Wireless sensor networks (WSNs) may consist of tiny, energy efficient sensor nodes communicating via wireless channels, performing distributed sensing and collaborative tasks for a variety of monitoring applications. One of the critical problems in sensor applications is detecting boundary sensors in a complex sensor network environment where sensed data is often required to be associated with spatial coordinates. In [2] a COBOM protocol that monitors the boundary of a continuous object was proposed. Sensor nodes are assigned with a Boundary sensor Node (BN) array to store BN information. The boundary monitoring is based on the changes to the observations in the BN array. As a updated version, [1] presented the DEMOCO protocol that enhanced COBOM by considering sensor nodes on one side of the boundary line called the “IN” range, and ignoring those on the other side of the boundary line called the “OUT” range which theoretically reduces approximately by half of the number of the selected BNs. Others like [3–6] also involve two-dimensional (2D) sensor localizations. To address the issues of adaptive sensor coverage and tracking for dynamic network topology, the authors of [7] utilized a Gaussian mixture model to characterize the mixture distribution of object locations and proposed a novel methodology to adaptively update sensor node placement according to the ML estimates of mass object locations with a distributed implementation of an EM algorithm to reduce communication costs. Moreover, [8] discussed a flocking-base mobility model for Distributed Kalman Filtering (DKF) in mobile sensor networks and [9,10] demonstrated efficient boundary detection algorithms with only the connectivity information.

In fact, the boundary detection problem has been mostly considered for 2D sensor networks and the case of 3D sensor networks has gone practically unnoticed. Despite the fact that difference between the normal 2D and the more realistic 3D scenario is only one extra dimension, network topology could be much more complex and the location scheme has to be more robust towards network irregularities. Taking a step further to expand from 2D to 3D sensor applications, several neighborhood-measurement [11] based 3D range-free boundary detection models [12–17] have been proposed. However, their tight dependence on sensor node densities and availability of sufficient neighbors are too optimistic for real 3D sensor applications due to their non-uniform sensor node densities and topology randomization. On the other hand, a range-based model such as in [18] does not make any assumption about sensor node densities and network topology. Instead it introduced a strong entity called mobile location assistants (LAs) that enables each location-unaware sensor node to easily estimate its own position using the measurable AOAs [11] and RSS [19]. Similar approaches like [5,6,20] assume that a small fraction of sensor nodes called anchors or beacons have a priori knowledge of their location and [21] proposed a range-based positioning method using beacon signals, that doesn’t require time synchronization since the beacon sensor nodes estimate the range based on frequency differences instead of time differences. To conclude, all the aforementioned approaches either introduced strong entities or made irrational assumptions. Furthermore, [22] presented a new high precision WSN positioning method with reasonable implementation cost for a 3D case. Reference sensor nodes with known locations transmit linear frequency modulation continuous waves (FMCWs), while other sensor nodes estimate the range difference to them based on the received signals' frequency difference, called time frequency difference arrival (TFDA).

Motivated by all above observations, instead of introducing miraculous assisting entities, our range-free Gaussian Mixture Model (GMM)-based approach performs a connectivity information-based segmentation algorithm [23] that partitions an irregular sensor field into nicely shaped pieces, associated with an enhanced BN Array and efficient distributed in-network information extraction virtual Thick Section Model (TSM); to the best of our knowledge, this is the first work that presents a principled algorithmic approach integrating computational geometry constructs adopted simultaneously for boundary detection in both 2D and 3D network areas. It is promising that our new statistical Gaussian mixture model [24]-based method in this paper is capable of fusing multivariate real-valued sensor inputs to detect boundaries of events in a mathematically principled manner. More precisely, the distribution of sensor readings within each sensor node’s spatial neighborhood is mathematically formulated using most popular finite GMMs. The model selection techniques [25–28] can then effectively identify the correct number of modes for finite mixture models. Therefore, Boundary and Non-Boundary sensor nodes can be consequently distinguished from their neighboring sensor node data distributions.

The remainder of this paper is organized as follows: the next section details enhancement to the BN Array concept; Section 3 simply describes general problems in boundary detection; Section 4 presents the proposed robust Boundary Detection scheme for 3D (BD3D) sensor networks in detail; Section 5 proves BD3D by simulation results; Finally, Section 6 concludes the paper with future work.

In [29] three different schemes which can only take inputs of the 0/1 decision predicates from neighboring sensor nodes are proposed. [30] presents a noise-tolerant algorithm named NED for event and event boundary detection. In NED, the moving mean of the readings of the neighboring sensor node set is used as the estimate for a certain sensor node. The authors of [31] propose Median-based approaches for outlying classification and event frontline detection, where the median is a useful and robust estimator which works directly with continuous numbers, rather than binary 0/1 readings. An extra description of the BN-Array of COBOM [2] and DECOMO [1] is given in this section. Suppose we have a sensor node v (N_v) and its neighbors ξ ( N v ) = ∑ i = 0 k N u i (k is the potential number of neighbors) (k = 6 in Figure 1). Let us consider the BN array in [1, 2]:

In Figure 1, the sensor readings of ξ(N_v) only indicate the relative locations of its neighbors only. Correspondingly, there is no own sensor reading, as a result, N_v judges itself by inquiring ξ(N_v) in a time and energy consuming way. In our model, we applied a head with 1 byte more space for the BN Array to store its own sensor reading as well (see Table 2) for self-judgment as a EBN or non-EBN. Here, we denote a BN inside object as Event BN (EBN), and a BN outside object as non-EBN. That is very important for monitoring applications in the sensor network because an Event sensor Node (EN) is usually highly responsible for sending and receiving the aggregated data should be constantly aware of own status.

Figure 2(a,b) show the expected boundary lines in COBOM and DEMOCO, respectively. Despite the fact that the shape of the expected boundary line in the 2D model of BD3D (see Figure 3) is similar to that of DEMOCO, the knowledge about Boundary sensor Nodes (BNs) promises to be different because we can clearly distinguish EBN and non-EBN as well.

We first present the problems before outlining how our proposal can benefit dynamic boundary detection for 2D and 3D sensor networks in the coming sections. To generally analyze the existing problems for superior boundary detection in a 3D impediment scenario, sensor nodes in the network usually have slight mobility which makes it difficult to establish their locations. Figure 3 illustrates two possible boundary line changes in a 2D scenario when the object is shrunk or expanded. Case (a) is relatively easy to manage, while (b) becomes a big problem that involves frequent inquiries among BNs and massive modifications to BN arrays.

This section involves the main objective of achieving a flexible and energy-efficient 3D continuous boundary detection with a clear knowledge of EBN and non-EBN. Assume that sensor nodes are randomly deployed over 3-dimensional terrain. Each sensor node has limited resources (CPU, battery, etc), and is equipped with an omni-directional antenna. For the radio model, E_elec is for running the transmitter or receiver circuitry and ɛ_amp is for the transmit amplifier. To transmit a δ-bit message a distance l using this radio model, the radio expends (E_elec × δ + ɛ_amp × δ × l²), to receive the message, the radio expands (E_elec × δ) [32]. This energy model assumes a continuous energy consumption function. Moreover, we currently assume that sensor node failures are primarily caused by energy depletion. Note that in our model, no assumptions are made about (1) homogeneity of sensor node distribution; (2) network and BN density; (3) proximity of querying observers and sensor node synchronization.

Our major contribution could be creating a statistical property of the finite mixture model, especially the Gaussian mixture model (GMM) and adopting it to distributed sensing scenarios. Suppose that we have a set of data observations ψ_i = {χ₁, χ₂, ...., χ_n}, n ≤ N (N is the total number of sensor nodes in the network) with each χ_i representing a D-dimensional random vector. Assume that ψ_i follows a k-component finite mixture distribution [24] as follows: (1) 𝒫 ( χ i | θ ) = ∑ j = 1 k α j 𝒫 ( χ i | θ j ) , j = 1,2 , ... , k ; I = 1,2 , ... , n .subject to ∑ j = 1 k α_j=1

where α_j is the mixing weight or sometimes called the prior weight and θ_j is the set of parameters of the j^th mixture component 𝒫(χ_i|θ). Denote θ = {α₁, θ₁, α₂, θ₂,...., α_k, θ_k}. The objective function of estimating θ from ψ_i is to maximize the log-likelihood criterion as follows: (2) Log ∏ i = 1 N 𝒫 ( ψ i | θ ) = ∑ i = 1 n log ∑ j = 1 k α j 𝒫 ( χ i | θ j )

Therefore, the maximum likelihood estimator of θ is: (3) θ ^ ML =   arg   max ︸ θ { log ∏ i = 1 n 𝒫 ( ψ i | θ ) }

Obviously, θ̂_ML cannot be computed analytically from the above equation. Instead, GMM is applied as its general solver to iteratively find the maximum likelihood solution of θ̂_ML. GMM is the most important class of finite mixture densities. GMM is formulated by using a Gaussian density 𝒢(χ_i|μ_j, ∑_j) with its mean vector μ_i and covariance matrix Σ_j to replace the general probability density function 𝒢(χ_i|θ_j) in the finite mixture model: (4) 𝒫 ( χ i | θ ) = Σ j = 1 k α j 𝒢 ( χ i | μ j , Σ j )where a multi-dimensional multivariate Gaussian distribution is defined as: (5) 𝒢 ( χ | μ , Σ ) = 1 | ∑ | 1 2 ( 2 π ) D 2 exp { − 1 2 ( χ − μ ) ′ Σ − 1 ( χ − μ ) }

The Bayesian Information Criterion (BIC) [33] is one of the most popular model selection criteria based on penalty terms of model complexity. In this paper, we use BIC for GMM model selection: (6) BIC     ( θ ) = − 2 log     ( 𝒫 ( ψ i | θ ) ) + Klog     ( m )where, m is the data sample number, and K is the total number of parameters to be estimated in GMM.

In this paper, we provide an algorithm for classifying EBNs. Given a sensor network {S_i}, we assume that sensor nodes are deployed with moderate density in the spatial terrain. From a mathematical perspective, sensor readings provide a dense, but discrete sampling of the underlying continuous distribution. To check whether or not N_v is a sensor node lying on the boundary of an event, we put the data {χ_n} from readings of the sensor nodes in ξ(N_v) and then build our best GMM based on {χ_n}.

In more detail, we first set the upper bound of the mixture component number to be K. Then for each j = 1,2,…,K, the data set {χ_n}is fed into (5) (6) for estimation of θ(J). Let BM denote the number of mixture components of the best model. We select BM where BIC   ( θ ( BM ) ) = min J = 1 K BIC     ( θ ( J ) ). Therefore our final is θ(BM) or {μ_j, Σ_j, α_j}_{j=1,2,....,BM}.

To classify if N_v is a EBN, the conditional probability for χ_i given model θ′(BM) is computed by (7) 𝒫 ( χ i | θ ′ ( BM ) ) = Σ j = 1 BM α j 𝒢 ( χ i | μ j , Σ j )then 𝒫(χ_i|θ′(BM))< γ, N_v is classified as a EBN. Where γ is used as a threshold to measure EBN which has significantly low probability density values given the final model θ′(BM). The threshold is set as γ = 0.25, the upper bound of the component number is set as K = 5. These parameters are used as the default in Sections 4 and 5, unless otherwise stated.

To dynamically update the estimates of observations by conducting [33], we have the following dynamic evolvement and observation equations: (8) χ v i = f ( χ v i − 1 ) + w v i (9) χ v i = g ( χ v i − 1 ) + v v iwhere f(·) is the linear or nonlinear state evolvement function and g(·) is highly nonlinear observation function. w v i and v v i are the standard deviation (noise sequences). For example in static sensor node location, where χ v i remain the same after deployment because of the governance of Equation (8). Therefore, we get the expression: (10) χ v i + 1 = f ( χ v i ) + w v iwhere, w v i model the small position perturbation or other effects.

In this section, we evaluated the performance of BD3D 2D and 3D model implemented in Matlab respectively. The simulation parameters are given in the following table:

This paper has proposed a novel Gaussian Mixture Model-based BD3D scheme for boundary detection of continuously moving object in a 3D sensor network. We adequately presented the proposed protocol, and the simulation results shown support our allegation that the BD3D 2D model surely outperforms COBOM and DEMOCO in terms of average residual energy per sensor node and the number of selected BNs, and the BD3D 3D model achieves accurate boundary detections by soundly selecting EBN and non-EBN for both regular variation and irregular variation object cases. Our future work will include additional optimization desired to improve the performance of our algorithm and verification of the precision of the expected boundaries and invention of a new protocol that considers data losses and route failures due to unpredictable errors such as sensor node failures, contention, interference and fading [35,36]. Moreover, the more accurate energy and mobility model will be addressed in future work.