The Lippmann--Schwinger--Lanczos (LSL) algorithm has recently been shown to provide an efficient tool for imaging and direct inversion of synthetic aperture radar data in multi-scattering environments \cite{DrMoZa3}, where the data set is limited to the monostatic, a.k.a. single input/single output (SISO) measurements. The approach is based on constructing data-driven estimates of internal fields via a reduced-order model (ROM) framework and then plugging them into the Lippmann-Schwinger integral equation. However, the approximations of the internal solutions may have more error due to missing the off diagonal elements of the multiple input/multiple output (MIMO) matrix valued transfer function. This, in turn, may result in multiple echoes in the image. Here we present a ROM-based data completion algorithm to mitigate this problem. First, we apply the LSL algorithm to the SISO data as in \cite{DrMoZa3} to obtain approximate reconstructions as well as the estimate of internal field. Next, we use these estimates to calculate a forward Lippmann-Schwinger integral to populate the missing off-diagonal data (the lifting step). Finally, to update the reconstructions, we solve the Lippmann-Schwinger equation using the original SISO data, where the internal fields are constructed from the lifted MIMO data. The steps of obtaining the approximate reconstructions and internal fields and populating the missing MIMO data entries can be repeated for complex models to improve the images even further. Efficiency of the proposed approach is demonstrated on 2D and 2.5D numerical examples, where we see reconstructions are improved substantially.

翻译：近年来，Lippmann-Schwinger-Lanczos (LSL) 算法已被证明为多散射环境下合成孔径雷达数据的成像与直接反演提供了高效工具（参见文献\cite{DrMoZa3}），其中数据集仅限于单站（即单输入/单输出，SISO）测量。该方法基于通过降阶模型（ROM）框架构建数据驱动的内部场估计，并将其代入Lippmann-Schwinger积分方程。然而，由于缺失多输入/多输出（MIMO）矩阵值传递函数的非对角元素，内部解的近似可能产生更大误差，进而导致图像中出现多重回波。本文提出一种基于ROM的数据补全算法以缓解该问题。首先，如文献\cite{DrMoZa3}所述，对SISO数据应用LSL算法以获得近似重建结果及内部场估计。接着，利用这些估计值计算前向Lippmann-Schwinger积分以填补缺失的非对角数据（提升步骤）。最后，为更新重建结果，我们使用原始SISO数据求解Lippmann-Schwinger方程，其中内部场由提升后的MIMO数据构建。对于复杂模型，可重复执行获取近似重建结果与内部场、填补缺失MIMO数据项的步骤以进一步提升图像质量。通过在二维及2.5维数值算例中的验证，本文所提方法显著改善了重建效果，证明了其有效性。