 Open Access
 Total Downloads : 154
 Authors : Arvind Kumar, A. Bhattacharya, D. K. Singh
 Paper ID : IJERTV3IS20915
 Volume & Issue : Volume 03, Issue 02 (February 2014)
 Published (First Online): 01032014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
Inversion of Scattered Field Using Particle Swarm Optimizer
1. Arvind Kumar, 2.A. Bhattacharya, 3.D. K. SINGH
1. Department of ECE,BIT Sindri
2. Department of E&ECE,IIT Kharagpur
3. Department of ECE, NIT Patna
Abstract This paper presents multi view to microwave imaging based on contrast source inversion method. Contrast source inversion method recast the inverse problem as an optimization problem. Particle swarm optimizer (PSO) has been used to minimize this cost function and reconstruct the unknown dielectric profile of two dimensional scatterer. The contrast source inversion method recast the inverse problem as an optimization problem in which we seek not only the contrast source but also the contrast itself to minimize a cost functional We perform the inversion of electromagnetic scattered data, measured by Institut Fresnel, Marseile, France, from cylindrical objects for TM illumination. The result shows that the present inversion method enables wide class of scattered field data.
Index Terms Microwave Imaging, Particle Swarm Optimization (PSO) and scattered field

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].
In the inverse scattering modeling, one aims to determine the shape, location and constitutive parameter of the object under investigation from the scattered field. Since the wave field depends upon the materials parameters, the inverse scattering problem is essentially nonlinear. In inverse method the fields and the contrast (permittivity) are considered as fundamental unknown. The cost function consists of superposition the mismatch between the measured field data and the predicted data and the error in satisfying consistency in the interior of the object under investigation. In these relations, an integral operator acts on the contrast source being the product of the unknown field and the unknown contrast (constitutive parameter of the object). Therefore, a more efficient and versatile method has been developed in which the contrast source and contrast are considered as fundamental unknowns. In contrast source inversion method the minimization of two cost function is carried out [7].
Kennedy and Eberhart [8] proposed particle swarm optimization (PSO) technique in 1995, which is a robust stochastic search procedure inspired by the social behavior of insects swarm. The PSO is a multipleagent optimization procedure in which the individuals, called particles, change
their position (state) with time. The particles fly in the multidimensional solution space and adjust their positions according to their own experience and the experience of neighboring particles by exploiting the knowledge of positions encountered by themselves and their neighbor.

MATHEMATICAL FORMULATION
Let us consider a scattering object of an arbitrary cross section. It has been assumed that the unknown scatterer is contained in the investigating domain D as square as shown in Fig.1. The domain D may be chosen large enough to contain the scattering object but larger the size more the computational cost. We assume that the fields vary sinusoidally with frequency . We consider that we know the Green function as the fundamental solution in the embedding medium. Spatial points are denoted by p (xp,yp,zp) and q(xq,yq,zq).
Receivers
S
Scattering field
D
0
= 0
Incident
field
Fig.1 scattering object enclosed in the investigating domain D
j
The object is irradiated by a number of known incident fields E inc(p), j= 1,2,,J originating from different source positions. For each incident field, total field is denoted by Ej(p) in D and given by equation (1) exterior to D.
j j
Ej(p) = E inc(p) + E iscat(p) (1)
, 2 , 2
2
F (w , )= +
(10)
iscat
n j,n n
2
1
The scattering fields Ej (p), j=1,2,..,J is measured on
where
surface S outside the investigating domain D. The total field
= f – G w , = E in – w
+ G w
and the scattered field satisfy the following integral representation in the operator form as
, j
S j,n
, n j
j,n
n D j,n
(11)
j
Ej(p) = E inc(p) + GD
Ej
p D, (2)
The inversion algorithm construct sequences of
{ wj,n } and {n}which iteratively reduce the value of the cost
j
E iscat(p) = GS
Ej
p S, (3)
functional. Although the updating of contrast source and contrast can be carried out simultaneously an alternative
Where (q) denotes the contrast of investigating object, so
(q) = 0 at those points in D outside the actual scattering object. The operators are source type integral representations [9] and represents the field due to a superposition of contrast sources, wj = Ej. They are given by
updating scheme simplify the algorithm significantly.
Updating the contrast sources
Suppose wj,n1 and n1are known, we update wj in PSO
GS,D = , ()
p [ D, S] (4)
wj,n = wj,n1+ vj,n (12)
where G(p,q) is the fundamental point source solution in the embedding medium, denoted as Green function. The subscript S and D are append to the integral operator G to clarify where in the space the field point p is placed. The forward scattering for (2) and (3) has been solved in [10].
Let the scattered measured on S denoted as fj, then scattered field in (3)is represented as data equation,
where vj,n is the updated velocity of the particle in nth iteration. Updating process of velocity has been discussed in the next section.
Updating the contrast
We note that the contrast is only present in the second term of the cost functional (10) which can be rewritten as
, , 2
fj = GSEj p S, (5)
We observe that the data equation contains both unknown
FD()=
2
,
(13)
where Ej,n = E inc + G w . In view of the simplicity of the cost
field and the unknown contrast in the form of product wj(p) = (p)Ej(p) p D, (6)
j D j,n
functional, it can be minimize analytically by taking the contrast as
which can be considered equivalent source that produces the
measured scattered field. This equivalent source is called a contrast source and is measure for the contrasting scatterer in
D. The data equation becomes
fj = GS wj(p) p D, (7)
analyt= , ,
, 2

SWARM PARTICLE OPTIMIZER
(14)
while the object equation from (2) is rewritten as
The swarm particle optimization technique is global search technique proposed by Kennedy and Eberhart in 1995.The steps of algorithm are as:
Ej=E inc + G w p D, (8)
j D j
Substituting this equation in (6) we obtain an object equation for contrast source rather than for the field
Step 1: Initialize swarm (particle) with random position and velocity.
Step 2: Evaluate the fitness of each particle and select the best
E in = w – G w
(9)
position (pbest) and global best
j j D j
The data equation (7) is called source type interal equation. Although this equation is linear in the contrast source, it is a classic illposed equation. This source type integral equation has served as an essential ingredient in many inversion procedures [11]. The contrast source inversion method [12] recast the inverse problem as an optimization problem in which we seek not only the contrast source but also the contrast itself to minimize a cost functional consisting of two terms, the error norms in data equation and in the object equation.
(gbest).
Step 3: Check the convergence of the cost function i.e. F(gbest)< .
Step 4: If function does not converse, update the velocity and position of the particle.
wn = wn1 + vn
(15)
where vn = avn1+c1*r1(pbest wn1) + c2*r2(gbest wn1)
(16)
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 w(p). Here N is the population of the swarm and D is the number of unknowns (contrast source) in the investigating domain i.e. the dimension the inverse of the problem. The first part of the cost functional contains contrast source, the fitness of the first part is evaluated and minimum value is obtained and considered as pbest of that fitness. The value of contrast source, for which the value of first part of cost functional is minimum, is updated in the equation (14) to obtain contrast analytically. For this value of the contrast fitness of the cost functional is evaluated. Convergence of the cost functional is checked and iteration is either stopped (depending upon the convergence limit set) or velocity and position of the particle is updated.

NUMERICAL RESULTS
Fig.2 (a) The original profile of the cylinder for reference (b) the profile reconstructed using PSO
Inversion from experimental data
Institut Fresnel, Marseille, France has measured the electromagnetic data has published it. These data have been used by several researchers for inversion of the scattered field. In the experimental setup the fields are generated and measured by horn antenna. The transmitting horn antenna is moved on a circle around the object at 36 locations at an angle 00 to 3600 at step of 100 and fields are recorded around the object at 49 locations at an angle 600 to 3000. Due to the physical limitation data has been measured at 49 locations instead of 72 locations. We first consider the scatter field for target of single cylinder experiment file name dielTM_dec4f.exp. The relative permittivity of the target is 3Â±0.3 ( = 2Â±0.3). We consider the data measured at frequency of 4GHz. Secondly we consider the scatter field for target of two cylinder experiment file name twodielTM_4f.exp. The relative permittivity of the target in case is also 3Â±0.3 ( = 2Â±0.3).
25
2
A. Inversion from synthetic data
In the experiment a homogeneous circular cylinder has been taken as the scattering object. The assumed parameters are followings:
Considering Fig.1 diameter d= 0/4 (0 wavelength in free space); frequency f = 3GHz; =1.0; the investigation domain is taken equal to 0; D=625(discretization cell); measurements of field is taken at (j=18 points) at surface S at distance of 30.The other PSO parameter are (chosen according to suggestions in the literatures) N=30; c1=c2=1.49,amax=0.9 and amin=0.4.
20
15
10
5
0
0 5 10 15 20 25
(a)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
2
25
20
15
10
5
0
0 5 10 15 20 25
(a)
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
2.5
2
1.5
1
0.5
0
25
20
15
10
5
0 0 5
25
15 20
10
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
25
20
15
10
5
0
0 5 10 15 20 25
1
0.8
0.6
0.4
0.2
0
(b)
Fig.3 Reconstruction of offcentered cylinder from Fresenel data
dielTM_dec4f.exp
(b)
2.5
2
1.5
– Ideal profile
. Reconstruction using PSO
In Fig.4 the error between the ideal profile of the permittivity of single cylinder and reconstruction of the profile using PSO based on contrast source inversion method has been presented. There is some error but the step change in the permittivity profile is self evident.
1
0.5
2.5
– Ideal profile
. Reconstruction using PSO
0
2
0 5 10 15 20 25
Fig.4 this figure shows the error in reconstruction of offcentered cylinder
For minimization of the cost function (10) we need incident fields in the investigation area D. The transmitting antenna has been approximated line source parallel to the target cylinder. In column 6 and 7 of the Fresnel data set the incident field measured at receiver positions has been given. These incident fields have been measured at receiver positions. But incident fields are required to be calculated in the area of investigation. So, we choose to match our synthetically calculated incident field, due to unit line current source, to the measured incident field [13]. Some other methods can also be used to approximate the transmitting antenna.
1.5
1
0.5
25
2
1.8
20
1.6
1.4
15
1.2
0
0 5 10 15 20 25
10
5
0
0 5 10 15 20 25
(a)
1
0.8
0.6
0.4
0.2
Fig.7 this figure shows the error in the reconstruction
two offcentered cylinders
In Fig.7 reconstruction from Fresnel data for two offcentered cylinders has been presented and the position of two cylinders is self evident. For the inversion of the data the inversion area has been subdivided into 25×25 cells.

CONCLUSION
2.5
2
1.5
1
0.5
0
25
20
15
10
5
0
0 5 10 15 20 25
2
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
The result shows that the contrast source inversion method with PSO leads to an effective inversion technique. In view of the result obtained and the approximation made for the transmitting and the receiving antennas, the contrast source inversion with PSO is a robust inversion technique. Inversion results can further be improved if a better model of approximation is used for the transmitting and receiving antennas.
(b)
Fig6. Reconstruction of two offcentered cylinders from Fresnel data
twodielTM_4f.exp d
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