本論文呈現一個在半空間中多導體形狀重建的逆散射問題之研究。在第一區分別由三個不同方向發射TM平面波照射埋藏的雙導體。經由在導體表面的邊界條件及在物體外部量測到的散射電場，我們可以推導出一組非線性的積分方程式，之後，這些散射場積分方程式透過動差法求得散射電場相關資訊，將電磁成像問題轉化為最佳化的問題。在這裡我們選擇使用傅立葉級數（Fourier series）展開及描述物體的形狀，並在逆演算法中利用改良型基因演算法（Steady state genetic algorithm）重建埋藏雙導體的形狀。只要適當的選取參數，並結合所求的散射公式，可以得到每一個世代所計算的散射場值。跟以往以微分為基礎求取極值的梯度法比較下，更容易找到全域最小值，而不易陷入區域最小值的陷阱。在模擬的結果中，不管初始的猜測值如何，改良型基因演算法總是能收斂至全域極值，甚至，初始猜測的形狀函數跟實際形狀函數相差甚鉅，以及兩個導體之間多重散射效應是非常嚴重的，依然可以很精準的重建其形狀，並且得到精確的數值解。另外，在本研究中即使加入高斯雜訊，我們可看到重建的結果是非常良好的，在雜訊準位為0.01以下時錯誤率在3%，由此可證明其雜訊容忍能力是相當好的。在本研究中，兩個導體埋藏的深度大約為八倍的波長，甚至物體的埋藏深度不同，其形狀還原的效果非常好，除此，埋藏較深的物體形狀的重建效果比另一個物體差。由此可知，埋藏越深的物體較不易得到散射場的資訊。

This paper presents an inverse scattering problem for recovering the shapes of multiple conducting cylinders with the immersed targets in a half space by genetic algorithm. Two separate perfect-conducting cylinders of unknown shapes are buried in one half space and illuminated by the transverse magnetic (TM) plane wave from the other half space. Based on the boundary condition and the measured scattered field, a set of nonlinear integral equations are derived, and the electromagnetic imaging problem is reformulated into an optimization problem. The improved steady state genetic algorithm is used to find out the global extreme solution. Numerical results are given to demonstrate the performance of the inverse algorithm. Good reconstruction can be obtained even when the initial guesses are far different from the exact shapes, and then the multiple scattered fields between two conductors are huge. In addition, the effect of Gaussian noise on reconstruction results is investigated. We find that the effect of noise is negligible for the normalized standard deviation below 0.01.

目錄
第一章	簡介………………………………………………………1
1.1節	研究動機與相關文…………………………………1
1.2節	本研究之貢獻………………………………………5
1.3節	各章內容簡述………………………………………6
第二章	多導體在半空間中的逆散射理論………………………7
2.1節  正散射的理論公式推導………………………………7
2.2節  數值方法………………………………………………11
2.2.1節	動差法於積分方程式的應用………………………11
2.2.2節	基因演算法…………………………………………12
2.2.3節  逆散射問題…………………………………………19
2.2.4節  任意形狀函數圖形及三次方仿樣函數的描述……21
第三章	數值模擬結果……………………………………………30
3.1節    Fourier series描述重建的形狀之數值模擬……30
3.2節    三次方仿樣函數描述重建的形狀之數值模擬……34
3.3節	結論…………………………………………………38
第四章	結論………………………………………………………59
附錄一   計算格林函數的方法……………………………………61
參考文獻………………………………………………………………64

圖目錄
圖2-1  二維雙導體在半空間的示意圖……………………………26
圖2-2  基因演算法之流程圖………………………………………27
圖2-3  基因演算法單點交配示意圖………………………………28
圖2-4  三次方仿樣函數描述任意形狀物體示意圖………………29
圖3-1(a) 例子1雙導體在半空間的形狀函數的重建情形。實線代表實際的形狀函數，星狀線代表第3世代猜測的形狀，虛線代表最後所重建之圖形…………………………………40
圖3-1(b) 例子1物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………41
圖3-1(c) 例子1在不同準位的雜訊干擾下對DR值的影響………42
圖3-2(a) 例子2雙導體在半空間的形狀函數的重建情形。實線代表實際的形狀函數，星狀線代表初始猜測的形狀，虛線代表最後所重建之圖形………………………………………43
圖3-2(b) 例子2物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………44
圖3-3(a) 例子3雙導體在半空間的形狀函數的重建情形。實線代表
實際的形狀函數，星狀線代表基因世代第4代猜測的形狀，虛線代表最後所重建之圖形………………………45
圖3-3(b) 例子3物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………46
圖3-4(a) 例子4雙導體在半空間的形狀函數的重建情形。實線代表實際的形狀函數，星狀線代表基因世代第21代猜測的形狀，虛線代表最後所重建之圖形……………………47
圖3-4(b) 例子4物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………48
圖3-5(a) 例子5雙導體在半空間的形狀函數的重建情形。實線代表實際的形狀函數，星狀線代表初始猜測的形狀函數，虛線代表最後所重建之形狀函數……………………………49
圖3-5(b) 例子5物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………50
圖3-6(a) 例子6雙導體在半空間的形狀函數的重建情形。實線代表實際的形狀函數，星狀線代表初始猜測的形狀函數，虛線代表最後所重建之形狀函數……………………………51
圖3-6(b) 例子6物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………52
圖3-6(c) 例子6物體在不同頻率的入射波照射下，DR值隨著入射頻率的變化情形…………………………………………53
圖3-6(d) 例子6兩個導體之間距離約為半波長的重建情形。實線為
實際的形狀函數，虛線代表用最後所重建之形狀函數...54
圖3-6(e) 例子6兩個導體之間距離約為1.5倍波長的重建情形。實線為實際的形狀函數，虛線代表用最後所重建之形狀函數………………………………………………55
圖3-6(f) 例子6重建的形狀函數偏差量隨兩個導體之間的距離變化的情形……………………………………………………56
圖 3-7(a) 例子7雙導體在半空間的形狀函數的重建情形。實線代表實際的形狀函數，星狀線代表初始猜測的形狀函數，虛線代表最後所重建之形狀函數……………………………57
圖3-7(b) 例子7物體在基因演算法中每個世代重建的DR值的變化情形………………………………………………………58       
                                                  
表目錄
表 2-1  基因演算法相關名詞解釋與中英對照表…………………25

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