本論文呈現利用粒子群聚最佳化演算法重建半空間中埋藏完全導體的逆散射問題。在第一區分別由三個不同方向發射之TM平面波照射埋藏於第二區中之導體。經由在導體表面之邊界條件及在第一區所量測到的散射電場，我們可推導出一組非線性積分方程式。接著，透過動差法我們可求得正散射公式。利用正散射公式，我們可以得到散射電場的相關資訊。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粒子群聚最佳化演算法(Particle Swarm Optimization)和改良式粒子群聚最佳化法(Modified particle swarm optimization)來進行埋藏導體之模擬重建，重建埋藏導體的形狀。
   PSO的基本演算模式如下：先以均勻分佈，隨機產生初始粒子群，每一個粒子都是一個求解問題的候選解，粒子群會參考個體的最佳經驗，以及群體的最佳經驗，選擇修正的方式，經過不斷的修正之後，粒子群會漸漸接近最佳解。
    透過傅立葉級數(Fourier series)展開描述物體形狀及適當的選取演算法中的參數形式，同時結合所求的散射公式，我們可以得到每一代所計算的散射場值，並利用這些散射場值的相關資訊，將電磁成像問題轉化為最佳化問題，藉由粒子群聚最佳化演算法進行重建，求得埋藏導體之形狀。
    在模擬的結果中我們可以發現，不論是利用粒子群聚最佳化法或改良式粒子群聚最佳化法來進行重建，不管初始的猜測值如何，粒子群聚最佳化演算法總是能收斂至整體的極值，即使初始猜測的形狀函數與實際形狀函數相距甚遠，我們仍可求得準確的數值解，成功的重建出物體形狀，而當量測的散射場值中有雜訊存在時，透過數值模擬仍顯示，我們依然可以得到良好的重建結果。

This paper presents an inverse scattering problem for recovering the shape ofconducting cylinders in a half space by Particle Swarm Optimization. A perfect-conducting cylinder of unknown shape is buried in one half space and illuminated by the transverse magnetic (TM) plane wave from the other 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. Here we choose Fourier series to describe the shape of object. The particle swarm optimization and the modified particle swarm optimization are used to find out the global extreme solution.
  The basic process of PSO is as follows: firsta population of individuals defined as random guesses at the problem solutions is initialized. These individuals are candidate solutions, also known as the particles. The particles iteratively evaluate the fitness of the particles and remember the location where they had their best success. The particle's best solution is called the pbest and that of the best particle in the swarm, called gbest. Movements through the search space are guided by these successes and after generations,the PSO can find the best solution according to the best solution experience.
  Numerical results are given to demonstrate the performance of the inverse algorithm. Good reconstruction can be obtained even when the initial guess is far different from the exact shape. In addition, the effect of Guassian noise on reconstruction results is investigated and through the numerical simulation shows that we can still get good results of reconstruction.

目錄
中文摘要................................................III
英文摘要................................................IV
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......................................5
1.3節 各章內容簡述......................................5
第二章 完全導體之逆散射問題.............................6
2.1節 理論公式推導與數值方法............................6
2.1.1節 正散射的理論公式推導............................6
2.1.2節 動差法於積分方程式的應用........................10
2.2節 粒子群聚最佳化法(Particle Swarm Optimization).....11
2.3節 改良式粒子群聚最佳化法(Modified Particle Swarm Optimization)...........................................17
第三章 數值模擬結果.....................................21
3.1節 粒子群聚最佳化法重建之模擬數值結果................21
3.2節 改良式粒子群聚最佳化法重建之模擬數值結果..........32
第四章 結論.............................................43
附錄一 計算半空間格林函數的方法.........................45
參考文獻................................................48

圖目錄
圖2-1 二維導體在半空間的示意圖..........................7
圖2-2 粒子群聚最佳化法流程圖............................13
圖2-3 粒子群聚法中位置與速度更新示意圖..................14
圖2-4 在粒子群聚最佳化法中，三種不同邊界條件之示意圖....16
圖2-5 改良式粒子群聚法流程圖............................20
圖3-1埋藏在半空間之二維導體在 平面上的示意圖............22
圖3-2(a) 例子1完全導體在半空間中形狀函數的重建情形......24
圖3-2(b) 例子1物體在粒子群聚最佳化法中每個世代重建的目標函數值之變化情形..........................................25
圖3-2(c) 例子1物體在粒子群聚最佳化法中每個世代重建的DR值之變化情形................................................25
圖3-2(d) 例子1物體針對不同雜訊準位重建時之DR值變化情形..26
圖3-3(a) 例子2完全導體在半空間中形狀函數的重建情形......27
圖3-3(b) 例子2物體在粒子群聚最佳化法中每個世代重建的目標函數值之變化情形..........................................28
圖3-3(c) 例子2物體在粒子群聚最佳化法中每個世代重建的DR值之變化情形................................................28
圖3-3(d) 例子2物體針對不同雜訊準位重建時之DR值變化情形..29
圖3-4(a) 例子3完全導體在半空間中形狀函數的重建情形......30
圖3-4(b) 例子3物體在粒子群聚最佳化法中每個世代重建的目標函數值之變化情形..........................................31
圖3-4(c) 例子3物體在粒子群聚最佳化法中每個世代重建的DR值之變化情形................................................31
圖3-4(d) 例子3物體針對不同雜訊準位重建時之DR值變化情形..32
圖3-5(a) 例子1完全導體在半空間中形狀函數的重建情形......34
圖3-5(b) 例子1物體在改良式粒子群聚最佳化法中每個世代重建的目標函數值之變化情形....................................35
圖3-5(c) 例子1物體在改良式粒子群聚最佳化法中每個世代重建的DR值之變化情形..........................................35
圖3-5(d) 例子1物體針對不同雜訊準位重建時之DR值變化情形..36
圖3-6(a) 例子2完全導體在半空間中形狀函數的重建情形......37
圖3-6(b) 例子2物體在改良式粒子群聚最佳化法中每個世代重建的目標函數值之變化情形....................................38
圖3-6(c) 例子2物體在改良式粒子群聚最佳化法中每個世代重建的DR值之變化情形..........................................38
圖3-6(d) 例子2物體針對不同雜訊準位重建時之DR值變化情形..39
圖3-7(a) 例子3完全導體在半空間中形狀函數的重建情形(原始圖).....................................................40
圖3-7(b) 例子3完全導體在半空間中形狀函數的重建情形(放大圖).....................................................41
圖3-7(c) 例子3物體在改良式粒子群聚最佳化法中每個世代重建的目標函數值之變化情形....................................41
圖3-7(d) 例子3物體在改良式粒子群聚最佳化法中每個世代重建的DR值之變化情形..........................................42
圖3-7(e) 例子3物體針對不同雜訊準位重建時之DR值變化情形..42

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