本論文研究自由空間中二維金屬導體柱體的電磁影像重建。此研究以有限時域差分法 (FDTD) 為基礎，利用最佳化方法於時域中重建自由空間中二維金屬導體柱體之特性參數。其中，對於描述形狀的方法，於正散射我們使用傅立葉函數展開(Fourier series expansion) ，並於逆散射中使用三次仿樣函數展開（cubic spline），另外，為了使柱體的形狀更為圓滑我們使用了次網格技術。
為了探究自由空間中未知形狀的金屬導體柱體，概念上吾人可向散射體發射電磁脈波，並量測其周圍的散射電磁波，再針對此量測散射電磁波分別以改良式粒子群聚法（MPSO）、動態差異形演化法（DDE）將逆散射問題轉化為求解最佳化問題。藉由量測而得的散射場以及計算而得的散射場數值互相比較，進而重建介電散射體的形狀函數與位置。
本論文探討上述兩種最佳化方法對於自由空間的二維金屬導體柱體逆散射問題的適用性。模擬結果顯示，即使最初的猜測值與實際散射體位置相距甚遠，此兩種最佳化方法皆可以成功地重建出柱體的位置與形狀。在此兩種最佳化方法收斂速度部份，動態差異型演化法與粒子群聚法可以大幅減少計算正散射次數並減少逆散射問題收斂時間，本研究模擬之數值結果中的金屬物體之電磁特性，可以得到良好的重建結果。

Microwave image problems of a two-dimensional metallic cylinder in free space based on the time-domain technique (finite difference time domain, FDTD)are investigated by modified particle swarm optimization (MPSO) and dynamic differential evolution (DDE).For the forward scattering the FDTD method is employed to calculate the scattered E fields, while for the inverse scattering modified modified particle swarm optimization (MPSO) and dynamic differential evolution (DDE) methods are utilized to determine the shape and location of the cylindrical scatterer with arbitrary cross section. The subgirdding technique is implemented for the FDTD code in order to model the shape of the cylinder more smoothly. In order to describe an unknown cylinder with arbitrary cross section more effectively during the course of searching, the closed cubic-spline expansion is adopted to represent the scatterer contour instead of the frequently used trigonometric series. The former is still used in the forward scattering part. 
    In order to explore the unknown metallic cylinder in free space, an electromagnetic pulse can be conducted to illuminate the cylinder, for which the scattered E fields can then be measured. The inverse problem is then resolved by an optimization approach. The idea is to perform the image reconstruction by utilization of two optimization schemes to minimize the discrepancy between the measured and calculated scattered field data. Modified particle swarm optimization (MPSO) and dynamic differential evolution (DDE) are tested and employed to search the parameter space to determine the shape and location of the metallic cylinder. 
  The suitability and efficiency of applying the above methods for microwave imaging of a 2D metallic cylinder are examined in this thesis. Numerical results show that even when the initial guesses are far away from the exact one, good reconstruction can be obtained by both these optimization methods. These optimization methods are tested by several numerical examples, and it is found that the performance of the MPSO and DDE are robust for reconstructing the metallic cylinder. Numerical results show that satisfactory reconstruction has been obtained.

目錄
中文摘要 ………………………………………………………………Ⅰ
英文摘要 ………………………………………………………………Ⅲ
第一章  簡介	                                       P1
1.1 研究動機與相關文獻	                              P1
1.2 本研究之貢獻	                                       P7
1.3 各章內容簡述	                                       P7
第二章	時域有限差分法	                              P8
2.1 馬克斯威爾方程式	                              P8
2.2 馬克斯威爾方程式於FDTD方法中差分離散實現	           P11
2.2.1 Yee單胞(Yee cell)的空間解析方法與蛙跳式(leap-frog)時間步進計算方法	                                      P11
2.2.2 FDTD更新方程式	                             P12
2.3  數值色散現象與Courant穩定準則	                    P13
2-4 吸收邊界條件(Absorbing Boundary Conditions)          P15
2-5 次網格方法(subgrid FDTD)	                    P16
第三章 改良式粒子群聚法與動態差異型演化法	           P20
3.1改良式粒子群聚最佳化法(Modified Particle Swarm Optimization)...	                                      P20
3.2動態差異型演化法(Dynamic Differential Evolution)     P28
第四章 數值模擬結果	                             P38
4.1模擬環境與相關參數設定	                             P38
4.1.1模擬環境配置與參數設定 	                    P38
4.1.2 散射體形狀描述方法	                             P40
4.1.3 目標函數與最佳化方法搜尋參數             	 P42
4.2最佳化方法重建自由空間中二維金屬導體柱體影像	 P43
4.2.1 以改良式粒子群聚法重建自由空間中二維金屬導體柱體	 P44
4.2.2 以動態差異型演化法重建自由空間中二維金屬導體柱體	 P54
4.2.3 最佳化方法重建自由空間中二維金屬導體柱體收斂速度及柱體影像	                                               P64
第五章 結論	                                      P70
參考文獻                                           	 P72
圖目錄
圖2.1 FDTD中二維Yee單胞於TMz模態(左)與TEz模態(右)表示圖。	11
圖2.2 FDTD中電磁場計算時序圖。	                     12
圖2.3 次網格結構示意圖。	                              18
圖2.4 次網格與大網格的電磁場更新動作時序圖。	            19
圖2.5 次網格方法流程圖。	                              19
圖3.1 粒子群聚法流程圖。	                              21
圖3.2 粒子群聚法中於二維目標函數等位線圖。	            23
圖3.3 二維問題中，三種不同邊界條件示意圖。 與 表示更新後的粒子位置與速度。	                                       25
圖3.4 改良式粒子群聚法流程圖。	                     27
圖3.5 差異型演化法流程圖。	                              29
圖3.6 差異型進化法中突變方法一的示意圖。              	   31
圖3.7 差異型進化法中突變方法二的示意圖。	            32
圖3.8 差異型進化法中突變方法三的示意圖。	            32
圖3.9 差異型進化法中交配向量結構示意圖。	            34
圖3.10 差異型進化法中的交配向量於一個二維目標函數等位線圖描述的示意圖。	                                       35
圖3.11 動態差異型演化策略法流程圖。	                     37
圖4.1 自由空間中任意形狀金屬導體柱體模擬環境示意圖	   39
圖4.2 入射電場波形與頻譜分佈。(a)入射電場時域波形，(b) 入射電場頻譜分佈。	                                       39
圖4.3 三次仿樣函數描述任意形狀散射體示意圖	            41
圖4.4 MPSO重建例子一柱體形狀函數的情形，實線代表真正的形狀函數，其他類型的線條則代表不同的世代中所計算出的形狀函數。	45
圖4.5 MPSO重建例子一柱體的特性參數過程中目標函數隨代數變化圖。	46
圖4.6 MPSO重建例子一柱體的特性參數過程中相對誤差變化趨勢圖。	46
圖4.7 MPSO重建例子一柱體特性參數隨相對雜訊位準變化的情形。	47
圖4.8 MPSO重建例子二柱體形狀的情形，實線代表真正的形狀函數，其他線條類型代表不同的世代中所計算出的形狀函數。	48
圖4.9 MPSO重建例子二柱體的特性參數過程中目標函數隨代數變化圖..............................................49
圖4.10 MPSO重建例子二柱體的特性參數過程中相對誤差變化趨勢圖。	49
圖4.11 MPSO重建例子二柱體特性參數隨相對雜訊位準變化的情形。	50
圖4.12 MPSO重建例子三柱體形狀的情形，實線代表真正的形狀函數，其他線條類型代表不同的世代中所計算出的形狀函數。	51
圖4.13 MPSO重建例子三柱體的特性參數過程中目標函數隨代數變化圖…………………………………………………………52
圖4.14 MPSO重建例子三柱體的特性參數過程中相對誤差變化趨勢圖。	52
圖4.15 MPSO重建例子三柱體特性參數隨相對雜訊位準變化的情形。	53
圖4.16 MPSO重建例子二之柱體影像隨機取5次的目標函數與function calls比較圖	53
圖4.17 DDE重建例子一柱體形狀函數的情形，實線代表真正的形狀函數，其他類型的線條則代表不同的世代中所計算出的形狀函數。	55
圖4.18 DDE重建例子一柱體的特性參數過程中目標函數隨代數變化圖。	56
圖4.19 DDE重建例子一柱體的特性參數過程中相對誤差變化趨勢圖。	56
圖4.20 DDE重建例子一柱體特性參數隨相對雜訊位準變化的情形。	57
圖4.21 DDE重建例子二柱體形狀的情形，實線代表真正的形狀函數，其他線條類型代表不同的世代中所計算出的形狀函數。	58
圖4.22 DDE重建例子二柱體的特性參數過程中目標函數隨代數變化圖	59
圖4.23 DDE重建例子二柱體的特性參數過程中相對誤差變化趨勢圖。	59
圖4.24 DDE重建例子二柱體特性參數隨相對雜訊位準變化的情形。	60
圖4.25 DDE重建例子三柱體形狀的情形，實線代表真正的形狀函數，其他線條類型代表不同的世代中所計算出的形狀函數。	61
圖4.26 DDE重建例子三柱體的特性參數過程中目標函數隨代數變化圖…………………………………………………………..62
圖4.27 DDE重建例子三柱體的特性參數過程中相對誤差變化趨勢圖。	62
圖4.28 DDE重建例子三柱體特性參數隨相對雜訊位準變化的情形。	63
圖4.29  DDE重建例子二之柱體影像隨機取5次的目標函數與function calls比較圖	63
圖4.30 兩種最佳化方法重建葫蘆形影像的目標函數與function calls比較。(a) Linear scale (b) Log scale	65
圖4.31 兩種最佳化方法重建三凹形柱體影像的目標函數與function calls比較。(a) Linear scale (b) Log scale	66
圖4.32  兩種最佳化方法重建四凹形影像的目標函數與function calls比較。(a) Linear scale (b) Log scale	67
圖4.33 兩種最佳化方法重建葫蘆形柱體影像的形狀比較	68
圖4.34 兩種最佳化方法重建三凹形柱體影像的形狀比較	69
圖4.35 兩種最佳化方法重建四凹形柱體影像的形狀比較	69



表目錄

表4.1 最佳化方法重建自由空間中金屬導體散射體相關錯誤率表…….69

[1]	F. Cakoni and D. Colton, “Open problems in the qualitative approach to inverse electromagnetic scattering theory,” European Journal of Applied Mathematics, pp. 1–15, 2004.
[2]	C.E. Baum, Detection and Identification of Visually Obscured Targets, Taylor and Francis, Philadelphia, Oct. 1998.
[3]	B. Borden, Radar Imaging of Airborne Targets, IOP Publishing, Bristol 1999.
[4]	X. Li, S. K. Davis, S. C. Hagness, D. W. van der Weide, and B. D. Van Veen, “Microwave imaging via space-time beamforming: Experimental investigation of tumor detection in multilayer breast phantoms,” IEEE Transactions on Microwave Theory and Techniques, vol. 52, no. 8, pp. 1856–1865, Aug. 2004.
[5]	Q. Fang, P. M. Meaney, and K. D. Paulsen, “Microwave imaging reconstruction of tissue property dispersion characteristics utilizing multiple-frequency information,” IEEE Transactions on Microwave Theory and Techniques., vol. 52, no. 8, pp. 1866–1875, Aug. 2004.
[6]	M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging. Bristol, U.K.: IOP Publishing Ltd., 1998.
[7]	S. Caorsi, A. Massa, M. Pastorino, and A. Rosani, “Microwave medical imaging: Potentialities and limitations of a stochastic optimization technique,” IEEE Transactions on Microwave Theory and Techniques, vol. 52, no. 8, pp. 1908–1916, Aug. 2004.
[8]	A. G. Ramm, “Uniqueness result for inverse problem of geophysics: I,” Inverse Problems, vol. 6, pp. 635-641, Aug.1990.
[9]	V. Isakov, “Uniqueness and stability in multidimensional inverse problems,” Inverse Problems, vol. 9, pp. 579–621, 1993.
[10]	O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: Retrievable information and measurement strategies,” Radio Science, vol. 32, pp. 2123–2138, Nov.–Dec. 1997.
[11]	D. Colton and L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagnetic waves,” Archive for Rational Mechanics and Analysis, vol. 119, pp. 59–70, 1992.
[12]	S. Caorsi, M. Donelli, D. Franceschini, and A. Massa, “A new methodology based on an iterative multiscaling for microwave imaging,” IEEE Transactions on Microwave Theory and Techniques, vol. 51, no. 4, pp. 1162-1173, Apr. 2003.
[13]	M. Bertero and E. R. Pike, Inverse Problems in Scattering and Imaging, ser. Adam Hilger Series on Biomedical Imaging. Bristol, MA: Inst. Phys., 1992.
[14]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996.
[15]	K. Belkebir, R. Kleinmann, and C. Pichot, “Microwave imaging-Location and shape reconstruction from multifrequency data,” IEEE Transactions on Microwave Theory and Techniques, vol. 45, pp. 469–475, April 1997.
[16]	O. M. Bucci, L. Crocco, T. Isernia, and V. Pascazio, “Inverse scattering problems with multifrequency data: Reconstruction capabilities and solution strategies,” IEEE Transactions on Geoscience and Remote Sensing, vol. 38, pp. 1749–1756, July 2000.
[17]	A. Baussard, “Inversion of multi-frequency experimental data using an adaptive multiscale approach,” Inverse Problems, vol. 21, pp. S15–S31, Dec. 2005.
[18]	T. H. Chu and D. B. Lin, “Microwave diversity imaging of perfectly conducting objects in the near-field region,” IEEE Transactions on Microwave Theory Tech., vol. 39, pp. 480-487, March 1991.
[19]	R. Car and C. R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Problems, vol. 10, pp. 1217–1229, 1994.
[20]	Charbonnier P, Blanc-Feraud L, Aubert G, Barlaud M. “Deterministic edge-preserving regularization in  computed  imaging,”  IEEE Transactions on Image  Processing, vol. 6, pp. 298-311, Feb. 1997.
[21]	Sarkar, T.; Weiner, D.; Jain, V., “Some mathematical considerations in dealing with the inverse problem,” IEEE Transactions on Antenna and Propagation., vol. 29, pp. 373-379, March 1981.
[22]	Chaturvedi, P.; Plumb, R.G., “Electromagnetic imaging of underground targets using constrained optimization,” IEEE Transactions on Antennas and Propagation, vol. 33, pp. 551-561, May 1995.
[23]	Y. S. Chung, C. Cheon, and S. Y. Hahn, “Reconstruction of dielectric cylinders using FDTD and topology optimization technique,” IEEE Transactions on Magnetics, vol. 36, no. 4, July 2000.
[24]	M. Moghaddam, W. C. Chew, “Nonlinear two-dimensional velocity profile inversion using time domain data,” IEEE Transactions on Geoscience and Remote Sensing, , vol. 30 , no. 1 , Jan. 1992.
[25]	Wenhua Yu, Zhongqiu Peng and Lang Jen, “A fast convergent method in electromagnetic inverse scattering,” IEEE Transactions on Antennas and Propagation,  vol. 44, no. 11 , Nov. 1996.
[26]	Y. M. Wang and W. C. Chew, “An iterative solution of two-dimensional electromagnetic inverse scattering problem,” International Journal of Imaging Systems Technology, vol. 1, pp. 100-108, 1989.
[27]	O. S. Haddadin and E. S. Ebbini, “Imaging strongly scattering media using a multiple frequency distorted Born iterative method,” IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, vol. 45, pp. 1485–1496, Nov. 1998.
[28]	Salvatore Caorsi, Antonio Costa, and Matteo Pastorino, “Microwave imaging within the second-order Born approximation: stochastic optimization by a genetic algorithm,” IEEE Transactions on Antennas and Propagation, vol. 49, no. 1, pp. 22-31, Jan. 2001.
[29]	R. M. Lewis, “Physical optics inverse diffraction,” IEEE Transactions on Antennas and Propagation, vol. 17, pp. 308-314, 1969.
[30]	N. N. Bojarski, “A survey of the physical optics inverse scattering identity,” IEEE Transactions on Antennas and Propagation, vol. 30, pp. 980-989, Sep. 1982.
[31]	J. B. Keller, “Accuracy and validity of Born and Rytov approximations,” Journal of the Optical Society of America, vol. 59, pp. 1003-1004, 1969.
[32]	K. P. Bube and R. Burridge, “The One-dimensional inverse problem of reflection seismology,” SIAM Review, vol. 25 no. 4, pp. 497-559, 1983.
[33]	J. Sylvester, “On the layer stripping approach to a 1-D inverse problem,” Inverse Problems in Wave Propagation, New York: Springer-Verlag, pp. 453-462, 1997.
[34]	J. Chang, Y. Wang, and R. Aronson, “Layer-stripping approach for recovery of scattering media from time-solved data,” in Inverse Problems in Scattering and Imaging, M. A. Fiddy, Ed. Bellingham, WA: SPIE, pp. 384-397, 1992.
[35]	F. Santosa and H. Schwetlick, “The inversion of acoustical impedance profile by methods of characteristics,” Wave Motion, vol. 4, pp. 99-1101, 1982.
[36]	T. M. Habashy, “A generalized Gel’fand-Levitan-Marchenko integral equation,” Inverse Problems, vol. 7, pp. 703-711, 1991.
[37]	R. F. Harrington, Field Computation by Moment Methods. New York, Macmillan, 1968.
[38]	H. Harada, D. J. Wall, T. Takenaka, M. Tanaka, “Conjugate gradient method applied to inverse scattering problem,” IEEE Transactions on Antennas and Propagation, vol. 43, no. 8, pp. 784-792, Aug. 1995.
[39]	S. Bonnard, P. Vincent, and M. Saillard, “Inverse obstacle scattering for homogeneous dielectric cylinders using a boundary finite-element method”, IEEE Transactions on Antennas and Propagation, vol. 48, no. 3, pp. 393-400, March 2000.
[40]	T. Takenaka, H. Jia, and T. Tanaka, “Microwave imaging of electrical property distributions by a forward-backward time-stepping method,” Journal of Electromagnetic Waves Application, vol. 14, pp. 1609–1625, 2000.
[41]	I.T. Rekanos, “Time-domain inverse scattering using Lagrange multipliers: an iterative FDTD-based optimization technique,” Journal of Electromagnetic Waves and Applications, vol. 17, no. 2, pp. 271-289, 2003.
[42]	S. Bonnard, P. Vincent and M. Saillard, “Cross-borehole inverse scattering using a boundary finite-element method”, Inverse Problems, vol. 14, pp. 521-534, 1998.
[43]	C. C. Chiu and Y. W. Kiang, “Microwave imaging of a Buried cylinder,” Inverse Problems, vol. 7, pp. 182-202, 1991.
[44]	A. Roger, “Newton-Kantorovitch algorithm applied to an electromagnetic inverse problem,” IEEE Transactions on Antennas and Propagation, vol. 29, pp. 232-238, March 1981.
[45]	C. C. Chiu and Y. W. Kiang, “Microwave imaging of multiple conducting cylinders,” IEEE Transactions on Antennas and Propagation, vol. 40, pp. 933-941, Aug. 1992.
[46]	F Hettlich, “Two methods for solving an inverse conductive scattering problem,” Inverse Problems, vol. 10, pp. 375-385, 1994.
[47]	A. Kirsch, R. Kress, P. Monk and A. Zinn, “Two methods for solving the inverse acoustic scattering problem,” Inverse Problems, vol. 4, pp.749-770, Aug. 1988.
[48]	S. Gutman and M. Klibanov, “Iterative method for multi-dimensional inverse scattering problems at fixed frequencies,” Inverse Problems, vol. 10, pp.573-599, Aug. 1984.
[49]	W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Transactions on Medical Imaging, vol. 9, pp. 218-225, 1990.
[50]	W. C. Chew and G. P. Otto, “Microwave imaging of multiple conducting cylinders using local shape functions,” IEEE Microwave Guided Wave Letters, vol. 2, issue 7 pp. 284-286, July 1992.
[51]	W. H. Weedon and W. C. Chew, “Time-domain inverse scattering using the local shape function (LSF) method,” Inverse Problems, vol. 9, pp.551-564, Oct. 1993.
[52]	C. C. Chiu and W. T. Chen, "Electromagnetic imaging for an imperfectly conducting cylinder by the genetic algorithm," IEEE Transactions on Microwave Theory and Techniques, vol. 48, Nov. 2000.
[53]	S. Caorsi, A. Massa, and M. Pastorino, “A computational technique based on a real-coded genetic algorithm for microwave imaging purposes”, IEEE Transactions on Geoscience and Remote Sensing, vol. 38, no. 4, pp. 1697-1708, July 2000.
[54]	C. C. Chiu, C. L. Li and W. Chien, “Image reconstruction of a buried conductor by the genetic algorithm, ” IEICE Transaction on Electronics, vol. E84-C, no. 7, pp. 961-966, Dec. 2001.
[55]	X.-M. Zhong, C Liao and W. Chen, “Image reconstruction of arbitrary cross section conducting cylinder using UWB pulse,” Journal of Electromagnetic Waves Application,, vol. 21, no. 1, pp. 25-34, 2007.
[56]	C. H. Huang, C. C. Chiu, C. L. Li, and Y. H. Li, “Image reconstruction of the buried metallic cylinder using FDTD method and SSGA,” Progress In Electromagnetics Research, PIER 85, 195-210, 2008.
[57]	C. H. Huang, S. H. Chen, C. L Li and C. C. Chiu, “Time domain inverse scattering of an embedded cylinder with arbitrary shape using nearly resonant technique,” 2004 International Conference on Electromagnetic Applications and Compatibility, Taipei, Taiwan, Oct. 2004.
[58]	M. Donelli and A. Massa, ,”Computational approach based on a particle swarm optimizer for microwave imaging of two-dimensional dielectric scatterers,” IEEE Transactions on Microwave Theory and Techniques, vol. 53,  issue 5, pp. 1761 – 1776, May 2005
[59]	T. Huang and A. S. Mohan, ” Application of particle swarm optimization for microwave imaging of lossy dielectric objects,” IEEE Transaction on Antennas and Propagation, vol. 1B,  pp. 852 - 855 ,Dec. 2005.
[60]	M. Donelli and G. Franceschini, A. Martini, A. Massa,” An integrated multiscaling strategy based on a particle swarm algorithm for inverse scattering problemsm,” IEEE Transactions on Geoscience and Remote Sensing, vol. 44,  issue 2,  pp. 298 – 312, Feb. 2006
[61]	G. Franceschini, M. Donelli, R. Azaro and A. Massa,;” Inversion of phaseless total field data using a two-step strategy based on the iterative multiscaling approach,” IEEE Transactions on Geoscience and Remote Sensing. vol. 44, issue 12, pp. 3527 - 3539, Dec. 2006.
[62]	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 Electromagnetics Research, PIER 82, pp. 381-400, 2008.
[63]	A. Qing, “Electromagnetic inverse scattering of multiple two-dimensional perfectly conducting objects by the differential evolution strategy,” IEEE Transactions on Antennas and Propagations, vol. 51, Issue 6, pp. 1251-1262, June 2003.
[64]	A. Qing, “Electromagnetic inverse scattering of multiple perfectly conducting cylinders by differential evolution strategy with individuals in groups (GDES),” IEEE Transactions on. Antennas and Propagations, vol. 52, issue 5, pp. 1223-1229, May 2004.
[65]	A. Qing, “Dynamic differential evolution strategy and applications in electromagnetic inverse scattering problems,” IEEE Transactions on Geoscience and Remote Sensing, vol 44, issue 1, pp.116 - 125, Jan. 2006.
[66]	D. Cherubini, A. Fanni, A. Montisci and P. Testoni, “Inversion of MLP neural networks for direct solution of inverse problems,” IEEE Transactions on Magnetics, vol. 41, issue 5, pp. 1784-1787, May 2005.
[67]	A. K. Hamid and M. Alsunaidi, “Inverse scattering by dielectric circular cylindrical scatterers using a neural network approach,” 1997 IEEE international symposium Antennas Propagation, Montreal, QC, Canada, pp. 2278-2281. July 1997, 
[68]	F. C. Morabito, A. Formisano and R. Martone, “Wavelet tools for improving the accuracy of neural network solution of electromagnetic inverse problems,” IEEE Transactions on Magnetics, vol. 34, pp. 2968-2971, May 1998.
[69]	T. Melamed, E. Heyman and L. B. Felsen, “Local spectral analysis of short-pulse excited scattering from weakly inhomogeneous media. II. inverse scattering”, IEEE Transactions on Antennas and Propagation, vol. 47, no. 7, July 1999.
[70]	S. Gutmant and M. Klibanov, “Three-dimensional inhomogeneous media imaging,” Inverse Problems, vol. 10, pp. 39-49, Aug. 1994.
[71]	A. Taflove and S. Hagness, “Computational Electrodynamics: The Finite-Difference Time-Domain Method,” Artech House, Boston, MA, 2000.
[72]	J. P. Benerger, “A perfectly matched layer for the absorption of electromagnetic waves,”Journal of Computational Physics, vol. 114, pp 185-200, 1994.
[73]	 Z. S Sacks, D. M. Kingsland, R. Lee, and J. F. Lee, “A perfectly matched anisotropic absorber for use as absorbing boundary condition,” IEEE Transactions on Antennas and Propagation, vol. 43, pp 1460- 1463, Dec. 1995.
[74]	C.  L. Li, C. W. Liu and S. H. Chen, “Optimization of a PML absorber's conductivity profile using FDTD,” Microwave and Optical Technology Letters, vol. 37 no. 5, pp. 69-73 , June 2003.
[75]	M. W. Chevalier, R. J.Luebbers and V. P. Cable, “FDTD local grid with materical traverse,” IEEE Transactions on Antennas and Propagation, vol. 45, no. 3, March 1997.
[76]	R. Storn and K. Price, “Differential Evolution – A Simple and Efficient Heuristic for Global Optimization over Continuous Space,” Journal of Global Optimization, vol. 11, pp. 341-359, 1997.
[77]	M. Clerc, J. Kennedy, “The particle swarm-explosion, stability, and convergence in a multidimensional complex space,” IEEE Transactions on Evolutionary Computation, vol. 6, issue 1. pp. 58~73, 2002
[78]	A. Carlisle and G. Dozier, “An off-the-shelf PSO,” Proc. of the Workshop on Particle Swarm Optimization, Indianapolis, April 2001.
[79]	T. Huang and A. S. Mohan, “A hybrid boundary condition for robust particle swarm optimization,” IEEE Antennas and Wireless Propagation Letters, vol. 4, pp. 112-117, 2005.
[80]	C. de Boor, A Practical Guide to Splines, Springer-Verlag, New York, 1978.