line, then the matrix

For simulative study of the FCAL method we developed and tested the 3-dimensional acoustic target tracking problem in a WSN with randomly distributed wireless sensor nodes. We used the ^{−9} seconds.

^{−9} (seconds) precision, (2) target localization by using the information of only four different sensing nodes in each set of simultaneous equations using the FCAL method, and (3) a simple formal majority voter algorithm. The accuracy of the best time synchronization algorithms in real cases were in the order of 10^{−6} seconds [^{4} m^{2}. A variation of formal majority voter presented in [

As

_{ij}_{ik}_{il}_{im}_{jk}_{jl}_{jm}_{kl}_{km}_{lm}_{ij}_{ik}_{il}_{im}

Therefore, from equations of ten intersection hyperplanes of five sensing hypercones, only four of them are linearly independent. The equations of the six remaining hyperplanes are linearly dependent on the equations of these four hyperplanes. The system of simulatenous equations of intersection hyperplanes has four unknown variables and four independent equations. Therefore the dimension of answer will be 4 – 4 = 0 implying that the intersection hyperplanes intersect on a common point. This point is our unique and feasible spatio-temporal information of the target object.

_{i}_{j}_{k}_{l}_{m}_{ijklm}

Let us now assume that we add a fifth sensing node P_{5} (750, 800, 175, 0.7802) to the example given in Part 3.2.

We extend the FCAL method and propose the Five Coverage Extent Point (FCEP) method. Based on Theorem 4 we make a system of four simultaneous equations of independent intersection hyperplanes as follows:

We can represent this system of linear equations in the following matrix form:

This method has higher computational cost as it requires the computation of the inverse of a 4 × 4 instead of 3 × 3 matrix of the FCAL method. The cost of computing the inverse of a matrix with dimension ^{2}) [

Based on Theorem 4 we propose a second extension to the FCAL method and call this method Five Coverage Extended Axis Line (FCEAL) method. In FCEAL we use the FCAL method for computing the spatio-temporal information of a target object. If case I or case III occurs we select the feasible answer and if case II occurs, we use the sensing information of a fifth sensor node. Only one of the two computed answers satisfies the equation of sensing hypercone of the fifth sensor node.

FCEP and FCEAL methods produce the same results but only differ in their computation method and computation cost. As ^{−11} m^{2} that is by far smaller than the 10^{4} m^{2} order of the square error of target tracking with four sensing coverage of the FCAL method shown in

Based on Theorem 3 and Theorem 4 we propose a new method relaying on the five degree sensing coverage called Five Coverage Redundant Axis Lines (FCRAL) method as yet another extension to the FCAL method. In FCRAL, each sensor node gathers the sensing information of four neighbor sensing nodes and uses the FCAL method for computation. Two different conditions occur; in case I or case III we can compute accurately the spatio-temporal of target object with 100% confidence degree. However, if case II occurs then the computing node sends both computed answers to sink node with 50% confidence degree. With five sensing coverage, we need at least two groups of sensing information to be constructed, wherein each group has the sensing information of at least four sensor nodes. Five sensing nodes of a target's sound may be located such that some of them cannot communicate with each other in single hop. Therefore, every sensing node sometimes needs to broadcast its sensing information to its neighbor nodes in two hops for making at least two sets of localization simultaneous equations. By this means at least two sets of sensing information of four sensor nodes can be constructed; this must be guaranteed by the management procedures that are enforced on sensor nodes.

Sensor nodes that are placed in the routing path to the sink node and we call them the fusing nodes, can use a modified formal majority voter algorithm [^{−11} m^{2}, which is comparatively smaller than the square error of the FCAL method in ^{4} m^{2} using four sensing coverage.

As

^{−11} m^{2}. As mentioned in Part 5.2, if at least three sensing nodes are located on a straight line then the coefficient matrix

Kalman and Particle filters are special types of Bayesian filters that use a measurement model beyond the system model for tracking a target object. Commonly, the measured signals of sensor nodes are used in measurement models [

Given that wireless sensor networks based solutions to 3-dimensional acoustic target tracking with four sensing coverage do not always compute the feasible spatio-temporal information of target objects, we investigated this weakness in a formal setting in this paper. To do so we first combined geometry and algebra for modeling the basics of 3-dimensional acoustic target tracking. These basics are valid for all variations of 3-dimensional acoustic target tracking methods like Bayesian filters. We converted the classic 3-dimensional acoustic target tracking problem to a form combining algebraic and geometric reasoning. This allowed us to study and prove some of the inherent and interesting properties of the problem. Based on these proven properties, we proved that four sensing coverage only under certain conditions guarantees to compute two dual answers, one of which is a feasible answer and another is infeasible answer. We then proved that four sensing coverage does not always guarantee to clarify feasible answer of the 3-dimensional acoustic target localization problem. This was achieved by using a set of lemmas and theorems we proved before applying them to our proposed four coverage axis line (called, FCAL) method for 3-dimensional acoustic target tracking.

We proved that five sensing coverage guarantees to always yield the spatio-temporal information of target objects in 3-dimensional acoustic target tracking. We extended our FCAL method to five sensing coverage in three ways and proposed three methods called five coverage extent point (FCEP) method, five coverage extended axis line (FCEAL) method, and five coverage redundant axis lines (FCRAL) method. We showed that the computational and memory usage overheads of all four methods on average and in the worst cases are equal to

We did not consider the time synchronization, sensor node localization error, and the sensing and environmental noises in our studies reported in this paper. In real applications though, these factors greatly influence the accuracy and precision of the results. In our previous works the basics of two dimensional target tracking were presented [

The aim of this paper was to prove some facts about the basics of 3-dimensional acoustic target tracking regardless of the above parameters. These facts are valid for all types of 3-dimensional acoustic target tracking methods. Bayesian methods like Kalman filtering and Particle filtering can also use these facts and our proposed methods in their measurement models. Application of the results of this paper to the Bayesian filters in real world applications is currently under investigation by the authors.

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 localization in 3-dimensional space.

A general right spherical double hypercone in 4-dimensional space mapped to 3-dimensional space at five different points in time.

(a) The intersection hyperplanes of two double hypercones at three different points in time. (b) The intersection of two 4-dimensional double hypercones of target localization in 3-dimensional space.

Possible quadric surfaces built from the intersection of a hypercone with a hyperplane [

The intersection surfaces, the pencil and the axis plane of three sensing hypercones (a). At time t = +1.5, (b). At time t = +2.0, and (c). At time t = + 2.5, (d). The axis plane of the pencil passes through point that representing real target object’s spatio-temporal information.

The axis plane of sensor nodes 1, 2 and 3, alongside their intersection curve, in the range of [−4,4] seconds.

(a). Triple combination of four different sensing nodes’ information making a bundle of hyperplanes that pass through an axis line. (b). Intersection curves of four sensing nodes passing from two different points on the axis line of a bundle of intersection hyperplanes in the time range of [−4,4].

Pitfalls of the FCAL method in computing the accurate spatio-temporal information of a target object when both answers are related to past time.

Square error of 3-dimensional acoustic target tracking of FCAL method.

Ten axis planes and five axis lines of their bundle of hyperplanes and their extent point of five sensing nodes.

Square error of 3-dimensional acoustic target tracking using FCEP method.

Square error of 3-dimensional target tracking using FCRAL method.