A family of new integral equations (NIE) is proposed in this paper, which are transformed from the original Lippmann-Schwinger integral equation. It can be shown that the NIE can effectively reduce the nonlinearity of inverse scattering problems by reducing the global nonlinear effects, introduced by the multiple scattering behaviors, in estimating the contrast. Equipping the previously proposed twofold subspace-based optimization method with such NIE, the new inversion method is able to solve inverse scattering problems with strong scatterers, like with high contrast and/or large dimensions (in terms of wavelength) ones. Furthermore, such a family of NIE could provide a convenient tool to appraise reconstructed results. Several representative numerical tests are carried out, using both synthetic and experimental data, to verify the efficacy of the new inversion method.

The Foldy-Lax equation is used to derive a multiple scattering model for the multiple-scattering small anisotropic spheres. By this model, if the number of the non-zero singular values of the multistatic response (MSR) matrix is smaller than the number of the antennas, the range space of the MSR matrix is found to be spanned by the background Green's function vectors corresponding to the and components of the electric and magnetic dipoles induced in each scatterer, which indicates that the multiple signal classification (MUSIC) method could be implemented to obtain the locations of the scatterers. After estimating the positions of the scatterers, a non-iterative analytical method is proposed for retrieving the polarization strength tensors as well as the orientations of the principle axes of each scatterer. Two numerical simulations show that, the MUSIC method and the non-iterative method are efficacious for the nonlinear inverse scattering problem of determining the locations and polarization strength tensors of multiple-scattering small anisotropic spheres. Such methods could also be applied to the inversion of small isotropic spheres or extended to the inversion of small bianisotropic spheres.

A fast Fourier transform (FFT) twofold subspace- based optimization method (TSOM) is proposed to solve electromagnetic inverse scattering problems. As mentioned in the original TSOM (Y. Zhong, et al, Inverse Probl., vol. 25, p. 085003, 2009), one is able to efficiently obtain a meaningful coarse result by constraining the induced current to a lower-dimensional subspace during the optimization, and use this result as the initial guess of the optimization with higher-dimensional current subspace. Instead of using the singular vectors to construct the current subspace as in the original TSOM, in this paper, we use discrete Fourier bases to construct a current subspace that is a good approximation to the original current subspace spanned by singular vectors. Such an approximation avoids the computationally burdensome singular value decomposition and uses the FFT to accomplish the construction of the induced current, which reduce the computational complexity and memory demand of the algorithm compared to the original TSOM. By using the new current subspace approximation, the proposed FFT-TSOM inherits the merits of the TSOM, better stability during the inversion and better robustness against noise compared to the SOM, and meanwhile has lower computational complexity than the TSOM. Numerical tests in the two-dimensional TM case and the three-dimensional one validate the algorithm.