Inversion of Scattered Field Using Particle Swarm Optimizer

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

1. 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 multiple-agent 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.

2. 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,n-1 and n-1are known, we update wj in PSO

   GS,D = , ()

   p [ D, S] (4)

   wj,n = wj,n-1+ 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

3. 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 ill-posed 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 = wn-1 + vn

   (15)

   where vn = avn-1+c1*r1(pbest wn-1) + c2*r2(gbest wn-1)

   (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.

4. 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 3±0.3 ( = 2±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 3±0.3 ( = 2±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 off-centered 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 off-centered 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 off-centered cylinders

   In Fig.7 reconstruction from Fresnel data for two off-centered 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.

5. 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 off-centered cylinders from Fresnel data

twodielTM_4f.exp d

REFERENCES

1. X. Li, S . K . Davis, S . C . Hagness, D . W . Van der Weide, and B . D . Van Veen, Microwave Imaging via. Space-time beam forming: Experimental investigation of tumor detection in multilayer breast phantoms IEEE Trans. Micro. Theory Tech., vol. 52, pp 1856-1865, 2004.

2. A.E.Bulyshev, S.Y. Semenov, A.E. Souvorov, R. H. Sevenov, A.G. Nazarov, Y.E.Sizov and G.P. Tatsis, Computaional modeling of three dimensional microwavetomography of breat cancer IEEE Trans. Biomed. Eng., Vol. 48,pp. 1053-1056,2001.

3. J.C. Bolomey , A Izadnegahdar, LJofre, C.Pchot, G. Peronnet and M. Soaimani ,Microwave diffraction tomography for biomedical applications, IEEE trans.Microwave Theory Tech.,vol. MTT-82 no. 11

   , pp. 1998-2000, Nov. 1982.

4. J.C. Bolomey, Recent Europian developments in active microwave imaging for industrial, scientific and medical applications, IEEE Trans. Microw. Theory Tech.,vol. 37, no.12, pp. 2109-2117, Dec. 1989.

5. A.C. Dubey, I. Cindrich, M. Ralston, and K.A. Rigano, Ditection technology for mines and mine like targets, in proc. SPIE, vol. 2496, Ornaldo, FL, pp. 568-569, Jun. 1995.

6. T. M.Habashy, M.L. Oristaglio, and A.T. de Hoop, Simultaneous nonlinear reconstruction of two- dimensional permittivity and conductivity, Radio Sci., vol. 29, pp. 1101-1118. 1994.

7. Peter M. Van den Berg and Aria Abubakar, Inverse Scattering algorithms Based on Contrast Source Integral Representations Taylor & Francis, vol. 10 No. 6, pp 559-576, 2002.

8. J. Kennedy and R. C. Eberhart, Particle swarm optimization, in Proc. IEEE Int. Neural Networks Conf., vol. IV, perth , Austrelia, pp. 1942- 1948, Nov./Dec. 1995.

9. A. T. De Hoop, 1995, Handbook Of Radiation and Scattering of Waves. Academic Press, London.

10. J.H. Richmond, Scattering by a dielectric cylinder of arbitrary cross section shape, IEEE Trans. Antenna propag., vol. AP-13, no. 5, pp 334- 341, May 1965.

11. T.M. Habashy, M.L.Oristagilo and A. T. De Hoop Simultaneous nonlinear Reconstruction of two dimensional permittivity and onductivity Radio Science, vol.-29, pp-1101-1118, 1995.

12. P.M. Van den Berg and R.E. Kleinman A Contrast Source Inversion Method , Inverse Probles, vol.-13, pp-L5-L10, 1997.

13. Richard F Bloemenkamp, Aria Abubakar and Peter M Van den Berg, Inversion of experimental multi-frequency data using contrast source inversion method Inverse Problem, vol.-17, pp-1611-1622, 2001.