Abbasi Mahdi
Department of Computer Engineering, Enginering Faculty, Bu-Ali Sina University, Hamedan, Iran.
J Med Signals Sens. 2014 Jan;4(1):43-52.
Planar D-bar integral equation is one of the inverse scattering solution methods for complex problems including inverse conductivity considered in applications such as Electrical impedance tomography (EIT). Recently two different methodologies are considered for the numerical solution of D-bar integrals equation, namely product integrals and multigrid. The first one involves high computational burden and the other one suffers from low convergence rate (CR). In this paper, a novel high speed moment method based using the sinc basis is introduced to solve the two-dimensional D-bar integral equation. In this method, all functions within D-bar integral equation are first expanded using the sinc basis functions. Then, the orthogonal properties of their products dissolve the integral operator of the D-bar equation and results a discrete convolution equation. That is, the new moment method leads to the equation solution without direct computation of the D-bar integral. The resulted discrete convolution equation maybe adapted to a suitable structure to be solved using fast Fourier transform. This allows us to reduce the order of computational complexity to as low as O (N (2)log N). Simulation results on solving D-bar equations arising in EIT problem show that the proposed method is accurate with an ultra-linear CR.
平面D - 巴积分方程是用于解决复杂问题的逆散射解法之一，这些复杂问题包括在诸如电阻抗断层成像（EIT）等应用中所考虑的逆电导率问题。最近，针对D - 巴积分方程的数值解考虑了两种不同的方法，即乘积积分法和多重网格法。第一种方法计算量很大，而另一种方法收敛速度（CR）较低。本文引入了一种基于辛克基的新型高速矩量法来求解二维D - 巴积分方程。在该方法中，D - 巴积分方程中的所有函数首先用辛克基函数展开。然后，它们乘积的正交特性消除了D - 巴方程的积分算子，得到一个离散卷积方程。也就是说，新的矩量法无需直接计算D - 巴积分就能得到方程的解。得到的离散卷积方程可以适配到合适的结构，以便使用快速傅里叶变换求解。这使我们能够将计算复杂度降低到低至O(N²log N)。在求解EIT问题中出现的D - 巴方程的仿真结果表明，所提出的方法具有超线性收敛速度且准确。
