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Application of Genetic Algorithm for of a Partially Immersed Non-uniform Conductivity Cylinder Wei Chien*, Hua-Pin Chen**, Chi-Hsien Sun***, Chien-Ching Chiu***, and Yi Sun**** *Electronic Engineering Department, De Lin Institute of Technology Tu-Cheng, Taiwan, R.O.C. **Department of Electronic Engineering and Institute of Electronic Engineering Ming Chi University of Technology, Taiwan, R.O.C. ***Electrical Engineering Department, Tamkang University Tamsui, Taiwan, R.O.C. ****School of Electrical Engineering, Beijing Jiaotong University, Beijing, China Abstract- We consider the inverse problem of determining both the shape and the conductivity of a partially immersed non-uniform conductivity cylinder from knowledge of the far-field pattern of TM waves by solving the ill posed nonlinear equation. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations is derived and the imaging problem is reformulated into an optimization problem. The genetic algorithm is then employed to find out the global extreme solution of the object function. As a result, the shape and the conductivity of the conductor can be obtained. Key words- inverse problem, partially immersed, non-uniform conductivity cylinder, steady-state genetic algorithm 1. INTRODUCTION Microwave imaging of the electromagnetic properties of unknown scatterers by inverting scattered field measurements is of great interest because it is associated with numerous applications in biomedical imaging, nondestructive testing, geophysical exploration, etc. In general, inverse scattering is a nonlinear and ill-posed problem [1]. Recently, many methods have been proposed to reconstruct the shape of a 2-D perfect conductor cylinder. General speaking, two main kinds of approaches have been developed. The first is based on gradient searching schemes such as the Newton-Kantorovitch method [2] and the Levenberg-Marguart algorithm [3]. These methods are highly dependent on the initial guess and tend to get trapped in a local extreme. In contrast, the second approach is population-based evolutionary algorithms, such as genetic algorithm [4], particle swarm optimization [5]. Most of the conducing objects are placed in a homogeneous space, while a buried imperfect conductor is reconstructed using GA by Chiu [6]. In this paper, the scattering object is not immersed in a single medium, but instead is located right at the interface of two mediums, the theoretical and numerical analysis of the scattering problem become much more difficult. To the best of our knowledge, there are no investigations on the electromagnetic imaging of partially immersed non-uniform conductivity cylinder. In this paper, the electromagnetic imaging of a partially immersed non-uniform conductivity cylinder is first reported using GA. In section II, the relevant theory and formulation are presented. In section III, the details of the improved SSGA are given. Numerical results for reconstructing objects of different shapes and conductivies are shown in section IV. Finally, some conclusions are drawn in section IV. 2. THEORETICAL FORMULATION An imperfectly conducting cylinder with conductivity ( ) is partially immersed in a lossy homogeneous half-space, as shown in Fig. 1. Media in regions 1 and 2 are characterized by permittivities and conductivities ( 1 , 1 ) and (2 , 2 ) respectively. A non-uniform conductivity cylinder is illuminated by a transverse magnetic (TM) plane wave. The cylinder is of an infinite extent in the z direction, and its cross-section is described in polar coordinates in the x, y plane by the equation F , i.e., the object is a star-like shape. We assume that time dependence of the field is harmonic with the factor exp( incident field from region 1 with incident angle jt ). Let E inc denote the 1 . Owing to the interface between regions 1 and 2, the incident plane wave generates two waves that would exist in the absence of the conducting object. Thus, the unperturbed field is given by jk ( x sin1 ( y a ) cos1 ) i Re jk1 ( x sin1 ( y a ) cos1 ) , y a E1 ( x, y ) e 1 E ( x, y ) i jk2 ( x sin 2 ( y a ) cos 2 ) , y a E 2 ( x, y ) Te i R cos 2 2 1 n n T cos 1 1 n , 1 n , (1) 2 j 2 1 j 1 ki2 2 i 0 j0 i Im( ki ) 0 k1 sin 1 k2 sin 2 i 1,2 , , , Since the cylinder is partially immersed, the equivalent current exists both in the upper half space and the lower half space. As a result, the details of Green’s function are given first as follows: (1) When the equivalent current exists in the upper half space, the Green’s function for the line source in the region 1, can be expressed as , y a G21 ( x, y; x' , y' ) G1 ( x, y; x' , y' ) G11 ( x, y; x' , y' ) G f 11 ( x, y; x' , y' ) G s11 ( x, y; x' , y' ) , y a (2) Where G 21 ( x, y; x ' , y ' ) G f 11 ( x, y; x' , y ' ) G s11 ( x, y; x' , y' ) 1 2 j e j 2 ( y a ) e j 1 ( y ' a ) e j ( x x ') d 1 2 j ( 2) H 0 [k1 ( x x' ) 2 ( y y ' ) 2 ] 4 1 2 j 2 1 ( 1 2 j )e 1 2 1 ( y 2a y ') (2.1) (2.2) e j ( x x ') d (2.3) i2 ki2 2 , i 1,2 , Im( i ) 0 , y ' a (2) When the equivalent current exists in the lower half space, the Green’s function for the line source in the region 2, is , y a G12 ( x, y; x' , y ' ) G2 ( x, y; x' , y ' ) G22 ( x, y; x' , y ' ) G f 22 ( x, y; x' , y ' ) Gs 22 ( x, y; x' , y ' ), y a (3) Where G12 ( x, y; x ' , y ' ) 1 2 G f 22 ( x, y; x' , y ' ) G s 22 ( x, y; x' , y ' ) j j ( y a ) j ( y ' a ) j ( x x ' ) d 1 2 e 1 e 2 e (3.1) j ( 2) H 0 [k 2 ( x x' ) 2 ( y y ' ) 2 ] 4 1 2 j 2 2 ( 2 1 j )e 2 1 2 ( y y ' 2 a ) (3.2) e j ( x x ') d (3.3) i2 ki2 2 , i 1,2 , Im( i ) 0 , y ' a For the direct scattering problem, the scattered field E S is calculated by assuming that the shape is known. For the inverse problem, we assume the approximate center of the scatterer, which in fact can be any point inside the scatterer, is known. Then, the shape function F ( ) and conductivity function ( ) can be expanded as: N /2 N /2 n 0 n 1 F ( ) Bn cos( n ) C n sin( n ) N /2 N /2 n 0 n 1 (4) ( ) Dn cos( n ) E n sin( n ) where (5) B n , C n , D n and En are real numbers to be determined, and 2(N+1) is the number of unknowns for the shape function and conductivity function. 4. NUMERICAL RESULTS Let us consider a non-uniform conductivity cylinder which is partially immersed in a lossless half-space ( 1 2 0 ) and the parameter a is set to zero. The permittivity in region 1 and region 2 is characterized by 1 0 and 2 2.56 0 , respectively. The frequency of the incident wave are chosen to be 1 GHz, with incident angles 1 equal to 45 and 315 , respectively. For each incident wave 8 measurements are made at the points equally separated on a semi-circle with the radius of 3m in region 1. Therefore, there are totally 16 measurements in each simulation. The number of unknowns is set to be 18 (i.e., 2(N+1)=18), to save the computation time. The population size of 100 is chosen and rank selection scheme is used with the top 30 individuals being reproduced accords to the rank. The search range for the unknown coefficient of the shape function is chosen to be from 0 to 0.1 and the unknown coefficient of the conductivity is chosen to be from 1 to 200S/m. The extreme values of the coefficient of the shape function can be determined by the prior knowledge of the objects. The crossover rate is set to 0.1 such that only 10 iterations are performed per generation. The mutation probability is set to 0.05 and the value of in (11) is chosen to be 0.001. In the examples the size of scatter is about the wavelength, so the frequency is in the resonance range. In the first example, the shape and conductivity function are chosen to be F 0.03 0.02 sin 2 m and ( ) (100 15 cos 2 20 sin ) S/m. The reconstructed shape function and conductivity function for the best population member are plotted in Fig. 2(a) and Fig. 2(b). The errors for the reconstructed shape DR and the reconstructed conductivity DSIG are shown in Fig. 2(c), of which DR and DSIG are defined as DR { 1 N ' cal [ F (i ) F (i )]2 / F 2 (i )}1 / 2 N ' i 1 DSIG { where N' (6) 1 N ' cal [ ( i ) ( i )] 2 / 2 ( i )}1 / 2 N ' i 1 (7) cal is set to 100. Quantities DR and DSIG provide measures of how well F ( ) approximates F ( ) cal and ( ) approximates ( ) , respectively. From Fig. 2(a), Fig. 2(b) and Fig. 2(c), it is clear that the reconstruction of the shape and the conductivity function are quite good. 5. CONCLUSIONS We have presented a study of applying the genetic algorithm to reconstruct the shape and conductivity of a partially immersed metallic object through the measured of scattered E fields. Based on the boundary condition and the measured scattered fields, we have derived a set of nonlinear integral equations and reformulated the imaging problem into an optimization one. By using the genetic algorithm, the shape and conductivity of the object can be reconstructed, even when the initial guess is far from exact oneNumerical results also illustrate that the conductivity reconstruction is more sensitive to noise than the shape reconstruction is ACKNOWLEDGEMENT This work was supported by National Science Council, Republic of China, under Grant NSC-98-2221-E-237-001. REFERENCES [1] P. C. Sabatier, “Theoretical considerations for inverse scattering,” Radio Science, vol. 18, pp. 1–18, Jan. 1983. [2] A. Roger, “Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 232-238, Mar. 1981. [3] D. Colton and P. Monk, “A novel method for solving the inverse scattering problem for time-harmonic acousticwaves in the resonance region II,” SIAM J. Appl. Math., vol. 46, pp. 506–523, Jun. 1986. [4] W. Chien; Chiu, Chien-Ching, “Using NU-SSGA to Reduce the Searching Time in Inverse Problem of a Buried Metallic Object,” IEEE Transactions on Antennas and Propagation, v 53, n10, p 3128-3134, October, 2005. [5] C. H. Huang, C. C. Chiu, C. L. Li, and K. C. Chen, “Time Domain Inverse Scattering of a Two-Dimensional Homogenous Dielectric Object with Arbitrary Shape by Particle Swarm Optimization,” Progress In Electromagnetic Research. PIER 82, pp. 381-400, 2008. [6] C. C. Chiu and W. T. Chen, “Electromagnetic imaging for an imperfectly conducting cylinder by the genetic algorithm,” IEEE Trans. Microwave Theory and Tec., vol. 48, pp. 1901 -1905, Nov. 2000. E inc 0.1 Region 1(y<-a) e xa c t in it ial 2 00 th final 0.08 1 2 , 2 2 0.06 F( ) 0.04 y=-a x 1 0.02 m 1, 1 0 - 0. 0 2 - 0. 0 4 Region 2(y>-a) - 0. 0 6 - 0. 0 5 0 0.05 m 0.1 y Fig. 1(a) Geometry of the problem in (x,y) Fig. 2(a) Shape function for example 1. The star curve represents the exact shape, while the solid curves are calculated shape in iteration process. 0.4 220 200 180 S/m 160 140 120 100 0.3 0.25 0.2 0.15 0.1 80 0.05 60 40 0 DR DSIG 0.35 relative error exact initial 200th final 100 200 degree 300 Fig. 2(b) Conductivity function for example 1. The star curve represents the exact conductivies, while the solid curves are calculated conductivies in iteration process. 0 0 100 200 300 400 generation 500 600 Fig. 2(C) The shape and conductivity function errors versus generation.