Optimization for the sensor placement problem in 3D environments

vinhlt@vnu.edu.vn Abstract—In this paper, we propose a novel 3D sensing model for the sensor placement optimization problem given a three dimensional environment.. The model take into ac

Optimization for the sensor placement problem in

3D environments

Nguyen Thi Tam

VNU University of Science,

Vietnam National University,

334 Nguyen Trai, Thanh Xuan,

Ha Noi, Viet Nam

tamnt@vnu.edu.vn

Hai Dang Thanh University of Dalat,

01 Phu Dong Thien Vuong , Da Lat, Viet Nam haidt@dlu.edu.vn

Le Hoang Son VNU University of Science, Vietnam National University,

334 Nguyen Trai, Thanh Xuan,

Ha Noi, Viet Nam

sonlh@vnu.edu.vn

Vinh Trong Le VNU University of Science, Vietnam National University,

334 Nguyen Trai, Thanh Xuan,

Ha Noi, Viet Nam vinhlt@vnu.edu.vn

Abstract—In this paper, we propose a novel 3D sensing model

for the sensor placement optimization problem given a three

dimensional environment The model take into account the angles

of a sensor, the distance between the sensor and a given point in

the terrain, the Line-of-Sight (visibility) capability, the constraints

of the terrain and the number of sensors needed to maximize the

coverage over the terrain In order to generate optimal solutions

to the model, we firstly present a novel Line-of-Sight (LoS)

method aiming to determine the number of obstacles between a

given sensor and a point in the region of interest using the ideas

of adaptive lengths and linear regression Secondly, we propose a

modification of PSO algorithm, where particles (sensors) update

their velocity by using only local information coming from their

neighbors The comparison and analyses of experimental results

reveal that optimal solutions achieved from the 3D sensing model

are better than those of the related work

I INTRODUCTION Wireless Sensor Network (WSN) consists of a number of

sensor nodes, which can sense, measure and gather information

from the environment and they can transmit the sensed data to

the user [16] Each sensor node can be attached to some special

types of sensors such as thermal, biological, chemical, optical,

and magnetic sensors for the measurement of properties of

the environment These measured data are transferred to a

base station by a mean of radio wave in wireless

com-munication WSN could be applied to various applications

involving environmental based public safety hazards such as

brush fires, biochemical accidents or attacks to obtain real

time and accurate information about the hazards for immediate

prevention

A Previous works

The Sensor Placement Optimization (SPO) problem

con-tributes a great impact to WSN involving a large number of

researches in recent years It can be represented as the

maxi-mization of the global coverage of WSN to a region of interest

The basic principle is that each location in the region of interest

should be within the sensing range of at least one of the

sensor nodes (or in short sensors) Thus the interaction between

points in the region of interest is always covered by WSN The

maximization of such the interaction can be regarded as the

assurance of high speed connection between those points A

number of methods were proposed in the literature to handle the SPO problem Deterministic methods [5], [8], [10], [13], [15], [17] aimed to select a minimum size connected K-cover, which is defined as a set of sensors such that each point in the sensor network is covered by at least K different sensors, and their locations to guarantee that an area is K-covered and the network is connected Approximation algorithms were used

to deliver sub-optimation solutions for this task Nonetheless, deterministic methods often rely on oversimplified sensing models and environmental factors Therefore the theoretical coverage shown in the deterministic methods may not hold true in practice Extensions of deterministic methods in 3D en-vironments were presented to overcome the limitations above Dhillon et al [3] combined terrain modeling and a probabilistic sensing model for SPO Huang et al [6] formulated the SPO problem in 3D space and proposed a polynomial-time method for this problem Ma et al [9] utilized a virtual force mechanism and simulated annealing Topcuoglu et al [11] proposed a new formulation for the deployment of sensors

in 3D environments Unaldi et al [12] proposed an algorithm based on a probabilistic sensing model, the Bresenham’s line

of sight (LoS) algorithm and a guided wavelet transform (WT)

in which the sensor movements are carried out within the mutation phase of the genetic algorithms Zhao et al [18] proposed a new coverage model called surface coverage in which the targeted Field of Interest is a complex surface in 3D space and sensors can be deployed only on the surface Akbarzadeh et al [2] developed a probabilistic sensing model for sensors with LoS-based coverage consisting of membership functions for sensing range and sensing angle, which takes into consideration sensing capacity probability as well as critical environmental factors such as terrain topography to tackle the SPO problem Other solutions could be referenced in the overviews [1], [4], [6]

B Limitations of the previous works The existing 3D models designed for the SPO problem have some limitations such as,

• They do not take obstacles into consideration;

• The binary sensing coverage was used;

Proceedings of 2015 IEEE 12th International Conference on Networking, Sensing and Control

Howard Civil Service International House, Taipei, Taiwan, April 9-11, 2015

• Constraints related to the 3D terrain were not taken into

account

C Contributions of the article

Firstly, a novel 3D sensing model for the SPO problem that

remedies the disadvantages, as pointed out above, are proposed

in Section 2 Those models are the generalization of that of

Akbarzadeh et al [2] where constraints are provided in the

problem, and the formulas to calculate membership values are

adjusted to express more accurately the placement problem in

a terrain and to be of parameter-free;

Secondly, a novel Line-of-Sight (LoS) method aiming to

determine the number of obstacles between a given sensor and

a point in the region of interest using the ideas of adaptive

lengths and linear regression is presented in section III-B;

Thirdly, a modification of PSO algorithm using the ideas of

Particle Swarm Optimization (PSO) [7] and local information

coming from their neighbors is introduced in section III-C

to generate optimized solutions for the proposed models

PSO works in the same way as genetic algorithms and other

evolutionary algorithms Similar to evolution algorithm, PSO

algorithm adopts a strategy based on particle swarm and

par-allel global random search PSO differs from these algorithms

by simulating the social behavior and moment dynamics of a

swarm Each swarm always moves to the own local optimum

solution and the global optimum solution Finally, swarm finds

the good optimum solution However, it has better performance

than early intelligent algorithms on calculation speed and

memory occupation, and has less parameter

D Organization of the article

The rest of the paper is organized as follows In section II

& III, we present the main contributions Section V presents

the experimental results and Section VI makes conclusions and

future works

Giving a target area and sensor network include N sensors,

the problem need be solved is that how to cover this area If

all the points in the target area are covered by sensor network,

the target area is covered On the other hand, there are infinite

number of points to deal with within the single target area To

overcome this, sampling method was used, where only a fix

number of points are used to evaluate the coverage One the

commonly used sampling methods is grid In grid method, the

target area is divided into uniform size grid If all grid points

are covered then the entire target area is covered

The proposed 3D sensing model is stated as follows

Suppose that we have:

• T is a Digital Elevation Model (DEM) terrain, which is a

matrix whose values representing for the elevations of grid

points as shown in Fig 1 Some parameters are:

◦ cellsize: the size of grid cell;

◦ nrows and ncols: the number of rows and columns of DEM respectively

Fig 1 Represent a point in terrain

• W SN = {s1, s2, , sN} is a sensor network where:

sj = {(xsj, ysj), hsj(xsj, ysj), αj, θj, ξj, βj} ∀j ∈ [1, 2, , N ]

(1)

◦ (xs

j, ys

j) is the coordinate of sj in Oxy;

◦ hj(xs

j, ys

j) is the heigh of sj in position (xs

j, ys

j)

◦ rs

j is the sensing radius of sj;

◦ θj is the pan angle of sj around the vertical axis (X direction);

◦ αj is the angle to define the orientation of the directional sensor sj around X direction, 0 ≤ αj ≤ 2π;

◦ ξj is the tilt angle sj around the horizonal axis (Z direc-tion);

◦ βj is the angle to define the orientation of the directional sensor sj around Z direction, 0 ≤ βj ≤ 2π The angles of

a sensor is represented as shown in Fig 2

Fig 2 Represent angles of the sensor

• R0 = {r1, r2, , rH} is a set of physical holes where can not put sensors where,

ri= {(xsi, yis), (xfi, yif)} ∀i ∈ [1, 2, , H] (2)

◦ Each hole is represented by a rectangle as shown in Fig 3;

◦ (xs

i, ys

i) is coordinate of top vertex of rectangle at left;

◦ (xfi, yfi) is coordinate of bottom vertex of rectangle at right

• E = {e1, e2, , eM} is the sampling set,

ei= {(xei, yie), hei(xei, yie), wi}, ∀i ∈ [1, 2, , M ] (3)

◦ M is the number of sampling points which is not in physical holes;

◦ (xs

i, yis) is coordinate of point ei in Oxy;

◦ wi is the weight of ei

Fig 3 Represent holes in terrain

• A point ei is said to be convered by sensor si if and only if

the following conditions are satisfied:

◦ The Euclidean distance between the location of sensor sj

and point ei less than or equal sensing radius of sj;

◦ The angle between the sensor sj and point ei along the X

direction less than or equal the pan angle of sj;

◦ The angle between the sensor sj and point ei along the Z

direction less than or equal the tilt angle of sj;

◦ Visibility from the sensor sj to point ei

Therefore, the sensing model mainly depends on distance,

orientation, and visibility

◦ µd is the binary function to measure the distance between

sj and ei:

µd=



1, d(sj, ei) ≤ rjs

d(sj, ei) =k (xe

i, ye

i, he

i) − (xs

j, ys

j, hs

j) k

◦ µpis the binary function to measure the coverage

capabil-ities of sensor sj to the point ei by angle of the sensors

along vertical axis;

µp=

(

1, arctan(y

e

i −y s j

x e

i −x s j

) ∈ [αj, αj+ θj]

where arctan(y

e

i −y s j

x e

i −x s

j) is the angle between the sensor sj and the point ei along the X direction

◦ µtis the binary function to measure the coverage

capabil-ities of sensor sj to the point ei by angle of the sensors

along horizontal axis;

µt=

(

1, arctan(h

e

i −h s j

d(sj,ei)) ∈ [ξj, βj+ ξj]

where arctan(h

e

i −h s j

d(sj,ei)) is the angle between the sensor sj and the point ei along the Z direction

◦ vij represent visibility between sj and ei;

vij=

 0, µd= 0 or µt= 0 or µd= 0

1 1+num Obstacles(sj,ei), otherwise (7)

where num Obstacles(sj, ei) is the number of obstacles

between sensor sj and point ei, it is determined by LoS

method,

• The coverage C(sj, ei) of sj at point ei can be defined as

functions of distance µd, pan angle µp, tilt angle µt and

visibility vij from sensor;

C(sj, ei) = µd× µt× µp× vij (8)

Given a set of sensors and a point, the probability that the sensor will detect an event at a given point can be calculated A point can be covered by one or more sensors C(sj, ei) represents the probability of coverage, and hence,

1 − C(sj, ei) gives the probability of non-coverage When two or more sensors will happen at the same time and sen-sors are independent then the special rule of multiplication law is used to find the joint probability To calculate the probability that sensors would not cover at target point, the multiplication law of probability is used to define the miss probability Q

i=1,N (1 − C(si, q)) In case, we are looking for probability of coverage Then, the probability of the environment that covers point ei is

Cei(W SN, ei) = 1 − Y

i=1,N (1 − C(si, q)) (9)

• The global coverage

Cg(W SN, E) =

P

ei∈E

wiCei(W SN, ei)

• Global coverage is maximum

• Constraints:

(xsj, yjs) /∈ R, ∀j ∈ [1, 2, , N ] (12)

SOLUTIONS

A Interpolation high of a point

We will use bilinear interpolation to determine height of point p having coordinate (x, y) To determine height of any point in DEM terrain, we will perform following steps: Step 1: We find coordinate of four grid points which are

in grid cell contain p To determine coordinate of four grid points, we will perform following:



x1= f loor(x/cellsize) × cellsize

y1= f loor(y/cellsize) × cellsize



x2= x1+ cellsize

y2= y1



x3= x1

y3= y1+ cellsize



x4= x1+ cellsize

y4= y1+ cellsize where, f loor(a) returns the largest integer value less than or equal to a

Step 2: Assume h1, h2, h3, h4are height of four grid points which is given by DEM matrix For example, to determine the height hiat (x, y), the elevations at y on the vertical boundaries

of the grid cell can be linearly interpolated between h1and h3

at ha, and h2 and h4 at hb:

ha≈ y × h1+ (1 − y) × h3

hb≈ y × h2+ (1 − y) × h4 Step 3: The required elevation at (x, y) can be linearly interpolated between ha and hb

h = ha× x + hb× (1 − x)

The bilinear function is akin to fitting a hyperbolic

paraboloid to the four vertices of the grid cell It is usually

written as:

h = a00+ a10x + a01y + a11xy, where,

a00= h1

a10= h2− h1

a01= h3− h1

a11= h1− h2− h3+ h4

B The LoS method

The aim of this section is to answer the question “how

many obstacles are there between a sensor and a point in

the region of interest?” This requires the understanding of

the characteristics or more specifically the morphology of the

DEM terrain If an answer is determined, it can be used to

calculate vij in equation (7) In fact, the main factor that

affects the visibility between a point and a sensor is the

elevations of all points in the straight line connecting them

This information is provided by DEM, which is basically a

two dimensional matrix, where each cell stores the elevation

of the corresponding location in the real environment

1) The old LoS method: In order to calculate the visibility

between a point and a sensor, a list of cells in the

line-of-sight matrix between them should be specified Some main

steps are drawn as follows Firstly, we divide the line segment

connecting the point and the sensor into several split points

according to the size of grid cell Secondly, each point in the

list is checked versus the cell containing it If its elevation

is smaller than the elevation of the cell then the number of

obstacles is increased In Fig 4, the line of sight is shown as

a line which connect between sj and ei

2) The new LoS method: However, the drawback of this

method is its computational complexity In order to achieve

high accuracy, a large number of split points must be checked;

thus increasing the computational time of the algorithm If

less number of split points is used then some obstacles could

be omitted and the calculation of vij is inaccurate For this

reason, our idea is to use the adaptive lengths between split

points to reduce the number of those points but still keeping

the accuracy In the old approach, two consecutive split points

are equally apart by the size of grid cell Our observation is

that if there is no obstacle measured in a split point then it is

likely no obstacle in the next split point taking into account the

differences between the altitudes of the split points and the cell

in two consecutive split points Thus, a linear regression could

be used to predict when the line connecting the point and the

sensor could intersect the obstacle If so, record the distance

at that point and the beginning point as the new length, set the

new point as the beginning one and continue to find another

adaptive lengths until the new point is over the sensor point

The pseudo-code in Table I describes the idea in details In this

algorithm, h(cellP oint) denotesthe height of grid cell contain

the split point, h(splitP oint) denotes the height of split point,

numberOf P oints denotes the number of split points, dij is

distance between sensor sj and point ei, k1, k2 are the user-defined

Fig 4 (a) LoS query returning visible, (b) LoS query returning not visible

TABLE I P ROCEDURE OF NEW L O S METHOD

1 numberOf P oints ← cellsizedij

2 while numberOf P oints > 0

3 X1← h(splitP oint) − h(cellP oint)

4 numberOf P oints ← numberOf P oints − 1

5 X2← h(splitP oint) − h(cellP oint)

6 if X1< 0 AND X 2 < 0

7 X3← k 1 × X 1 + k2× X 2

8 While X3> 0 AND numberOf P oints > 0

10 X2← X3

11 X3← k1× X1 + k2× X2

12 numberOf P oints ← numberOf P oints − 1

13 Else if X1 > 0

14 count ← count + 1

15 Else if X 2 > 0

16 count ← count + 1

C Finding optimized solutions by PSO This section presents the optimization method to determine the optimized solutions of the 3D sensing model (1-12) by Particle Swarm Optimization (PSO) [7] PSO is a population based stochastic optimization technique developed by Dr Eberhart and Dr Kennedy in 1995, inspired by social behavior

of bird flocking or fish schooling Generally, it is based on the principle: “The best strategy to find the food is to follow the bird which is nearest to it” Indeed, in PSO, each single solution is a “bird” or “particle” in the search space All particles have fitness values which are evaluated by the fitness function to be optimized, and have velocities which direct the flying of the particles The particles fly through the problem space by following the current optimum particles Motivated

by the virtual forces algorithm (VFA) in [14], in this article,

we present the VFA based PSO algorithm to find the optimized solutions Some notations is used in PSO algorithm:

• Xi = (X1i, X2i, , XN i) and Vi = (V1i, V2i, , VN i) represent for the position and the velocity pi where, Xji= (xji, yji) and Vji= (vji, v0ji) represent for the position and the velocity of sensor sj in particle pi ∀j = 1, N ,

• pbesti = {pbest1i, pbest2i, , pbestN i} denotes the best particle of particle ith, where pbestji the best position of sensor sj in particle pi,

• gbest = {gbest1, gbest2, , , gbestN} the best particle in the swarm, where gbestjthe best position of sensors sjin history

of the swarm,

• f (gbest) is the best fitness value of the swarm.

Details of this algorithm are shown below

Step 1: Initialization

The beginning population is initiated with Npop particles,

where Npop is a designated parameter Each particle is

ran-domly initiated their position and velocity

Step 2: Calculate fitness values of all particles The

proce-dure is shown as in Table II

TABLE II C ALCULATE FITNESS

1 For i = 0 to M

2 For j = 0 to N

3 Calculate µ d according to (4).

4 Calculate µ p according to (5).

5 Calculate µ t according to (6).

6 Calculate v ij according to (7).

7 C(s j , e i ) ← µ d × µ t × µ t × v ij

9 C (W SN, e i ) ← 1 − Q

j=1,N

(1 − C(s j , e i ))

10 Cg(W SN, E) ← Cg(W SN, E) + Cei(W SN, ei)

11 return Cg (W SN,E)M

Step 3: Update pbest and gbest The procedure is shown

as in Table III

TABLE III U PDATE pbest AND gbest PROCESS

1 For i = 1 to Npop

2 If f (p i ) > f (pbest i )

3 f (pbest i ) ← f (p i )

4 For j = 1 to N

5 pbest ji ← X ji

1 For i = 1 to Npop

2 If f (gbest) < f (pbesti)

3 f (gbest) ← f (pbesti)

4 For j = 1 to N

5 gbestj← pbest ji

Step 4: Update the velocities and positions of particles by

virtual forces The procedure is shown as in Table IV Some

terms is used

• d(si, sj) is the Euclidean distance between sensors,

• adj(si) is the adjacency set of sensor si, sensor sj is called

adjacency of si sensor if and only if d(si, sj) ≤ rc, where

rc is communication radius, rc= 2 × rs,

• Fij is the virtual force exterted by the neighborhood sj on

si,

• Fi is the total virtual force action on sensor si,

• daveis the average distance between two sensors when they

are evenly distributed in the area,

In modification of PSO algorithm, sensors update their velocity

and position by using information coming from their

neigh-bors Virtual Force approach has ability to ”position” sensors

with no overlap, by using attractive and repulsive forces based

on the distance between sensors The basic idea of the virtual

force based PSO algorithm is based on attributes of sensors

which are electromagnetic particles: when two electromagnetic

particles are too close to each other, a repulsive force calculated

as in equation (13) pushes them apart

Fij =







d ave −d ij

2×dij (xi− xj, yi− yj) if dave> dij

−dave −d ij

2×dij (xi− xj, yi− yj) if dave< dij

(13) Finally, the total virtual force action on a sensor is,

si∈adj(s j )Fij (14) TABLE IV U PDATE THE VELOCITIES AND POSITIONS OF PARTICLES

1 For i = 1 to Npop

2 For j = 1 to N

3 V ij = w × V ij + r 1 × c 1 × (pbest ij − Xij) + r 2 × c 2 × (gbest j − X ij )

r 3 × c 3 × F j

4 X ij =

 Xij+ Vij if Xij+ Vij∈ R /

Xij otherwise Step 5: Repeat the whole process from Step 2 to Step 4 until the maximal interation step (PSO MaxIter) is reached

IV ANALYSIS OF COMPLEXITY 1) The LoS method

The largest number of split points num is calculated as following:

num = p(cellsize × nrows)2+ (cellsize × ncols)2

cellsize

(15) where nrows and ncols are the number of rows and columns of DEM matrix, respectively It is clear that in this case two points are located on main diagonal of the DEM matrix The computational time complexity of the LoS algorithm in the worst case is O(num) because each node is checked only once

2) The PSO algorithm

• Step 1: Generation Npop particle, each particle includes

N sensors, each sensor is not in holes Then, complexity

of this step is T1= O(Npop× N × L), where L is the number of holes

• Step 2: The computational time complexity of this step

is T2= O(M × N × num), where M is the number of sampling points

• Step 3: The computational time complexity of this step

is T3= Npop× N × M × num

• Step 4: The computational time complexity is T4 = O(Npop× N × L)

The worst-case computational time complexity of the PSO algorithm in each iteration is:

T = T1+T2+T3+T4= O(Npop×N ×num×(M +L)) (16)

A LoS method

In the following section, we implemented the proposed LOS method using the C programming language We per-formed with 100, 200, 500, 1000, 5000, 10000, 20000 couple

of points These methods were run 30 times and the giving

average results in Table V

TABLE V R ESULT OF NEW METHOD L O S

The number of couple of points Accuracy

Fig 5 Comparison time of new method LoS with old method LoS

Fig 5 compares time of new method LoS with old method

LoS It is observed that as the number of couple of points

increases, time of old method LoS tend to climb very fast

B PSO Algorithm

We have implemented the proposed algorithms using the C

programming language and executed them on a Linux Cluster

1350 with eight computing nodes of 51.2GFlops Each node

contains two Intel Xeon dual core 3.2GHz, 2GB Ram These

algorithms were run against the DEM terrains of

Bolzano-Bolzen province, Italy in 2005 Parameters of the algorithm

are set as:

• rs

j = 10m, αsj = 0, βjs= 0, θsj = 180o, ξsj = 90o, ∀j =

1, , N ,

• Npop= 100, P SO M AXIT ER = 100

We made several tests with different numbers of sensors (N )

and numbers of points in the restricted region (L) as in Table

VI In each test, each model is run 10 times and the coverage

percentages in Table VI are the average result

Some remarks extracted from the experiments are shown

as follows

• The probabilistic model given in [2]

◦ Size of target area is 200m × 200m;

◦ Using 131 senros, each sensor have sensing radius of 30m;

◦ Coverage percentages is up to 95.98%

• The proposed models use 71 and 232 sensors, each sensor have sensing radius of 10m achieving the coverage percent-ages of 93.339% and 92.194% respectively The optimized solutions achieved from 3D sensing models are better than those of the relevant works

• Sensing radius of sensors is used 10m in in our model while one model [2] is 30m When we increase sensing radius of sensors to 30m, we only use 105 sensors and achieving the coverage percentange of 97.845% with DEM size of 225m × 225m It is obvious that our proposal uses a smaller number

of sensors than the probabilistic model [2]

• The proposed models are more generalized than that of the probabilistic model [2] They could achieve a higher coverage percentage if a suitable number of sensors and the sensing range are provided

This paper has made several contributions to the sensor placement optimization problem, such as i) designing some novel 3D sensing models that are the generalization of the existing models; ii) proposing a novel Line-of-Sight (LoS) method aiming to determine the number of obstacles between

a given sensor and a point in the region of interest; iii) designing a modification PSO algorithm using the ideas of Particle Swarm Optimization and local information coming from neighbors of sensors The experimental results show that the optimized solutions achieved from the 3D sensing models are better than those of the relevant works

For future works, we may consider the following ap-proaches to further extend our proposed models

• Considering the region of interest and the restricted regions

in the model as different polygon shapes;

• Examining various types of DEM terrains and morphologies;

• Taking the hybrid sensors/BS into the wireless networks;

• Designing variants of the LoS/ PSO methods

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DEM size: 120m × 120m

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DEM size: 225m × 225m

N c1= c2= c3= 2 c1= c2= 2, c3= 1 c1= 1, c2= c3= 2 c2= 1, c1= c3= 2

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