1. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
INTERNATIONAL JOURNAL OF ELECTRONICS AND
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 6, November - December (2013), © IAEME
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Volume 4, Issue 6, November - December, 2013, pp. 57-61
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MICROWAVE IMAGE RECONSTRUCTION OF TWO DIMENSION
DIELECTRIC SCATTERERS USING SWARM PARTICLE OPTIMIZATION
Arvind Kumar
Department of ECE
BIT Sindri
A Bhattacharya
Department of E&ECE
IIT Kharagpur
D K Singh
Department of ECE
NIT Patna
ABSTRACT
In this paper multi-view approach to microwave imaging is proposed which is based on
stochastic optimization algorithm, particle swarm optimization (PSO). The inverse problem is recast
in an optimization problem. This paper is aimed at assessing the effectiveness of proposed approach
in reconstructing the dielectric parameter of known two dimensional scatterers. Such an analysis is
carried out by comparing performance of PSO based approach with genetics algorithm (GA).
Index term: Inverse Scattering, Microwave Imaging, Particle Swarm Optimization (PSO),
Genetics Algorithm (GA).
I.
INTRODUCTION
In recent years microwave imaging techniques have found considerable attention by
researcher since these techniques can be used for a number of engineering applications such as
biomedical diagnosis of human physiologies [1], [2] non-destructive evaluation [3], [4] subsurface
detection [5] and dielectric properties of scaterers [6].
It is well known that traditional deterministic techniques [7], [8] used for fast reconstruction
of microwave images suffers from major drawback, where the final image is highly dependent upon
the initial trial solution. In addition, the use of the iterative procedures often the reconstruction
process computationally expensive. To overcome this obstacle, population based stochastic methods
such as genetics algorithm (GA) [9] and particle swarm optimization have immersed as alternative to
reconstruct microwave image. Kennedy and Eberhart [10] proposed particle swarm optimization
(PSO) technique in 1995, which is a robust stochastic search procedure inspired by the social
behavior of insects swarm. In this technique the original inverse microwave imaging problem is
recast as a global optimization problem and successively solved by means of a minimization
technique.
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2. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 6, November - December (2013), © IAEME
II.
MATHEMATICAL FORMULATION
Let us consider an investigating domain as square AI with side a x a shown in fig.1 containing
a cylindrical dielectric scatterer of circular cross-section an modeled by following object functions:
Fig.1
߯ሺ‫ݕ ,ݔ‬ሻ ൌ ሾߝ௥ ሺ‫ݕ ,ݔ‬ሻ െ 1ሿ (x,y)‫ א‬I
(1)
where ߝ௥ ሺ‫ݕ ,ݔ‬ሻ is the relative dielectric permittivity. The investigating domain is successfully
illuminated by set of V-incident TM wave characterized by z-directed electric field given by
௏
௏
‫ܧ‬௜௡௖ ሺ‫ݎ‬ሻ ൌ ‫ܧ‬௜௡௖ ሺ‫ݕ ,ݔ‬ሻࢠ
(2)
The scattered field is given by
௏
௏
‫ܧ‬௦௖௔௧ ሺ‫ݎ‬ሻ ൌ ‫ܧ‬௦௖௔௧ ሺ‫ ,ݔ‬ሻ
ሺ‫ݕ ,ݔ‬ሻ ‫א‬AI
(3)
where v= 1,……V
The scattered field is arising from multiple-scattering interaction between incident wave and
the unknown object and is measured at V-different measuring points. V measurement points are
located in area called the observation domain AO, external to the investigating domain AI. The
background medium is assumed to be homogeneous and lossless with dielectric permittivityߝ଴ . The
imaging process is aimed at retrieving the distribution of object function given by equation (1) and of
electric Etot starting from the knowledge of scattering data Escatt and Einc. By modeling the nonlinear
electromagnetic interaction through well known Lippmann-Schwinger integral equations
௏
ଶ
‫ܧ‬௦௖௔௧ ሺ‫ݎ‬ሻ ൌ ݇଴ ‫׬‬஺ ‫ܩ‬଴ ሺ‫ ′ ݎ ,ݎ‬ሻ߯ሺ‫′ݎ ,ݎ‬ሻ
బ
௏
× ‫ܧ‬௧௢௧ ሺ‫′ݎ‬ሻ݀‫ ݎ‬r‫א‬AO
(4)
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3. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 6, November - December (2013), © IAEME
௏
௏
‫ܧ‬௧௢௧ ሺ‫ݎ‬ሻ ൌ ‫ܧ‬௜௡௖
ଶ
௏
൅݇଴ ‫׬‬஺ ‫ܩ‬଴ ሺ‫ ′ ݎ ,ݎ‬ሻ߯ሺ‫ ′ ݎ‬ሻ‫ܧ‬௧௢௧ ሺ‫′ݎ‬ሻ݀‫ א ݎ ′ݎ‬AI
బ
(5)
where G0 is the two dimensional free space Green function. The forward scattering in (4) and (5) has
been solved using Richmond’s method [12].
The inverse problem is then recast as the global minimization problem. The cost function is
given in (6).
మ
F(χ)=
೔
೔
∑ೇ ∑ಿ ቛாೞ೎ೌ೟ ሺ௣ೕ ሻି ீ ೄ ఞா೔೙೎ ሺ௣ೕ ሻቛ
೔సభ ೕ
(6)
మ
೔
∑ೇ ∑ಿ ฮாೞ೎ೌ೟ ฮ
೔
ೕ
Where χ is the unknown parameter, GS is radiation operator which relates internal source to
the scattered field, N is the total number of discretization cells and V is the measuring points.
III.
PARTICLE SWARM OPTIMIZATION
The swarm particle optimization technique is global search technique proposed by Kennedy
and Eberhart in 1995. The steps of algorithm are as:
Step 1: Initialize swarm (particle) with random position and velocity.
Step 2: Evaluate the fitness of each particle and select the best position (pbest) and global best
(gbest).
Step 3: Check the convergence of the cost function i.e. F(gbest)< ϵ.
Step 4: If function does not converse, update the velocity and position of the particle.
Step 5: Repeat the steps from 2 to 4 until the function converses or maximum number of iteration is
reached.
In the initialization process N x D swarm has been generated randomly for χ(x, y). Here N is
the population of the swarm and D is the number of unknowns in the investigating domain i.e. the
dimension the inverse of the problem. Prior knowledge of χ can be used in selecting the range of χ(x,
y).Fitness of cost function is evaluated for each value of χ(x,y) and Fmin{χ(x,y)}is obtained . The
value of χ(x, y) for which cost function is minimum is considered as pbest for that fitness. The value
of χ(x, y) for which cost function has least value among Fmin {χ(x,y)}of all iteration has been
considered best .Convergence of F(gbest) is checked and iteration is either stopped (depending upon
convergence limit) or velocity and position of the particle is updated.
IV.
NUMERICAL RESULTS
In the experiment a homogeneous circular cylinder has been taken as the scattering object.
The assumed parameters are followings:
diameter d= λ0/4 (λ0 wavelength in free space); χ =1.0; a= λ0; D=625(discretization cell) ; V=18
(measurement points) and b= 3λ0.The other PSO parameter are (chosen according to suggestions in
the literature) N=30; c1=c2=1.49,wmax=0.9 and wmin=0.4.
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4. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 6, November - December (2013), © IAEME
In the table 1 the quantitative reconstruction errors, in different optimization techniques, are
expressed through the followings parameters: the percentage errors on the reconstruction of the
object (∑o), the percentage errors on the reconstruction of the back ground (∑b), maximum value the
χ. The percentage errors in the reconstruction of back ground are better in the case of PSO based
procedure while the reconstruction of object profile is good in the case of GA based procedure. The
shape of the scatterer (homogeneous cylinder) has been retrieved better in PSO because of the better
reconstruction of the back ground. The reconstruction the dielectric scatter has be shown in fig 2.
25
1
25
20
0.8
20
1
15
0.6
10
0.4
5
0.2
5
0
0
0
10
20
10
0.4
5
0.2
1
0.8
0.2
0.6
0
0
0
30
15
0.4
10
20
0.6
15
25
0.8
10
20
0
0
30
0
(a)
(b)
10
20
30
(c)
Fig.2 Reconstructed image of test area (a) Ideal reconstruction for reference,
(b) reconstruction using PSO (χ=1), (c) reconstruction using GA
Table 1
Optimization process
∑b
∑o
χmax
GA
1.51%
34.48%
1.1
PSO
0.84%
41.37%
1.1
∑b = Percentage error on the reconstruction of the back ground
∑o = Percentage error on the reconstruction of the object
χmax= Maximum value of χ reconstructed
In fig 2. (a) Shows the original dielectric scatterer for reference while in (b) it has been reconstructed
using PSO procedure. The reconstructed image of the dielectric scatterer in (b) part give very clear
information when sharp change in the dielectric value at the boundary of the background and the
cylindrical scatterer. The maximum value of the χ is 1.1 while in the reference it is 1.0. From
equation(2) maximum value of the relative dielectric constant is 2.1 instead of 2.0. So the error is 5%
in retrieving the relative dielectric constant using PSO technique for inverse imaging.
V.
CONCLUSION
In this paper particle swarm optimization (PSO) technique has been presented for high
dimensional microwave imaging problem. It has been found that PSO based optimization technique
is suitable for reconstruction of dielectric profile of the scatterer. This technique has been found more
efficient than GA based technique because of its capability of escaping from local minima and
convergence speed. It has the capability to include the priori information in the computational
technique which enhances the rate of convergence. The computational cost increases with number of
particle and dimensionality of the inverse imaging problem. Because of the simplicity and ease of
implementation of its algorithm this optimization technique can further be integrated with some
procedures to reduce the cost of computation.
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5. International Journal of Electronics and Communication Engineering & Technology (IJECET), ISSN 0976 –
6464(Print), ISSN 0976 – 6472(Online) Volume 4, Issue 6, November - December (2013), © IAEME
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