利用粒子群聚演算法串疊牛頓法重建完全 導體

Image Reconstruction of Perfectly Conducting Objects by a Cascaded Method

李卓儒(Jhuo-Ru Li)

指導教授 ： 丘建青

淡江大學/工學院/電機工程學系碩士班/碩士(2009年)

https://doi.org/10.6846/TKU.2009.01133

摘要

本論文提ㄧ數值方法，主要目的為重建二維完全導體之影像。 於逆散射方面，將逆散射問題轉換為最佳化問題之後，省先利用改 良式粒子群聚法，得ㄧ最佳化解或可接受之解，再串疊 Newton-Kantorovitch 迭代法，以快速收歛至更精確之解。 所採之逆散射法則是基於嚴謹的數學方法，利用接收的散射場 及適當的邊界條件導出非線性積分方程組，接著應用 Newton-Kantorovitch 迭代法及動差法將此非線性方程組化成矩陣形 式，再以虛反運算法克服逆散射過程中所遭遇到的不良情況，最後 得到一組收斂而穩定的解。 不論初始的猜測值如何，改良式粒子群聚法總會收歛到整體的 極值(global extreme)，因此，在數值模擬顯示中，即使最初的猜測值與實際值相距甚遠，我們仍可求得準確的數值解，成功的重建出物體形狀函數，而以微分為基礎求取極值的方法(calculus-based method)，卻常常會陷入區域極值(local extreme)的陷阱裡。 本論文以改良式粒子群聚法所得之解，當作牛頓法之初始猜測 值。藉由改良式粒子群聚法之全域搜尋特性，以求得可接受之解， 期望此解對於區域性搜尋之牛頓法而言，可能為適當之初始猜測 值。串疊之方法比較單一改良式粒子群聚法或者單一牛頓迭代法， 其解之精確度勢必較高。在論文中以數值模擬的方法，驗證了此串 疊方法的準確性和可行性。

關鍵字

逆散射 ； 牛頓法 ； 粒子群聚最佳化法

並列摘要

In this paper, we propose a method, which combines a particle swarm optimization (PSO) algorithm with a Newton-Kantorovitch algorithm for image reconstruction of
perfectly conducting Objects. First, the inverse problem is recast as a global nonlinear optimization problem, which is solved by a PSO. Then, the solution obtained by the PSO is taken as an initial guess for the Newton Kantorovitch algorithm to obtain the more accuracy solution in a few iterations.
The inversion algorithm which is based on the rigorous mathematics makes use of the received scattered field and appropriate boundary condition to derive a set of nonlinear
integral equations. The Newton-Kantorovitch algorithm and the moment method are used to transform the nonlinear integral equations into matrix form. Then the pseudoinverse
transformation is employed to overcome the ill-posedness to obtain a convergent and stable solution.
The particle swarm optimization algorithm is employed to find out the global extreme solution of the object function. Numerical results demonstrated that, even when the initial guess is far away from the exact one, good reconstruction has been obtained. In such a case, the gradient-based methods often get trapped in a local extreme.
Numerical simulations are conducted to demonstrate that our cascaded method is accurate and practical. Numerical results show that the performance of this cascaded
method is better than the individual PSO and the individual Newton-Kantorovitch algorithm. Satisfactory reconstruction has been obtained by using this cascaded method.

並列關鍵字

Inverse scattering ； Newton-Kantorovitch method ； Particle swarm optimization

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