Johnson S A, Zhou Y, Tracy M K, Berggren M J, Stenger F
Ultrason Imaging. 1984 Jan;6(1):103-16. doi: 10.1177/016173468400600109.
olving the inverse scattering problem for the Helmholtz wave equation without employing the Born or Rytov approximations is a challenging problem, but some slow iterative methods have been proposed. One such method suggested by us is based on solving systems of nonlinear algebraic equations that are derived by applying the method of moments to a sinc basis function expansion of the fields and scattering potential. In the past, we have solved these equations for a 2-D object of n by n pixels in a time proportional to n5. In the present paper, we demonstrate a new method based on FFT convolution and the concept of backprojection which solves these equations in time proportional to n3 X log(n). Several numerical examples are given for images up to 7 by 7 pixels in size. Analogous algorithms to solve the Riccati wave equation in n3 X log(n) time are also suggested, but not verified. A method is suggested for interpolating measurements from one detector geometry to a new perturbed detector geometry whose measurement points fall on a FFT accessible, rectangular grid and thereby render many detector geometrics compatible for use by our fast methods.
在不采用玻恩近似或里托夫近似的情况下求解亥姆霍兹波动方程的逆散射问题是一个具有挑战性的问题，但已经提出了一些缓慢的迭代方法。我们提出的一种这样的方法是基于求解非线性代数方程组，这些方程组是通过将矩量法应用于场和散射势的 sinc 基函数展开而推导出来的。过去，我们求解这些方程时，对于一个 n×n 像素的二维物体，所需时间与 n^5 成正比。在本文中，我们展示了一种基于快速傅里叶变换（FFT）卷积和反投影概念的新方法，该方法求解这些方程的时间与 n^3×log(n) 成正比。针对尺寸高达 7×7 像素的图像给出了几个数值示例。还提出了在 n^3×log(n) 时间内求解里卡蒂波动方程的类似算法，但未进行验证。提出了一种方法，用于将测量值从一种探测器几何结构插值到一种新的受扰动探测器几何结构，其测量点落在一个可通过 FFT 访问的矩形网格上，从而使许多探测器几何结构适用于我们的快速方法。
