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 [17], 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 [17] 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：李普曼-施温格-兰佐斯算法近期被证明可为多散射环境下的合成孔径雷达数据成像与直接反演提供高效工具[17]，其中数据集仅限于单站（即单输入/单输出）测量。该方法基于降阶模型框架构建数据驱动的内部场估计，并将其代入李普曼-施温格积分方程。然而，由于缺失多输入多输出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。本文提出一种基于降阶模型的数据补全算法以缓解该问题。首先，我们如文献[17]所述对单输入单输出数据应用李普曼-施温格-兰佐斯算法，获得近似重建结果及内部场估计。随后，利用这些估计值计算前向李普曼-施温格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，我们使用原始单输入单输出数据求解李普曼-施温格方程，其中内部场由提升后的多输入多输出数据构建。对于复杂模型，可重复执行获取近似重建结果与内部场、填补缺失多输入多输出数据项的步骤以进一步提升图像质量。通过二维及2.5维数值算例验证了所提方法的有效性，重建结果得到显著改善。