Inverse scattering solutions of scalar Helmholtz wave equation by a multiple source moment method

1983 ◽  
Vol 19 (4) ◽  
pp. 130 ◽  
Author(s):  
S.A. Johnson ◽  
Yoon Tae-Hoon ◽  
Ra Jung-Woong
Keyword(s):  
Wave Equation ◽  
Inverse Scattering ◽  
Moment Method ◽  
Multiple Source ◽  
Helmholtz Wave Equation

Inverse Scattering Solutions by a Sinc Basis, Multiple Source, Moment Method -- Part III: Fast Algorithms

1984 ◽  
Vol 6 (1) ◽  
pp. 103-116
Author(s):  
S. A. Johnson ◽  
Y. Zhou ◽  
M. K. Tracy ◽  
M. J. Berggren ◽  
F. Stenger
Keyword(s):  
Wave Equation ◽  
Inverse Scattering ◽  
Scattering Problem ◽  
Inverse Scattering Problem ◽  
Algebraic Equations ◽  
The Past ◽  
Nonlinear Algebraic Equations ◽  
Detector Geometry ◽  
Scattering Potential ◽  
Helmholtz Wave Equation

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.

Inverse Scattering Scheme Based on the Moment Method in the Spectral Domain, Part I: Theory

1992 ◽  
Vol 14 (1) ◽  
pp. 16-28 ◽  
Author(s):  
Se-Yun Kim ◽  
Hyun-Chul Choi ◽  
Jae-Min Lee ◽  
Jung-Woong Ra
Keyword(s):  
Inverse Scattering ◽  
Basis Function ◽  
Moment Method ◽  
Spectral Domain ◽  
Scattering Problems ◽  
Actual Mechanism ◽  
Spectral Components ◽  
Key Factor ◽  
The Moment ◽  
Measurement Location

Recently, electromagnetic and ultrasonic imaging of inhomogeneous objects by applying the moment-method procedures of forward scattering problems in the reverse sequence have been developed. In this paper, the inverse scattering formulation has been modified to be applicable in the spectral domain. Compared to previous schemes, the suggested formulation illustrates clearly the actual mechanism of the inverse scattering process by explicit separation of the contributions from several variables, such as the measurement location, basis function, and geometry of objects. The ill-posedness inherent in inverse scattering problems was also explained easily in this spectral scheme by the exponentially-decaying behavior of high-frequency spectral components of the scattered field. It implies that enlargement of the discretized cell size is a key factor in regularizing the ill-posedness. In particular, since the singular kernel to be integrated on each cell became regular in the modified scheme, various types of basis functions instead of pulse function were adopted without additional difficulties. This advantage is expected to play an important role in regularizing the noise effect by selecting polynomial basis function on the enlarged cells of discretization in the spectral inverse scattering scheme.

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