本論文擬模擬研究雙軸性掩埋在半空間與三層空間中複雜柱體(即包含雙軸性介質物體與完全導體)的電磁影像重建。設一未知的不均勻雙軸性介質複雜物體掩埋在其中一空間中(不論是在半空間或三層空間中)，吾人可在另外的空間中適當的安排一組具有不同入射和極化方向的無關聯波照射物體並量測在此之散射場，利用簡單的矩陣運算，我們就可以克服非線性和不良情況的發生的困擾，進而重建雙軸性複雜物體的介電常數分佈。在理論部份，主要是根據邊界條件導出一組非線性的積分方程組，接著利用以動差法與無關聯照射法計算其散射場，再根據電磁成像法則，重建出介質物體內部的介電常數。在數值結果方面，將證明理論部份的正確性。此結果亦顯示即使物體的介電常數很大時，我們能成功的重建介電常數的分佈。而且即使在在量測的散射場中有高斯雜訊的存在，依然可以得到良好的重建結果。除此之外，我們也會在文中探討雜訊對重建結果的影響程度。

In this thesis, inverse scattering of a biaxial complex cylinder which is buried in half space and three layers structure is investigated. Dielectric cylinders of unknown permittivities are buried in one space and scatter a group of unrelated waves incident from another space where the scattered field is recorded. By proper arrangement of the various unrelated incident fields, the difficulties of ill-posedness and nonlinearity are circumvented, and the permittivity distribution can be reconstructed through simple matrix operations. For theoretical formulation, based on the boundary condition, a set of integral equations is derived and solved by the moment method as well as the unrelated illumination method. Numerical results show that the permittivity tensor distribution of the materials can be successfully reconstructed even when the permittivity is fairly large. Good reconstruction is obtained even in the presence of additive Gaussian noise in measured data. In addition, the effect of noise on the reconstruction result is also investigated.

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………………………………………………7
　　1.3　內容簡述……………………………………………………8
第二章　電磁成像理論…………………………………………...…..9
　　2.1　理論推導…………………………………………………...10
    2.1.1 正散射的理論推導 ……………………………………….10
2.2　數值方法……………………………………………………16
         2.2.1 動差法於求解積分方程式之應用 …………………16
         2.2.2應用無關聯照射法於逆散射問題……………………20
    2.3 數值模擬結果   ……………………………………………26
    2.4 結 論  ………………………………………………………39
第三章 三層介質中雙軸性介質物體之逆散射………………….…..40
　　3.1　理論推導…………………………………………………...41
3.2　數值方法……………………………………………………48
         3.2.1 動差法於求解積分方程式之應用 …………………48
         3.2.2 應用無關聯照射法於逆散射問題 …………………50
    3.3 數值模擬結果   ……………………………………………55
    3.4 結 論  ………………………………………………………68

第四章　本論文之總結………………………………………..…..…..69
　
附錄一  計算半空間格林函數的方法…………………..……………70
附錄二  矩陣[G1]~[G19]元素的計算…………………….……………73
附錄三  計算三層空間格林函數的方法 ……………………………77
附錄四  矩陣[G20]~[G27]元素的計算…………………………………83
參 考 文 獻  ………………………………………………….…...…85
Publication of C. J. Lin    …………………………………………… 92

圖目錄 圖2-1 在半空間中，雙軸性複雜物體在X-Y平面上的示意圖 ..….24 
圖2-2 無關聯照射法之波束聚焦法示意圖 ………………….……...25 
圖2-3 第一個例子(長方形)的原始介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε)……………………………………..…29 
圖2-3 第一個例子(長方形)的原始介電常數分佈 (c) ),(yx 3 ε ……………………………………………..……...30 
圖2-4 第一個例子(長方形)的重建介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε……………………….………………..31 
圖2-4 第一個例子(長方形)的重建介電常數分佈 (c) ),(yx 3 ε….………………………………………………...32 
圖2-5 第二個例子(正方形)的原始介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε………………………….…………….33 
圖2-5 第二個例子(正方形)的原始介電常數分佈 (c) ),(yx 3 ε…………………………………………………….34 
圖2-6 第二個例子(正方形)的重建介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε…...……………………………………35 
圖2-6 第二個例子(正方形)的重建介電常數分佈 (c) ),(yx 3 ε……………………………………………………...36
圖2-7 第一個例子(長方形)介電常數的重建誤差對雜訊階的模擬結果…………………………………………………………….37 
圖2-8 第二個例子(正方形)介電常數的重建誤差對雜訊階的模擬結果…………….………………………………………………38 
圖3-1 在三層空間中，雙軸性介質物體在X-Y平面上的示意圖………………………………………………………………41 
圖3-2 第一個例子(長方形)的原始介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε………………………………..………58 
圖3-2 第一個例子(長方形)的原始介電常數分佈 (c) ),(yx 3 ε……………………………………………………..59 
圖3-3 第一個例子(長方形)的重建介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε………………………………………...60 
圖3-3 第一個例子(長方形)的重建介電常數分佈 (c) ),(yx 3 ε.……………………………………..………………61 
圖3-4 第二個例子(正方形)的原始介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε………………………………………...62 
圖3-4 第二個例子(正方形)的原始介電常數分佈 (c) ),(yx 3 ε……………………………………………………...63 
圖3-5 第二個例子(正方形)的重建介電常數分佈 (a) ),(yx 1 ε (b) ),(yx 2 ε………………………………………...64
圖3-5 第二個例子(正方形)的重建介電常數分佈 (c) ),(yx 3 ε………………………………………......................65 
圖3-6 第一個例子(長方形)介電常數的重建誤差對雜訊階的模擬結果……………………………………………………………….66 
圖3-7 第二個例子(正方形)介電常數的重建誤差對雜訊階的模擬結果……………………………………………………………….67

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