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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/).

This paper proposes a high precision Gaussian Mixture Model-based novel Boundary Detection 3D (BD3D) scheme with reasonable implementation cost for 3D cases by selecting a minimum number of Boundary sensor Nodes (BNs) in continuous moving objects. It shows apparent advantages in that two classes of boundary and non-boundary sensor nodes can be efficiently classified using the model selection techniques for finite mixture models; furthermore, the set of sensor readings within each sensor node’s spatial neighbors is formulated using a Gaussian Mixture Model; different from DECOMO [

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 [

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 [

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 [

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 [_{v}) and its neighbors

In _{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

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.

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 _{amp} is for the transmit amplifier. To transmit a δ-bit message a distance _{elec} × δ + _{amp} × δ × ^{2}), to receive the message, the radio expands (E_{elec} × δ) [

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} = {χ_{1}, χ_{2}, ...._{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 [_{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 = {α_{1}, θ_{1}, α_{2}, θ_{2},...., α_{k}, θ_{k}}. The objective function of estimating θ from ψ_{i} is to maximize the log-likelihood criterion as follows:

Therefore, the maximum likelihood estimator of θ is:

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:

The Bayesian Information Criterion (BIC) [

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
_{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
_{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 [

Suppose

Although the determination of sensor node status e.g., EBN or non-EBN

Expanding from 2D to 3D, we find that not only the sensing area of sensor node but the network topology is getting more complex, therefore when talking about the relative position of sensor nodes, we need 3D sense of space to construct the model.

Define the state variable as 3D position for a specific sensor node modeled in

In a 3D sensing space, sensor nodes are randomly distributed to form a network. To simplify the complicated operations in dealing with sensor node localization in 3D model, we apply a new concept of 2D plane that each 3D space can be divided into

Randomly pick up to three sensor nodes {

Suppose

Strictly speaking, a 2D plane is definitely as a 2D section (see

Suppose the 3D network area is ζ^{3} cube (ζ is pre-determined in programming) and

The 2D section in our simulation model actually is a (ζ^{2}*d) area [see

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:

Network Area (2D,3D) | (100 m)^{2}(100 m)^{3} |

Number of sensor nodes(2D,3D) | 2,500,10,000 |

The sink (2D,3D) | (50,175), (50,175,50) |

Transmission range(2D) | 10 m |

Time slots | 100 seconds |

Initial Energy | 2J/battery |

Message size | 100 Bytes |

E_{elec} |
50 nJ/bit |

E_{fs} |
10 pJ/bit/m2 |

δ_{amp} |
0.0013 pJ/bit/m4 |

E_{DA} |
5 nJ/bit/signal |

Sensor nodes make local observations every 2 time slots

(2D model)

Design a regular variation object: a circle initially centered at (50, 50) and continually expand it by increasing its radius by 10 meters every 10 time slots. (see

Design an irregular variation object: the initial ENs that adequately covers a circle area {(x − 50)2+(y−50)2 = Rcircle2} to initiate the event. At every time slot, EN propagates by picking up a random number of neighbors to join the event (non-EN→EN). In this way, the network is guaranteed to be fully connected. (see

(3D model)

Design a regular variation 3D object: the object center is (50, 50, 50) and continually expand its radius by 10 meters every 10 time slots.

Design an irregular variation 3D object: the initial ENs are within a spherical area {(x − 50)^{2}+(y − 50)^{2}+(z − 50)^{2} = R_{sphere}^{2}. EN propagates in a similar way as that used for the irregular variation object in 2D model.

The BD3D is flexible enough to be used in a clustered network or a non-clustered network since it does not put any constraints on cluster architecture. However, BNs are usually heavily utilized to send aggregated data associated with the object/network boundary information to cluster head (in clustered networks) or the sink (non-clustered networks), they would run out of energy more quickly. Therefore, achieving a reasonable amount of BNs (the less the better) benefits energy saving.

This section discusses the performance evaluations based on BD3D 2D model.

Comparison of the number of BNs for a regular variation object with COBOM and DEMOCO is shown in

However, due to the elusive ways proposed to expand the irregular variation object, we can hardly do comparison with COBOM and DEMOCO anymore.

In this section, we modeled the BD3D 3D with different values of

Meanwhile, we set the same parameter environment in the BD3D 3D model for evaluating the number of BNs in the network in

We hereby conclude that our BD3D for continuous boundary detection in 3D case works well especially when

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 [

This research was supported by Waseda University Global COE Program International Research and Education Center for Ambient SoC sponsored by MEXT, Japan. The authors would also like to greatly thank the anonymous reviewers for their constructive comments.

Readings of neighbors in BN Array of N_{v}.

Expected boundary lines [

Possible boundary line changes when the object shrunk or expanded.

EBN and non-EBN on BL in BD3D 2D model when object expanded or shrunk.

Position of N_{v} in 3D co-ordinate.

Concept of 2D plane for 3D sensing space.

TSM concept with

Possible 2D sections in 3D network area and (c) is the model used in simulations.

Sample of BD3D 2D model with regular variation and irregular variation object.

Rebound and boundary distances for BD3D.

Average energy level status of 2,500 sensor nodes after 50 and 100 time slots operation.

Performance evaluation by using BD3D 2D when

A combinational vertical section view of 3D sensor network with {

Comparison for regular variation object case using BD3D 3D model.

Performance comparison for irregular variation object case using BD3D 3D model.

BN Array of N_{v} [Note: “0” and “1” are sensor readings (sample)].

BD3D BN Array of N_{v}

BD3D BN Array of N_{v}.

Sensor reading of N_{v} (head) |
Sensor readings of ξ(N_{v})(rear) |

Note:

Head &HR based sensor node status determination.

| ||||||
---|---|---|---|---|---|---|

Head | 1 | 0 | 1 | 1 | 0 | 1 |

HR | random | random | All 0 & random | All 1 | All 0 | All 1 |

EBN ∈ EN, EBN ∪ non-EBN = BN (see

Example of BD3D BN Array of

1 | 0 | 1 | 1 | 1 | 1 | 1 | 0 | 1 |