本論文就柱型導體逆散射問題做了三個面向的探討。
第一個部分，考慮在不同環境下一個未知形狀及表面可變導電係數
的非完全導體，對於非完全導體之邊界條件，可藉由表面阻抗的概念配
合導體表面感應電流的觀念，可導出非線性積分方程式，繼而利用動差
法求得正散射公式。利用正散射公式，我們可以得到散射場的相關資料。
對於逆散射部分，我們引進了基因法則(genetic algorithm)。利用基因
法則時，我們適當地選取參數形式，同時結合所求的散射公式，由此即
可求出散射場的相關資料，藉以求得柱體的形狀函數與導電係數。
第二個部分，藉由傳統的Fourier series 以及cubic-spline 描述
物體外觀形狀，探討在基因法則下，對於不同環境逆散射問題的形狀重
建的適用性，實驗顯示，利用Fourier series 描述複雜形狀往往造成無
法得到收斂解，cubic-spline 的描述方式則可在較少的變數個數，獲得
良好的形狀描述結果。
第三個部分，藉由使用改良的基因法則(NU-SSGA)與傳統GA 比較，
NU-SSGA 可藉由大幅減少計算正散射的次數，獲取遠優於傳統GA 的效
能，以往利用全域解演算法解逆散射問題最令人詬病的收斂時間，實驗
結果顯示，利用NU-SSGA 求解可獲得大幅改善。

The thesis presents three related aspects of computational approach to
the imaging of a conducting cylinder. In the first one, an imperfect
conducting cylinder of unknown shape and variable conductivity is
considered. Two different cases of inverse problem in free space and half
space have done respectively. In the second one, cubic-spline method and
trigonometric series for shape description are used and compared in several
different situations (half space, partially immersed, slab medium, and
periodic conductor in free space). In the third one, the inverse scattering
problem is addressed to discuss the CPU time for reconstructing a perfectly
conducting cylinder for two different cases (half space and slab medium). It
is solved by the improved Steady State Genetic Algorithm (SSGA) and
Simple Genetic Algorithm (SGA) and the consuming time in finding out the
global extreme solution of the objective function is compared. Based on the
boundary conditions and the measured scattered field, a set of nonlinear
integral equations is derived and the imaging problem is reformulated into an
III
optimization problem. In the first one, the genetic algorithm is employed to
find out the global extreme solution of the objective function. Numerical
results demonstrate that, even when the initial guess is far away from the
exact one, a good reconstruction has been obtained. In the second one, the
shape of the scatterer described by using cubic-spline method can be
reconstructed. In such case, Fourier series expansion will fail. Numerical
results show that the shape description by using cubic-spline method is much
better than that Fourier series. In the third one, it is found that the searching
ability of SSGA is much powerful than that of the SGA. The consuming time
for converging to a global extreme solution by using SSGA is much less than
that SGA. Numerical results show that the image reconstruction problem by
using SSGA is much better than by SGA in time consuming.

第一章 簡介.……………………………...……………..….………………1
1.1 節 研究動機與相關文獻.………….……………..………………1
1.2 節 本研究之貢獻..………………………………..………………4
1.3 節 各章內容簡述…...……………………………………………..6
第二章 不完全導體之逆散射問題..………...…………….……………….6
2.1 節 自由空間理論推導與數值方法…………………………...6
2.2 節 半空間理論推導與數值方法……………………………10
2.3 節 基因演算法則……………………………………………13
2.4 節 模擬結果……………………………………………....22
第三章 導體之形狀函數適應性問題…………..……………………….46
3.1 節 各種空間理論推導與數值方法………………………...46
3.2 節 任意形狀函數圖形及三次方仿樣函數的描述.…………...62
3.3 節 模擬結果..........…………………………………………65
VI
第四章 導體逆散射之演算法最佳化問題……....……………....………101
4.1 節NU-SSGA……………………………………………………101
4.2 節 模擬結果……………………………………………………102
第五章 結論…………………….………………………………………119
附錄一 計算半空間格林函數收斂的方法……………………………121
附錄二 計算三層結構格林函數收斂的方法…………………………124
附錄三 計算半掩埋結構格林函數收斂的方法.………………………129
附錄四 計算週期性結構格林函數收斂的方法….……………………133
附錄五 當y ≈ y' 時改進週期性格林函數收斂的方法….………………135
參考文獻…..…………………………………………………………….136
圖目錄
圖2.1 導電率為 之二維不完全導體在 平面上的示意圖。………………………………………………………………29
圖2.2 埋藏在半空間導電率為 之二維不完全導體在 平面上的示意圖。……………………………………………………… 30
圖2.3 基因法則流程圖。…………………………………………………31
圖2.4(a) 例子1自由空間的形狀函數的重建情形。星狀線代表真正的形狀函數，其他類型的線條則代表不同的基因世代所計算出的形狀函數。…………………………………………………32
圖2.4(b) 例子1自由空間的導電率函數的重建情形。星狀線代表真正的導電率函數，其他類型的線條則代表不同的基因世代所計算出的導電率函數。…………………………………………33
圖2.4(c) 例子1自由空間的形狀函數及導電率函數隨相對雜訊位準變化的情形。……………………………………………………34
圖2.5(a) 例子2自由空間的形狀函數的重建情形。星狀線代表真正的形狀函數，其他類型的線條則代表不同的基因世代所計算出的形狀函數。…………………………………………………35
圖2.5(b) 例子2自由空間的導電率函數的重建情形。星狀線代表真正的導電率函數，其他類型的線條則代表不同的基因世代所計算出的導電率函數。…………………………………………36
圖2.6(a) 例子3自由空間的形狀函數的重建情形。星狀線代表真正的形狀函數，其他類型的線條則代表不同的基因世代所計算出的形狀函數。…………………………………………………37
圖2.6(b) 例子3的自由空間導電率函數的重建情形。星狀線代表真正的導電率函數，其他類型的線條則代表不同的基因世代所計算出的導電率函數。…………………………………………38
圖2.7(a) 例子4半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，其他類型的線條則代表不同的基因世代所計算出的形狀函數。……………………………………………………39
圖2.7(b) 例子4半空間的導電率函數的重建情形。星狀線代表真正的導電率函數，其他類型的線條則代表不同的基因世代所計算出的導電率函數。……………………………………………40
圖2.7(c) 例子4半空間的形狀函數及導電率函數隨相對雜訊位準變化的情形。………………………………………………………41
圖2.8(a) 例子5半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，其他類型的線條則代表不同的基因世代所計算出的形狀函數。……………………………………………………42
圖2.8(b) 例子5半空間的導電率函數的重建情形。星狀線代表真正的導電率函數，其他類型的線條則代表不同的基因世代所計算出的導電率函數。……………………………………………43
圖2.9(a) 例子6半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，其他類型的線條則代表不同的基因世代所計算出的形狀函數。……………………………………………………44
圖2.9(b) 例子6半空間的導電率函數的重建情形。星狀線代表真正的導電率函數，其他類型的線條則代表不同的基因世代所計算出的導電率函數。…………………………………………… 45
圖3.1 埋藏在半空間之二維完全導體在 平面上的示意圖。……72
圖3.2 埋藏在三層結構之二維完全導體在 平面上的示意圖。………………………………………………………………73
圖3.3 線電流源位於區域一時的示意圖。………………………………74
圖3.4 線電流源位於區域二時的示意圖。………………………………75
圖3.5(a) 當a>0時，部份埋藏之二維完全導體在 平面上的示意圖。……………………………………………………………76
圖3.5(b) 當a<0時，部份埋藏之二維完全導體在 平面上的示意圖。………………………….…………………………………77
圖3.6 週期長度d之二維週期性完全導體在 平面上的示意圖。………………………………………………………………78
圖3.7 任意形狀結構示意圖。……………………………………………79
圖3.8(a) 例子1半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。……………80
圖3.8(b) 例子1半空間的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。…………………………81
圖3.9(a) 例子2半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。……………82
圖3.9(b) 例子2半空間的形狀函數隨相對雜訊位準變化的情形，S-S表示利用cubic-spline展開法表示欲估測圖形，也用cubic-spline展開法做重建。………………………………83
圖3.10(a) 例子3半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。………84
圖3.10(b) 例子3半空間的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。……………………85
圖3.11(a) 例子4三層結構的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。……86
圖3.11(b) 例子4三層結構的形狀函數隨相對雜訊位準變化的情形，S-S表示利用cubic-spline展開法表示欲估測圖形，也用cubic-spline展開法做重建。……………………………87
圖3.12(a) 例子5三層結構的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。……88
圖3.12(b) 例子5三層結構的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。……………………89
圖3.13 例子6三層結構的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用7項Fourier展開法所重建之圖形，長虛線代表11項Fourier展開法所重建之圖形，實線代表15項Fourier展開法所重建之圖形，點狀線代表cubic-spline展開法所重建之圖形。………………………………………………90
圖3.14(a) 例子7半掩埋結構的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。91
圖3.14(b) 例子7半掩埋結構的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。…………………92
圖3.15(a) 例子8半掩埋結構的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用Fourier展開法所重建之圖形，長虛線代表用cubic-spline展開法所重建的圖形。……93
圖3.15(b) 例子8半掩埋結構的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。…………………94
圖3.16 例子9半掩埋結構的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用7項Fourier展開法所重建之圖形，長虛線代表11項Fourier展開法所重建之圖形，實線代表15項Fourier展開法所重建之圖形，點狀線代表cubic-spline展開法所重建之圖形。…………………………………………95
圖3.17(a) 例子10週期性結構的形狀函數的重建情形。星狀線代表真正的形狀函數，長虛線代表用Fourier展開法所重建之圖形，短虛線代表用cubic-spline展開法所重建的圖形。…96
圖3.17(b) 例子10半掩埋結構的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。…………97
圖3.18(a) 例子11半掩埋結構的形狀函數的重建情形。星狀線代表真正的形狀函數，長虛線代表用Fourier展開法所重建之圖形，短虛線代表用cubic-spline展開法所重建的圖形。…98
圖3.18(b) 例子11半掩埋結構的形狀函數隨相對雜訊位準變化的情形，F-F表示利用Fourier展開法表示欲估測圖形，也用Fourier展開法做重建，F-S表示利用Fourier展開法表示欲估測圖形，而用cubic-spline展開法做重建。………99
圖3.19 例子12半掩埋結構的形狀函數的重建情形。星狀線代表真正的形狀函數，實線代表用7項Fourier展開法所重建之圖形，短虛線代表11項Fourier展開法所重建之圖形，長虛線代表15項Fourier展開法所重建之圖形，點狀線代表cubic-spline展開法所重建之圖形。……………………………………… 100
圖4.1 單點交配示意圖。………………………………………………107
圖4.2 Beta分布的機率密度函數。……………………………………108
圖4.3(a) 例子1半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用SGA在經過6000次function call後所重建之圖形，長虛線代表用NU-SSGA在經過6000次function call後所重建的圖形。…………………………109
圖4.3(b) 例子1半空間的形狀誤差函數與function call的關係。虛線代表SGA的收斂情形，實線代表用NU-SSGA的收斂情形。……………………………………………………………110
圖4.3(c) 例子1半空間的形狀誤差函數與function call在相對雜訊強度為 情況下的關係。虛線代表SGA的收斂情形，實線代表用NU-SSGA的收斂情形。………………………………111
圖4.4(a) 例子2半空間的形狀函數的重建情形。星狀線代表真正的形狀函數，短虛線代表用SGA在經過20000次function call後所重建之圖形，長虛線代表用NU-SSGA在經過20000次function call後所重建的圖形。…………………………112
圖4.4(b) 例子2半空間的形狀誤差函數與function call的關係。虛線代表SGA的收斂情形，實線代表用NU-SSGA的收斂情形。……………………………………………………………113
圖4.5(a) 例子3三層結構的形狀函數的重建情形。星狀線代表真正的形狀函數，點狀線代表用SGA在經過6000次function call後所重建之圖形，實線代表用交配率為0.1之NU-SSGA在經過6000次function call後所重建的圖形。……………………………………………………………114
圖4.5(b) 例子3的形狀誤差函數與function call的關係。點虛線代表SGA的收斂情形，星虛線代表用交配率為0.2之NU-SSGA的收斂情形，實線代表用交配率為0.1之NU-SSGA的收斂情形，虛線代表用交配率為0.2之NU-SSGA的收斂情形。……………………………………………………………115
圖4.5(c) 例子3的形狀誤差函數與function call在相對雜訊強度為 情況下的關係。點虛線代表SGA的收斂情形，星虛線代表用交配率為0.2之NU-SSGA的收斂情形，實線代表用交配率為0.1之NU-SSGA的收斂情形，虛線代表用交配率為0.2之NU-SSGA的收斂情形。……………………………………116
圖4.6(a) 例子4的形狀函數的重建情形。星狀線代表真正的形狀函數，點狀線代表用SGA在經過6000次function call後所重建之圖形，實線代表用交配率為0.1之NU-SSGA在經過6000次function call後所重建的圖形。…………………………117
圖4.6(b) 例子4的形狀誤差函數與function call的關係。點虛線代表SGA的收斂情形，星虛線代表用交配率為0.2之NU-SSGA的收斂情形，實線代表用交配率為0.1之NU-SSGA的收斂情形，虛線代表用交配率為0.2之NU-SSGA的收斂情形。……………………………………………………………118
表目錄
表2.1 基因演算法相關名詞解釋與中英對照表……………………… 15

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