本論文研究存在於自由空間中二維金屬導體的頻域電磁逆散射問題。在正散射的分析部分，本研究以動差法(Method of Moments, MoM)為基礎，至於逆散射則被轉換為最佳化問題以進行求解，吾人使用非同步粒子群聚法(APSO)、非同步粒子群聚法結合平滑變化機制(APSO+Smoothvary)以及非同步粒子群聚法結合直交表和平滑變化機制(OA+APSO+Smoothvary)所求的解做比較。  　
　　本論文先探討非同步粒子群聚法、非同步粒子群聚法結合平滑變化機制與非同步粒子群聚法結合直交表和平滑變化機制在不同學習因子的情況下，以九種具不同特性之測試函數予以測試，維度方面皆設定為10維。每個測試函數最佳的學習因子和成功率不盡相同，經過實驗結果可以發現學習因子c1=2.8和c2=1.3是一組普遍通用的學習因子，在引進平滑變化機制之後，讓成功率大幅提昇之外，在收斂深度方面來的更深。
   另外，直交表因為具有均勻分布的特性，將其結合並應用在逆散射問題時，可展現出其優越性，例如在初始最佳物種、收斂深度和圖形重建方面，皆有明顯的改善。為此，我們可看到針對二維金屬導體的逆散射問題，結合直交表及/或平滑變化機制之後皆可以有良好的改善。

This thesis studies the electromagnetic inverse scattering problem in frequency domain for a two-dimensional inhomogeneous dielectric cylinder located in free space. The analysis of forward scattering part is based on the Method of Moments (MoM) , while the inverse scattering part is tackled by transforming the problem into an optimization one, of which the asynchronous particle swarm optimization(APSO) is chosen. The reconstructed results by APSO are compared with those obtained by  APSO plus certain kind of smooth variation for the control parameter and/or associated with the orthogonal array (OA).

At first, the convergence speed and results for nine benchmarked functions (with dimension 10) are tested as the control parameters are varied for the algorithm of APSO. It is found that not only the best values of the control parameters are different for each function, but also the best success rates are. Nevertheless, we do find one set of the learning factor, c1=2.8 and c2=1, that is suitable for all benchmarked functions tested in general. Then the introduction of certain kind of smooth variation for the control parameters is tested during the course of searching procedure. It is found that the mechanism of smooth variation for the control parameters can increase the success rate significantly in addition to the convergence depth. 
Finally, since the orthogonal array exhibits the characteristic of uniform appearance for each level of the experimental parameters, it do reveal its superiority as combined and applied for the inverse scattering problem. For examples, the initial best particle, convergence depth and reconstructed results can be significant improved. It is concluded that for the inverse scattering problem of a two-dimensional metallic conductor, the inclusion of orthogonal array and the mechanism of smooth variation of the control parameters is a helpful technique.

目錄
第一章 簡介	1
1.1 研究動機與相關文獻	1
1.2 本研究之貢獻	8
1.3 各章內容簡述	8
第二章  最佳化演算法	10
2.1直交表	10
2.2非同步粒子群聚最佳化法	12
2.3平滑變化機制	17
2.4最佳化方法測試	17
第三章 頻域自由空間中二維金屬導體影像重建	61
3.1自由空間理論推導與數值方法	61
3.2數值模擬結果	65
  3.3以非同步粒子群聚法重建自由空間中二維金屬柱體影像
 	65
  3.4非同步粒子群聚法重建自由空間中二維金屬柱體影像之討論
 	81
第四章 結論	82
參考文獻	84

















圖目錄
圖2.1改良式粒子群聚法流程圖	14
圖2.2二維問題空間中，PSO的邊界條件示意圖	16
圖2.3學習因子C1和C2的平滑變化示意圖	18
圖2.4九種驗證函數的圖形	18
圖2.5應用APSO及其衍生算法於搜尋SPHERE函數之收斂測試	26
圖2.6應用APSO及其衍生算法於搜尋AXIS PARALLEL HYPER-ELLIPSOID函數之收斂測試	27
圖2.7應用APSO及其衍生算法於搜尋Quadric函數之收斂測試	28
圖2.8應用APSO及其衍生算法於搜尋Griewank函數之收斂測試
	29
圖2.9應用APSO及其衍生算法(p=16)於搜尋Rosenbrock函數之收斂測試
	30
圖2.10應用APSO及其衍生算法(P=64)於搜尋ROSENBROCK函數之收斂測試
	31
圖2.11應用APSO及其衍生算法於搜尋Ackley函數之收斂測試
	32
圖2.12應用APSO及其衍生算法於搜尋Generalized Schwefel’s Problem函數之收斂測試
	33
圖2.13應用APSO及其衍生算法於搜尋RASTRIGIN函數之收斂測試
	34
圖2.14應用APSO及其衍生算法於搜尋Weierstrass函數之收斂測試
	36
圖2.15應用APSO及其衍生算法於多種驗證函數之混合比較
	38
圖3.1二維完全導體在 平面上的示意圖
	64
圖3.2 (a) 例子一 f=0.03的適應值曲線變化圖(
使用APSO)	67
圖3.2 (b) 例子一 f=0.03的適應值曲線變化圖(
使用APSO+smoothvary） 
	68
圖3.2 (c) 例子一 f=0.03的適應值曲線變化圖(
使用APSO+smoothvary+OA）	68
圖3.3 (a) 例子一 f=0.03的形狀錯誤率曲線變化圖(使用APSO）
	68
圖3.3 (b) 例子一 f=0.03的形狀錯誤率曲線變化圖(使用APSO+smoothvary）
	69
圖3.3 (c) 例子一 f=0.03的形狀錯誤率曲線變化圖(使用APSO+smoothvary+OA） 
	69

圖3.4例子一 f=0.03的兩條最佳適應值曲線之比較圖(分別使用apso+smoothvary和apso+smoothvary+OA)   
	70
圖3.5例子一 f=0.03的兩條形狀錯誤率曲線之比較圖(分別使用apso+smoothvary和apso+smoothvary+OA)  
	70
圖3.6針對例子一f=0.03，在c1=2.8to0.1下的影像重建圖(分別使用apso+smoothvary和apso+smoothvary+oa)  
	71
圖3.7(a) 例子二 f=0.03+cos3(θ)的適應值曲線變化圖(
使用APSO）
	75
圖3.7 (b) 例子二 f=0.03+cos3(θ)的適應值曲線變化圖(
使用APSO+smoothvary）
	75
圖3.7 (c) 例子二 f=0.03+cos3(θ)的適應值曲線變化圖(
使用APSO+smoothvary+oa）
	75
圖3.8 (a) 例子二 f=0.03+cos3(θ)的形狀錯誤率曲線變化圖(
使用APSO）
	76
圖3.8 (b) 例子二 f=0.03+cos3(θ)的形狀錯誤率曲線變化圖(
使用APSO+smoothvary）
	76
圖3.8 (c) 例子二 f=0.03+cos3(θ)的形狀錯誤率曲線變化圖(
使用APSO+smoothvary+oa）
	76
圖3.9例子二 f=0.03+cos3(θ)的兩條適應值收斂最快曲線之比較圖(分別使用apso+smoothvary和apso+smoothvary+OA) 
	77
圖3.10例子二 f=0.03+cos3(θ)的兩條形狀錯誤率曲線之比較圖—在 c1=2.8to0.1的情況下(分別使用apso+smoothvary和apso+smoothvary+oa）
	77
圖3.11針對例子二f=0.03+cos3θ，在c1=2.8to0.1下的影像重建圖(分別使用apso+smoothvary和apso+smoothvary+oa)
	78
圖3.12形狀誤差率隨雜訊位準的變化模擬	78
圖3.13針對例子三，分別使用apso、apso+smoothvary和apso+smoothvary+oa三種方法的適應值曲線之比較圖	81
圖3.14針對例子三，分別使用apso、apso+smoothvary和apso+smoothvary+oa三種方法的形狀錯誤率曲線之比較圖	80
圖3.15針對例子三，在c1=2.7下的影像重建圖(使用apso）	80
圖3.16針對例子三，在c1= c1=2.8to0.1下的影像重建圖(使用apso+smoothvary)	80
圖3.17針對例子三，在c1= c1=2.8to0.1下的影像重建圖(使用apso+smoothvary+oa)	81

 

表目錄
表2.1 田口直交表OA(18,5,3,2)	11
表2.2 九種驗證函數列表	19
表2.3 FTN1到FTN9的收斂速度排名列表（當C1值改變）	58
表2.4 FTN1到FTN9的成功率排名列表（當C1值改變）	59
表2.5 C1=2.8TO0.1(APSO+SMOOTH VARY and/or OA)和C1=2.8(APSO)的收斂速度和成功率比較表(FTN1~FTN9)
	59

[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]	A. Massa, M.Pastorino, A. Rosani and M. Benedetti “A Microwave Imaging Method for NDE/NDT Based on the SMW Technique for the Electromagnetic Field Prediction,” IEEE Transactions on Instrumentation and Measurement, Vol. 55, No. 1, pp.240 - 247, Feb. 2006.
[7]	O. Mudanyalı, S. Yıldız, O. Semerci, A. Yapar, and I. Akduman, “A Microwave Tomographic Approach for Nondestructive Testing of Dielectric Coated Metallic Surfaces.”, IEEE Geoscience and Remote Sensing Letters, Vol. 5, No. 2, pp. 180 - 184, Apr. 2008.
[8]	T. Rubak, O. S. Kim, P. Meincke, “Computational Validation of a 3-D Microwave Imaging System for Breast-Cancer Screening,” IEEE Transactions on Antennas and Propagation, vol. 57, No. 7, Jul. 2009.
[9]	M. Klemm, J. A. Leendertz, D. Gibbins, I. J. Craddock, A. Preece, R. Benjamin, “Microwave Radar-Based Breast Cancer Detection: Imaging in Inhomogeneous Breast Phantoms” IEEE Antennas and Wireless Propagation Letters, Vol. 8, 2009.
[10]	J. Bourqui, M. Okoniewski, E. C. Fear, “Balanced Antipodal Vivaldi Antenna With Dielectric Director for Near-Field Microwave Imaging.”, IEEE Transactions on Antennas and Propagation, Vol. 58, No. 7, Jul 2010.
[11]	Moghaddam, and M.; Chew, W.C, “Nonlinear two-dimensional velocity profile inversion using time domain data, ” IEEE Transactions on Geoscience and Remote Sensing, , Vol. 30 , No. 1 , Jan. 1992.
[12]	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.
[13]	Wenhua Yu and Raj Mittra, “An improved method for the reconstruction of lossy dielectric objects,” Microwave and Optical Technology Letters, Vol. 15, No. 5, August 1997
[14]	Milica Popovic’ and Allen Taflove, “Two-Dimensional FDTD Inverse-Scattering Scheme for Determination of Near-Surface Material Properties at Microwave Frequencies,” IEEE Transactions on Antennas Propagation, Vol. 52, No. 9, Spet. 2004
[15]	T. Moriyama, Z. Meng, and T. Takenaka, "Forward-backward time-stepping method combined with genetic algorithm applied to breast cancer detection", Microwave and Optical Technology Letters, Vol. 53, No. 2, pp.438-442, 2011.
[16]	R. Persico, R. Bernini, and F. Soldovieri, “The Role of the Measurement Configuration in Inverse Scattering From Buried Objects Under the Born Approximation,” IEEE Transactions on Antennas and Propagation, Vol. 53, No.6, pp. 1875-1887, Jun. 2005.
[17]	X. Chen, K. Huang and X.-B. Xu, “Microwave imaging of buried inhomogeneous objects using parallel genetic algorithm combined with FDTD method:” Progress In Electromagnetic Research. PIER 53, pp. 283-298, 2005.
[18]	‘A. Massa, D. Franceschini, G. Franceschini, M. Pastorino, M. Raffetto, and M. Donelli, “Parallel GA-Based Approach for Microwave Imaging Applications,” IEEE Transaction on Antennas and Propagation, Vol. 53, No. 10, pp. 3118 - 3127, Oct. 2005.
[19]	R A. Wildman and D S. Weile, “Greedy Search And A Hybrid Local Optimization/Genetic Algorithm For Tree-Based Inverse Scattering,” Microwave and Optical Technology Letters, Vol. 50, No. 3, pp. pp. 822-825, Mar. 2008.
[20]	A. Saeedfar, and K. Barkeshli, “Shape reconstruction of three-dimensional conducting curved plates using physical optics, number modeling, and genetic algorithm, ” IEEE Transaction on Antennas and Propagation, Vol. 54, No. 9, 2497-2507, Sep. 2006.
[21]	A. Semnani, I.T. Rekanos, M. Kamyab, T.G. Papadopoulos, “Two-Dimensional Microwave Imaging Based on Hybrid Scatterer Representation and Differential Evolution,” IEEE Transaction on Antennas and Propagation, Vol. 58, No. 10, pp. 3289 - 3298, Oct. 2010.
[22]	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
[23]	K. A. Michalski, “Electromagnetic Imaging of Circular-Cylindrical Conductors and Tunnels Using A Differential Evolution Algorithm,”   Microwave and Optical Technology Letters, Vol. 27, No. 5, pp. 330 - 334, Dec. 2000.
[24]	I. T. Rekanos, “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. 1967-1974, Jul. 2008.
[25]	A. Semnani and M. Kamyab, “An Enhanced Hybrid Method for Solving Inverse Scattering Problems,” IEEE Transactions on Magentics, Vol. 45, No. 3, pp. 1534-1537, Mar. 2009.
[26]	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, No.12, pp. 3527-3539, Dec. 2006.
[27]	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.
[28]	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, 2005.
[29]	M. Donelli, G.. Franceschini, A. Martini and A. Massa,” An integrated multiscaling strategy based on a particle swarm algorithm for inverse scattering problems” IEEE Transactions on Geoscience and Remote Sensing, Vol 44,  Issue 2, pp.298 – 312, Feb. 2006.
[30]	M. Donelli, D. Franceschini, P. Rocca and A. Massa,” Three-Dimensional Microwave Imaging Problems Solved Through an Efficient Multiscaling Particle Swarm Optimization” IEEE Transactions on Geoscience and Remote Sensing, Vol 47, No. 5, pp.1467 – 1481, May. 2009.
[31]	Y. Xia, G. Feng and J. Wang, “A Novel Recurrent Neural Network for Solving Nonlinear Optimization Problems With Inequality Constraints”, IEEE Transactions on Neural Network, Vol. 19, No. 8, pp. 1340 – 1353, Aug. 2008.
[32]	A. Awada, B. Wegmann, I. Viering, and A. Klein, ” Optimizing the Radio Network Parameters of the Long Term Evolution System Using Taguchi’s Method”, IEEE Transactions on vehicular technology, Vol. 19, No. 8, pp. 3825-3839, Oct. 2011.
[33]	M. Ashabani, Y. A.-R. I. Mohamed, and J. Milimonfared, “Optimum Design of Tubular Permanent-Magnet Motors for Thrust Characteristics Improvement by Combined Taguchi–Neural Network Approach” IEEE Transactions on Magnetics, Vol. 46, No. 12, pp. 4092-4100, Dec. 2010.
[34]	J. H. Ko, J. K. Byun, J. S. Park, and H. S. Kim, “Robust Design of Dual Band/Polarization Patch Antenna Using Sensitivity Analysis and Taguchi's Method” IEEE Transactions on Magnetics, Vol. 47, No, 5, pp. 1258-1262, May 2011.
[35]	J. C. Y. Huang," Reducing Solder Paste Inspection in Surface-Mount Assembly Through Mahalanobis–Taguchi Analysis", IEEE Transactions on Electronics Packaging Manufacturing, Vol. 33, No. 4, pp. 265-274, Oct. 2010
[36]	A. G. Ramm, “Uniqueness result for inverse problem of geophysics: I,” Inverse Problems, Vol. 6, pp. 635-641, Aug.1990.
[37]	V. Isakov, “Uniqueness and stability in multidimensional inverse problems,” Inverse Problems, Vol. 9, pp. 579–621, 1993.
[38]	O. M. Bucci and T. Isernia, “Electromagnetic inverse scattering: Retrievable information and measurement strategies,” Radio Sci., Vol. 32, pp. 2123–2138, Nov.–Dec. 1997.
[39]	D. Colton and L. Paivarinta, “The uniqueness of a solution to an inverse scattering problem for electromagnetic waves,” Arc. Ration. Mech. Anal., Vol. 119, pp. 59–70, 1992.
[40]	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..
[41]	M. Bertero and E. R. Pike, Inverse Problems in Scattering and Imaging, ser. Adam Hilger Series on Biomedical Imaging. Bristol, MA: Inst. Phys., 1992.
[42]	A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems. New York: Springer-Verlag, 1996.
[43]	A. M. Denisov, Elements of Theory of Inverse Problems. Utrecht, The Netherlands: VSP, 1999.
[44]	A. E. Eiben, R. Hinterding, and Z. Michalewicz, “Parameter control in evolutionary algorithms,”, IEEE Transactions on Evolutionary Computation, Vol. 3, No. 2, pp.124–141, Jul. 1999.
[45]	A. E. Eiben, R. Hinterding, and Z. Michalewicz, “Parameter control in evolutionary algorithms,”, IEEE Transactions on Evolutionary Computation, Vol. 3, No. 2, pp.124–141, Jul. 1999.
[46]	D. S. Weile and E. Michielssen, “Genetic algorithm optimization applied to electromagnetics: a review ,” IEEE Transactions on Antennas and Propagation, Vol. 45, No. 3, pp. 343- 353, Mar. 1997.
[47]	J. Robinson and Y. Rahmat-Samii, “Particle swarm optimization in electromagnetics,” IEEE Transactions on Antennas and Propagation, Vol. 52, No. 3, pp. 397–407, Feb. 2004.
[48]	P. Rocca, G. Oliveri, and A. Massa,“Differential Evolution as Applied to Electromagnetics ,” IEEE Antennas and Propagation Magazine, Vol. 53, No. 1, pp. 38–49, May. 2011.
[49]	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.
[50]	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.
[51]	N. J. A. Sloane, “A library of orthogonal array,” [Online]. Available: http://www.research.att.com/~njas/oadir/.
[52]	F. B. Hildebrand, Methods of Applied Mathematics, New Jersey:    Prentice-Hall, 1965.
[53]	T. B. A. Senior, “Approximation boundary conditions,” IEEE Trans. Antennas Propagat., vol. AP-29, pp. 826-829, Sept. 1981.
[54]	F. M. Tesche, “On the inclusion of loss in time domain solutions of electromagnetic interaction problems,” IEEE Trans. Electromagn. Compat., vol. EMC-32, pp. 1-4, Feb. 1990.