Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part III: Fast Algorithms
Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part III: Fast Algorithms
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Sinc 基、多源、矩量法的逆散射解 — 第三部分：快速算法
DOI：
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
Frank Stenger
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文献类型：
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作者：
S. A. Johnson;Y. Zhou;M. Tracy;M. Berggren;Frank Stenger
求解Helmholtz波动方程的逆散射问题而不采用Born或Rytov近似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系统的非线性代数方程组，该方程组是通过将矩量法应用于场和散射势的sinc基函数展开而得到的。在过去，我们已经在与n5成比例的时间内解决了n × n像素的2-D对象的这些方程。在本文中，我们展示了一种新的方法的基础上FFT卷积和反投影的概念，解决这些方程的时间成比例的n3 · log（n）。几个数值例子给出的图像高达7 × 7像素的大小。文中还提出了在n3 · log（n）时间内求解Riccati波动方程的类似算法，但未得到验证。一种方法建议从一个探测器的几何形状的测量插值到一个新的扰动探测器的几何形状，其测量点落在一个FFT访问，矩形网格，从而使许多探测器的几何形状兼容使用我们的快速方法。
Solving 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 • 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 • 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.