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J. Imaging 2019, 5(2), 27; https://doi.org/10.3390/jimaging5020027

Contraction Integral Equation for Three-Dimensional Electromagnetic Inverse Scattering Problems

1,*
and
2
1
Institute of High Performance Computing, Agency for Science, Technology and Research (A*STAR), Singapore 138632, Singapore
2
Key Lab of RF Circuits and Systems of Ministry of Education, Hangzhou Dianzi University, Hangzhou 310018, China
*
Author to whom correspondence should be addressed.
Received: 31 December 2018 / Revised: 30 January 2019 / Accepted: 31 January 2019 / Published: 8 February 2019
(This article belongs to the Special Issue Microwave Imaging and Electromagnetic Inverse Scattering Problems)
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Abstract

Inverse scattering problems (ISPs) stand at the center of many important imaging applications, such as geophysical explorations, industrial non-destructive testing, bio-medical imaging, etc. Recently, a new type of contraction integral equation for inversion (CIE-I) has been proposed to tackle the two-dimensional electromagnetic ISPs, in which the usually employed Lippmann–Schwinger integral equation (LSIE) is transformed into a new form with a modified medium contrast via a contraction mapping. With the CIE-I, the multiple scattering effects, i.e., the physical reason for the nonlinearity in the ISPs, is substantially suppressed in estimating the modified contrast, without compromising physical modeling. In this paper, we firstly propose to implement this new CIE-I for the three-dimensional ISPs. With the help of the FFT type twofold subspace-based optimization method (TSOM), when handling the highly nonlinear problems with strong scatterers, those with higher contrast and/or larger dimensions (in terms of wavelengths), the performance of the inversions with CIE-I is much better than the ones with the LSIE, wherein inversions usually converge to local minima that may be far away from the solution. In addition, when handling the moderate scatterers (those the LSIE modeling can still handle), the convergence speed of the proposed method with CIE-I is much faster than the one with the LSIE. Secondly, we propose to relax the contraction mapping condition, i.e., different contraction mappings are used in updating contrast sources and contrast, and we find that the convergence can be further accelerated. Several numerical tests illustrate the aforementioned interests. View Full-Text
Keywords: inverse scattering; nonlinear problem; contraction integral equation for inversion (CIE-I); imaging inverse scattering; nonlinear problem; contraction integral equation for inversion (CIE-I); imaging
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Zhong, Y.; Xu, K. Contraction Integral Equation for Three-Dimensional Electromagnetic Inverse Scattering Problems. J. Imaging 2019, 5, 27.

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