Through-wall Imaging of Conductors by Transverse Electric Wave Illumination

Transcription

1 Journal of Applied Science and Engineering, Vol. 20, No. 4, pp (2017) DOI: /jase Through-wall Imaging of Conductors by Transverse Electric Wave Illumination Wei Chien 1, Chien-Ching Chiu 2 * and Wei-Siang Gu 2 1 Department of College of Electric Information, Qinzhou University, Binhai Avenue, Qinzhou, Guangxi, P.R. China 2 Department of Electrical Engineering, Tamkang University, Tamsui, Taiwan 251, R.O.C. Abstract A novel method for through-wall imaging (TWI) illuminated by the transverse electric (TE) waves is presented. Most microwave inverse scattering algorithms developed are for transverse magnetic (TM) wave illumination in which vector problem can be simplified to a scalar one, which less works have been reported on the more complicated TE case. In the TE case, the presence of polarization charges makes the inverse problem more nonlinear. This paper uses the self-adaptive dynamic differential evolution (SADDE) algorithm to recover the shapes of the two dimensional conducting cylinders by TE plane wave illumination. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equation is derived and the imaging problem is reformulated into optimization problem. The SADDE algorithm is employed to find out the global extreme solution of the object function. Numerical results show that the shapes of the conductor are well reconstructed. In addition, the effect of Gaussian noise on the reconstruction is investigated. Key Words: Through-wall Imaging, Frequency-domain, Self-adaptive Dynamic Differential Evolution, Inverse Scattering 1. Introduction *Corresponding author. chiu@ee.tku.edu.tw The image problem of conducting objects has been a subject of considerable importance in nondestructive measurement, medical imaging, and biological application. Numerical inverse scattering studies found in the literature are based on either frequency or time domain approaches. However, it is well known that one major difficulty of inverse scattering is its ill-posedness in nature [1]. Another inverse scattering problem is the nonlinearity because it involves the product of two unknowns: the electrical property of object, and the electric field within the object. In general, the nonlinearity of the problem is coped with by applying iterative optimization techniques [2,3]. These algorithms based on stochastic strategies, offer advantages relative to local inversion algorithms including strong search ability simplicity, robustness, and insensitivity to ill-posedness. In contrast to traditional computation systems, evolutionary computation [4 8] provides a more robust and efficient approach for solving inverse scattering problems. The self-adaptive dynamic differential evolution (SADDE) algorithm is potentially able to obtain the global optimum of a functional whatever the initial guesses are [9 11]. Frequency domain inverse scattering by populationbased stochastic algorithms are published in the last ten years. Through-wall imaging (TWI) consists of imaging objects hidden behind an obstacle by using electromagnetic waves at microwave frequencies. This problem is of great interest, as the aim of detecting and localizing hidden objects is shared in many applicative contexts (i.e. both military and civilian) such as in search and rescue, security, peacekeeping and law enforcement operations [12 20]. Most microwave inverse scattering algorithms developed are for TM wave illumination in which vector

2 478 Wei Chien et al. problem can be simplified to a scalar one, which less works have been reported on the more complicated TE case [21,22]. In the TE case, the presence of polarization charges makes the inverse problem more nonlinear. As a result, the reconstruction becomes more difficult. To the best of our knowledge, a study about the performance of SADDE when applied to TE case has not yet been investigated for through-wall imaging in frequency domain. In this paper, the inverse scattering problem of conducting cylinders by transverse electric TE wave illumination is investigated on the application of SADDE. The objective function of SADDE is to minimize the discrepancy between measured and estimated scattered field data. In section 2, the solution of the forward scattering problem is presented. In section 3 and 4, inverse problem and the numerical results of the proposed inverse problem are given, respectively. Section 5 gives the conclusions. 2. Forward Problem Let us consider conducting cylinder is buried in region 3, as shown in Figure 1. Media in regions 1 and 2 and 3 are characterized by permittivities and conductivities ( 1, 1 ) and ( 2, 2 ) and ( 3, 3 ), respectively. A conducting cylinder is illuminated by a TE 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( ). We assume that the time dependence of the field is harmonic with the factor e j t. Let H inc denote the incident field form region 1 with incident angle 1 as shown in Figure 1. The scattered field, H H z can be expressed by s s With (5) Here G(x, y; x, y ) is the two-dimensional Green s function, and J sm ( ) is the induced surface magnetic current density which is proportional to the normal derivative of the magnetic field on the conductor surface. For a perfectly conducting scatterer, the total tangential electric field at the surface of the scatterer is equal to zero. (6) where n is the outward unit vector normal to the surface of the scatterer. 3. Inverse Problem For the inverse problem, assume the approximate center of scatterer, which in fact can be any point inside the scatterer, is known. Then the shape function F( ) can be expanded as: (7) where B n and C n are real coefficients to be determined, and N+1 is the number of unknowns for the shape function. In the inversion procedure, the SADDE is used to minimize the following objective function: (1) where (2) (3) (4) Figure 1. Geometry of the problem in (x, y) plane.

3 Through-wall Imaging of Conductors by Transverse Electric Wave Illumination 479 (8) where M t is the total number of the mimic measurement data points. H exp cal s ( rm) and Hs ( rm ) are the measured and calculated scattered fields, respectively. The SADDE are based on DDE scheme [23,24]. The SADDE algorithm starts with an initial population of potential solutions that is composed by a group of randomly generated individuals which represents the shape function and the conductivity of the cylinders. The flowchart of the SADDE algorithm is shown in Figure 2. SADDE algorithm goes through six procedures as follows: I. Initialize a starting population: Individuals in SADDE algorithm represent a set of D-dimensional vectors in the parameter space for the problem, {X i : i =1,2,, M p }, where D is the number of parameters to be optimized and M p is the population size. II. Evaluate the population using objective function: after initialization, SADDE algorithm evaluates the objective function (8) for each individual in the population. III. Perform mutation operation to generate trial vectors: The mutation operation of SADDE algorithm is performed by arithmetical combination of individuals. For each parameter vector X i of the parent generation, a trial vector V i is generated according to the following equation: trial vector V i. Perform selection operation to produce offspring: selection operation is conducted by comparing the parent vector X i g with the crossover vectors u i g 1. The vector with smaller objective function value is selected as a member of the next generation. The SADDE algorithm is carried out in a dynamic way: each parent individual will be replaced by his offspring if the offspring has a better objective function value than its parent individual does. IV. Stop the process and obtain the best individual if the termination criterion is satisfied, else go to step Numerical Results We illustrate the performance of the proposed inversion algorithm and its sensitivity to random noise in the scattered field. Let us consider a lossless three-layer structure ( 1 = 2 = 3 = 0) and a perfectly conducting cylinder buried in region 3. The permittivity in each region is characterized by 1 = 0 and 2 = and 3 = 0 respectively. The frequency of the incident wave is chosen to be 3 GHz with incident angles 1 equal to 45,90 and 135, respectively. For each incident wave, 8 measurements are made at the points equally separated on a semi-circle with the radius of 3 m in region 1. There are 24 measurement points in each simulation. We set the (9) where g i and g i are the scaling factors associated with g g g g the vector differences ( X best X i ) and ( X m X n ),re- spectively. The disturbance vector V due to the mutation mechanism consists of parameter vector X g i,the g best particle X best and two randomly selected vectors. Note that g i and g i are adjusted automatically. In SADDE, the basic idea is to have the control parameters evolve through generations. New vectors are generated by using the evolved values of the control parameters. These new vectors are more likely to survive and produce offspring during the selection procedure. In turn, the survived vectors carry the improved values of the control parameters to the next generation. Perform crossover operation with probability of crossover Crs to deliver crossover vectors: the crossover operation of SADDE algorithm is performed to increase the diversity of the parameter vectors. The crossover operation of SADDE algorithm allows to deliver the crossover vector u i by mixing component of the current vector X g i and the Figure 2. Flowchart for self-adaptive dynamic differential evolution (SADDE).

4 480 Wei Chien et al. generation to be 500 respectively. Number of unknowns for shape function is set to be 7 (i.e., N + 1 = 7). The search range for the unknown coefficient of the shape function is chosen to be from 0 to Our purpose is to reconstruct the shape of the object by using the scattered field at different incident angles. Two examples are investigated for the inverse scattering of the proposed structure by using the SADDE. In the first example, the shape function is chosen to be F( ) = ( cos 2 ) m. The reconstructed shape function for the best population member is plotted in Figure 3 with the shape error shown in Figure 4. The reconstructed shape error is 2%. For investigating the sensitivity of the imaging algorithm against random noise, we added the Gaussian noise to the real and imaginary parts of the simulated scattered fields. Normalized standard deviations of 10 3, 10 2 and 10 1 are used in the simulations. The shape error versus normalized noise level is plotted in Figure 5. It is found that the effect of noise to the shape reconstruction is negligible for normalized standard deviations below The reconstructed result is quite good. Here, the shape function discrepancy is defined as (10) In the second example, the shape function is chosen to be F( ) = ( cos sin 3 ) m. The reconstructed shape function for the best population member is plotted in Figure 6 with the shape error shown in Figure 7. The reconstructed shape error is 5%. For investigating the sensitivity of the imaging algorithm against random noise, we added the Gaussian noise to the real and imaginary parts of the simulated scattered fields. Normalized standard deviations of 10 3,10 2 and 10 1 are used in the simulations. The shape error versus normalized noise level is plotted in Figure 8. It is found that the effect of noise to the shape reconstruction is negligible for normalized standard deviations below Conclusions Figure 3. The reconstructed shape of the cylinder for example 1. The through-wall imaging of the shape reconstruction of perfectly conducting cylinder is investigated by applying SADDE technique, for which the inverse prob- Figure 4. The shape function error in each generation. Figure 5. The trend of relative errors with noise level.

5 Through-wall Imaging of Conductors by Transverse Electric Wave Illumination 481 Figure 8. The trend of relative errors with noise level. Figure 6. The reconstructed shape of the cylinder for example 2. through simulated results. Good reconstruction has been obtained from the scattered fields both with and without the additive Gaussian noise. References Figure 7. The shape function error in each generation. lem is reformulated into optimization one. Based on the boundary condition and the measured scattered field, we have derived a set of nonlinear integral equations and reformulated the imaging problem into an optimization one. The through-wall imaging of the perfectly conducting cylinder in the TE case, the presence of polarization charges makes the inverse problem more nonlinear. As a result, the reconstruction becomes more difficult. Since SADDE realizes the ideas of approaching the self-adaptive. Hidden targets of different profiles are well reconstructed even in the presence of significant amount of noise. The SADDE algorithm has good ability [1] Sabatier, P. C., Theoretical Considerations for Inverse Scattering, Radio Science, Vol. 18, No. 1, pp (1983). doi: /RS018i001p00001 [2] Storn, R. and Price, K., Differential Evolution - a Simple and Efficient Adaptive Scheme for Global Optimization over Continuous Spaces, Technical Report TR , International Computer Science Institute, Berkeley, Mar. (1995). [3] Kennedy, J. and Eberhart, R. C., Particle Swarm Optimization, Proceedings of the IEEE International Conference on Neural Network, pp (1995). doi: /ICNN [4] Rekanos, I. T., Shape Reconstruction of a Perfectly Conducting Scatterer Using Differential Evolution and Particle Swarm Optimization, IEEE Transactions on Geoscience and Remote Sensing, Vol. 46, No. 7, pp (2008). doi: /TGRS [5] Michalski, K. A., Electromagnetic Imaging of Circular-cylindrical Conductors and Tunnels Using a Differential Evolution Algorithm, Microwave and Optical Technology Letters, Vol. 27, No. 5, pp (2000). doi: / ( )27:5<330 ::AID-MOP13>3.0.CO;2-H [6] Lee, Y. H., Cheng, Y. T., Chiu, C. C. and Chang, S. P., Microwave Imaging for Half-space Imperfect Conductors, Nondestructive Testing and Evaluation, Vol. 30, No. 1, pp (2015). (EI) (SCI) doi: /

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