At high frequencies, phase measurement is difficult to meet the accuracy required for imaging, and it is more susceptible to noise pollution. Consequently, it is important to develop inversion models and algorithms for electromagnetic inverse scattering problems with phaseless data (PD-ISPs). Compared to the ISPs with full-data (FD-ISPs), the PD-ISPs are more nonlinear due to the lack of the phase information. Aiming at solving highly nonlinear PD-ISPs, in this article we present a contraction integral equation for inversion (CIE-I) regularized with Fourier subspace via contrast source inversion method, denoted as PD-CSI-CIE-I. Under the model of CIE-I, the nonlinearity of PD-ISPs can be effectively alleviated. In order to stabilize CIE-I, a nested optimization scheme with multi-round procedure is adopted by choosing the proper number of low-frequency components from unknown induced current spanned in the Fourier subspace. By the Fourier bases expansion of induced current, the computational cost can be further reduced. Numerical and experimental examples validate the efficiency of the proposed inversion method. Futher, it is shown that the proposed PD-CSI-CIE-I has a stronger inversion capability than the one with the Lippmann-Schwinger integral equation (LSIE) when tackling high contrast scatterers with phaseless data.

中文翻译：

---

用反演的收缩积分方程求解无相高度非线性逆散射问题

在高频下，相位测量难以满足成像所需的精度，更容易受到噪声污染。因此，针对无相数据 (PD-ISP) 的电磁逆散射问题开发反演模型和算法非常重要。与全数据ISP（FD-ISP）相比，PD-ISP由于缺乏相位信息而更加非线性。针对高度非线性的 PD-ISP，本文提出了一个通过对比源反演方法用傅立叶子空间正则化的收缩积分反演方程（CIE-I），记为 PD-CSI-CIE-I。在CIE-I模型下，可以有效缓解PD-ISPs的非线性。为了稳定 CIE-I，通过从跨越傅立叶子空间的未知感应电流中选择适当数量的低频分量，采用具有多轮过程的嵌套优化方案。通过感应电流的傅里叶基数展开，可以进一步降低计算成本。数值和实验示例验证了所提出的反演方法的效率。此外，表明在处理具有无相数据的高对比度散射体时，所提出的 PD-CSI-CIE-I 具有比使用 Lippmann-Schwinger 积分方程 (LSIE) 更强的反演能力。