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The Closed-Form Solution to the Reconstruction of the Radiating Current for EM Inverse Scattering | IEEE Journals & Magazine | IEEE Xplore

The Closed-Form Solution to the Reconstruction of the Radiating Current for EM Inverse Scattering


Abstract:

The closed-form solution to the reconstruction of radiating current with a line measurement configuration is presented in this paper. The analytical result of the continu...Show More

Abstract:

The closed-form solution to the reconstruction of radiating current with a line measurement configuration is presented in this paper. The analytical result of the continuous singular value decomposition of the scattering integral operator is derived and used to analyze the imaging resolution and reconstruct the radiating component of the equivalent current density. The main four advantages of the proposed closed-form solution are as follows: 1) The reconstruction of radiating current can be achieved in a very short computation time; 2) it is quite tolerant to different levels of noises; 3) it is very easy to realize the reconstruction of radiating currents within the obstacles embedded in layered medium; and 4) the imaging resolution kernel of the scattering operator can also be derived in a closed form. Numerical simulations show the high efficiency of the proposed method for the reconstruction of radiating current.
Published in: IEEE Transactions on Geoscience and Remote Sensing ( Volume: 47, Issue: 1, January 2009)
Page(s): 361 - 369
Date of Publication: 25 November 2008

ISSN Information:


I. Introduction

In The Past few years, the electromagnetic (EM)inverse scattering problem, whose goal is to find the location, shape, and the dielectric properties of the obstacles in a noninvasive fashion, has been addressed by many authors [1]–[11]. The three main particular difficulties in solving an inverse EM scattering problem are considerable computational cost, nonlinearity, and ill-posedness [1], [2]. To overcome nonlinearity, a common technique is to transform the original problem (nonlinear) into an inverse source one (linear) by the introduction of an equivalent current density inside the scatterers [10]. Although the inverse scattering problem is linearized by introducing the equivalent current density, the resulting operator is still ill posed in the sense of Hadamard, and its solution is usually nonstable and nonunique due to the presence of the so-called nonradiating currents [10]. So far, some regularization strategies, such as Tikhnonov regularization, truncated singular value decomposition (TSVD), and so on, have been developed to improve solution stability in ill-posed problems [1], [2]. When these approaches are employed, one must face two challenging issues: 1) the determination of the regularization parameter, which should be determined through considerable numerical experiments and a prior information [3]–[5]; and 2) expensive computation cost of the SVD because it resorts to numerical approach. The calculation of Green's function for layered medium case is complicated and time consuming due to the slow convergence and high oscillation kernel of the Sommerfeld integral [6], [7]. Therefore, a closed-form solution appears to be very appealing. When dealing with the SVD of Jacobian matrix for the medical tomographic problem where the transmitters/receivers are located on a circle, the analytical SVD approach is proposed and analyzed by using Zernike polynomials [8]. In [8], the additive formulations of Hankel or spherical Hankel functions are employed to realize the closed-form solution of SVD of the scattering integral operator, when the receivers are set on a circle for 2-D problem (sphere for 3-D case) [9], [10]. Due to the measured data on a closed line (or surface for 3-D case) and the use of additive formulation of Hankel (or spherical Hankel for 3-D) function, the singular values are in a discrete form. Another excellent study aimed at full vectorial inverse source problem has been made by Marengo et al. [11]–[13]. The closed-form solution to the reconstruction of radiating currents when the measured data are collected on an open line, which is a common fashion of data sampling in practice, has not been reported yet.

References

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