In this article, we have improved the quantitative inversion performance of the cross-correlated contrast source inversion (CC-CSI) method by incorporating the subspace optimization strategy. The proposed method is called the cross-correlated subspace optimization method (CC-SOM). Meanwhile, multifrequency data are used to improve the inversion performance of high-contrast scatterers, where the L-curve method is introduced to select the regularization parameters of each frequency point without relying on experience. Finally, a fast algorithm is implemented by using the property of singular value decomposition (SVD) to simplify the large-scale matrix, and the fast Fourier transform (FFT) to accelerate the calculation. Synthetic and experimental inversion results demonstrate that both CC-SOM and CC-CSI show better robustness than SOM. In comparison to CC-CSI, CC-SOM is superior in terms of inversion accuracy when the regularization parameters have been appropriately selected. However, these advantages come at the cost of higher computational complexity.

Diffraction tomography (DT) is a linear approach to solving electromagnetic inverse scattering problems. Based on the weak scattering assumptions (such as Born or Rytov approximations), the spatial spectrum of the contrast at one certain frequency is a linear mapping of the scattered field data. As is well known, using more frequencies means better performance of noise suppression. However, the spatial spectra are aliased in cases of multi-frequency data. In addition, the permittivity and the conductivity are coupled in the aliased multi-frequency spatial spectrum. In this paper, a quantitative DT for weak scatterers is proposed by firstly doing a coordinate transformation, then decoupling the permittivity and the conductivity, and finally modifying the aliased multi-frequency spatial spectrum. In doing so, the modified spatial spectra of the permittivity and conductivity are formulated respectively, leading to a quantitative diffraction tomography for weak scatterers based on aliasing modification of the multi-frequency spectrum. Inversion results with synthetic and experimental data demonstrate that the proposed method outperforms the conventional DT approach in terms of quantitative inversion accuracy for weak scatterers of multi-frequency data while maintaining the anti-noise performance.

In this article, a twofold physical model (PM) embedded 3-D U-Net with polarization fusion (PF), referred to as PUP-Net, is proposed for solving 3-D inverse scattering problems (ISPs) with a sparse planar array. To mitigate the high nonlinearity of ISPs, the twofold PM, consisting of the coherence factor back-projection algorithm (CF-BPA) and cross-correlated subspace-based optimization method (CC-SOM), first recovers the preliminary quantitative co-polarization images with high-fidelity geometry. The preliminary results are then input into the 3-D U-Net-based convolutional neural network (CNN), which outputs the fine contrast images of co-polarization (HH and VV). Finally, the upsampling PF achieves the co-polarization image fusion and generates higher accuracy and resolution results. PUP-Net follows the divide-and-conquer method: refine quantitative inversion based on good qualitative inversion. Synthetic and experimental inversion results demonstrate that PUP-Net achieves superior reconstruction accuracy and generalization ability compared with conventional iterative methods and current deep learning (DL) approaches in solving ISPs of half-space configurations.

A sparse planar array often suffers from the strong grating-lobes that deteriorate inversion performance due to spatial undersampling. In the paper, a diffraction tomography (DT) algorithm is proposed to solve three-dimensional (3-D) electromagnetic inverse scattering problems with a sparse planar array and polarization diversity. The method deals with multifrequency multipolarization data as follows: firstly, doing a polarization fusion; next, using the exact coordinate transformation; then decoupling and modifying the aliased multifrequency spatial spectrum of the permittivity and the conductivity; finally, applying the range-normalization two-dimensional coherence factor (2-D CF) weighting. The weak scatterers’ geometries and dielectric properties can be well reconstructed under low-contrast conditions, and the non-weak scatterers can also be detected. The proposed method outperforms the conventional DT approach regarding grating-lobes and noise suppression without compromising quantitative inversion performance, as demonstrated by the inverted results obtained from synthetic and experimental data.

Without phase information of the measured field data, the design complexity of the observation equipment can be greatly simplified. In the meantime, the phaseless data inverse scattering problems counter more serious nonlinearity and ill-posedness compared to the full data ones. In this paper, a multi-frequency cross-correlated contrast source inversion method using phaseless measurements of the total fields (referred to as PD-CC-CSI) is proposed. By introducing the cross-correlated error term to the cost functional, PD-CC-CSI shows better robustness against data noise and challenges in more complicated inversion scenarios. Inverted results with transversal magnetic (TM) polarized synthetic and experimental data demonstrate the advantages of PD-CC-CSI in comparison to the phaseless contrast source inversion method (PD-CSI) and its variant PD-MR-CSI. In addition, an empirical approach has been proposed to effectively determine the optimal inverted results. Since a fast algorithm can be straightforwardly applied, phaseless inversion with multi-frequency data can be done within a few minutes.

This article is devoted to solving full-wave electromagnetic inverse scattering problems (EM-ISPs), which determine the geometrical and physical properties of scatterers from the knowledge of scattered fields. Due to the intrinsic ill-posedness and nonlinearity of EM-ISPs, traditional non-iterative and iterative methods struggle to meet the requirements of high accuracy and real-time reconstruction. To overcome these issues, we propose a two-step contrast source learning approach, cascading convolutional neural networks (CNNs) into the inversion framework, to tackle 2D full-wave EM-ISPs. In the first step, a contrast source network based on the CNNs architecture takes the determined part of the contrast source as input and then outputs an estimate of the total contrast source. Then, the recovered total contrast source is directly converted into the initial contrast. In the second step, the rough initial contrast obtained beforehand is input into the U-Net for refinement. Consequently, the EM-ISPs can be quickly solved with much higher accuracy, even for high-contrast objects, almost achieving real-time imaging. Numerical examples have demonstrated that the proposed two-step contrast source learning approach is able to improve accuracy and robustness even for high-contrast scatterers. The proposed approach offers a promising avenue for advancing EM-ISPs by integrating strengths from both traditional and deep learning-based approaches, to achieve real-time quantitative microwave imaging for high-contrast objects.