Physically Unrolling Network Under Contraction Integral Equation for Limited-Aperture Inverse Scattering Problem

Inverse scattering problems (ISPs) with a limited aperture have higher nonlinearity owing to the fewer measurement data, which brings more challenges in the application of full-wave quantitative imaging. To solve the ISPs with limited aperture, an unrolling algorithm of subspace-based optimization method under physical contraction integral equation subspace optimization method (CIE-SOM), named CIE-SOM-NET, is proposed. The CIE model reduces nonlinearity, enabling CIE-SOM-NET to achieve better results in addressing regression problems. The proposed algorithm is composed of several submodules, and each submodule is constructed with a small convolutional neural network (CNN) and an operator of least square method, which are utilized to update the induced current and the modified contrast function, respectively. In addition, a weighted loss function, which is composed of the consistency function of the induced current, the scattered field, and the relative permittivity, is defined to constrain the training process in the physical CIE model. Compared to traditional iterative inversion methods, the proposed CIE-SOM-NET exhibits several merits in terms of imaging accuracy and computational cost. The proposed method also achieves better stability and robustness compared to the initial SOM-NET. Several numerical examples validate that the proposed CIE-SOM-NET has excellent inversion performance in the case of limited aperture.