本論文探討子空間演算法(Subspace-based algorithm)應用於自由空間中二維介質物體之逆散射問題。處理逆散射問題的方法中，子空間演算法特別不同的地方在於計算上使用到奇異值分解(Singular value decomposition , SVD)，運用子空間的概念，將感應電流分成確定性部分及不確定性部分，確定性部分對於逆散射提供良好初始猜測值，逆散射只針對不確定性部分做運算及最佳化，這部分是子空間演算法的精華，可以在計算上減少未知數的數量，有效降低計算成本及簡化計算過程。最佳化演算法方面再使用自我適應之動態差異型演化法(Self-Adaptive Dynamic Differential Evolution, SADDE)，避免像使用共軛梯度法(Conjugate Gradient method, CG)或牛頓法(Newton's Method)會容易陷入區域極值的問題，雖然避免了區域極值的問題，但計算時間卻會增加，因此利用子空間演算法本身簡化計算降低成本的優點再配合SADDE之強健性和搜尋速度，收斂至更佳的結果，並增加對雜訊的抗性。此外同時比較子空間演算法分別搭配SADDE和基因演算法(Genetic Algorithm, GA)之結果顯示子空間演算法在演算法方面搭配SADDE有較佳的重建結果。另外進一步討論子空間演算法對於複雜非均勻介電物體的重建及對於雜訊的優良抗性，研究模擬之數值結果顯示，此數值方法運用於重建複雜非均勻介電物體之材質特性分佈，皆能得到良好之重建結果，且無論加入雜訊等級的大小，儘管已使數據與正確值相差甚大，皆能藉由子空間演算法之參數調整，收斂至更良好之重建結果。

This thesis presents the two-dimensional electromagnetic imaging problem by Subspace-based algorithm. Subspace-based algorithm is different with methods of processing inverse scattering problem by contrast source inversion (CSI). The essence of the subspace-based optimization method is that part of the contrast source is determined from the spectrum analysis without using any optimization when the rest is determined by optimization method. By applying the singular value decomposition (SVD) to the field equation, the induced current is divided into the signal space and the noise space. This feature can reduce the number of unknowns and computing costs to speed up the convergence of the algorithm. We also transform the inverse scattering problem into optimization problem and solved by Self-Adaptive Dynamic Differential Evolution (SADDE). SADDE can process numerous unknowns of electromagnetic imaging problems. Different scatterers and environment will be used to investigate whether Subspace-based algorithm can keep stability of reconstruction or not. We will also compare Genetic Algorithm (GA) to show the robustness and the searching speed of SADDE.

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	11
1.3 各章內容簡述	12
第二章 非均勻介電物體之電磁成像	13
2.1 理論推導	13
2.2 數值方法	16
2.2.1 動差法於積分方程式之應用	16
2.2.2 散射場之計算與驗證	18
第三章  子空間演算法	23
第四章  隨機式全域最佳化演算法	28
4.1 自我適應之動態差異型演化法(Self-Adaptive Dynamic Differential Evolution)	28
4.2 基因演算法 (Genetic Algorithm)	37
第五章 數值分析及模擬結果	49
第六章 結論	76
參考文獻	78
圖目錄
圖2.1  二維介質物體在自由空間的示意圖	15
圖2.2  推導圓柱形均勻介電物體之解析散射場值示意圖	21
圖2.3  模擬正散射場值驗證之示意圖	22
圖4.1  自我適應之動態差異型進化法流程圖	30
圖4.2  自我適應之動態差異型進化法中突變方法一的示意圖	32
圖4.3  自我適應之動態差異型進化法中突變方法二的示意圖	33
圖4.4  自我適應之動態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4.5  遺傳演算法之流程圖	39
圖5.1  模擬之環境結構圖	50
圖5.2  例子一之介電係數分佈圖。 (a) (b) (c)	53
圖5.2  例子一之介電係數分佈圖。 (d) (e) (f)	54
圖5.3  矩陣G2之特徵值	55
圖5.4  (a)不同L值之目標函數值與呼叫函數次數比較圖	56
圖5.4  (b) SADDE和GA之目標函數值與呼叫函數次數比較圖	57
圖5.5  例子二之介電係數分佈圖。 (a) (b) (c)	60
圖5.5  例子二之介電係數分佈圖。(d)	61
圖5.6  SADDE和GA之目標函數值與呼叫函數次數比較圖	62
圖5.7  例子三之介電係數分佈圖。(a) (b)	64
圖5.8  子空間演算法配合SADDE加入不同雜訊等級之重建結果。  (a) (b)	66
圖5.8  子空間演算法配合SADDE加入不同雜訊等級之重建結果。  (c) (d)	67
圖5.9  加入雜訊等級0.1%並選擇以不同L值之重建結果。         (a) (b)	69
圖5.9  加入雜訊等級0.1%並選擇以不同L值之重建結果。         (c) (d)	70
圖5.10 例子四之介電係數分佈圖。(a) (b)	73
圖5.10 例子四之介電係數分佈圖。(c) (d)	74
圖5.11  SADDE和GA之目標函數值與呼叫函數次數比較圖	75


表目錄
表4.1　基因演算法相關之名詞解釋與中英對照表	47

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