ROM inversion of monostatic data lifted to full MIMO

TL;DR

The paper addresses SAR imaging with monostatic measurements, where missing off-diagonal MIMO information degrades image quality. It introduces a data-completion pipeline that lifts SISO ROM-based internal fields to a surrogate full MIMO transfer, enabling a more accurate MIMO ROM and improved reconstruction via the Lippmann-Schwinger equation. The approach comprises: (i) a SISO LSL stage to obtain approximate internal fields and $q$, (ii) a forward lifting step to fill off-diagonal data and construct a MIMO mass matrix, (iii) a MIMO LSL stage to produce improved internal fields, and (iv) optional iterations to refine the image for complex scenes. Numerical experiments in 2D and 2.5D demonstrate substantial image sharpening and ghost-artifact reduction, with robustness to moderate noise; the method offers a computationally efficient path to sharper SAR images in multi-scattering environments.

Abstract

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.

Figures (8)

• Figure 7.1: True model for imaging of two targets in homogeneous background. Red crosses show the locations of SAR sources/receivers.
• Figure 7.2: While the background solution (top middle) doesn't capture any reflections of the true solution (top left), the LSL solution without data completion captures them in spherical averages only (top right). The data completion step improved the internal solution reconstruction (bottom left) significantly. The second iteration (bottom right) sharpened the profile, though all of the reflections were already captured during the first iteration.
• Figure 7.3: 2D example. LSL solution without data completion (bottom left) improves upon the Born solution (top right) . One step of LSL data completion (bottom right) allowed us to improve the image significantly. The second iteration didn't improve the image for this simple model.
• Figure 7.4: 2D example: Red crosses in the true model (top left) show the locations of SAR sources/receivers. LSL solution without data completion (middle left) improves upon the Born solution (top right) . One step of LSL data completion (middle right) allowed us to resolve the hidden object. Adding 5% noise didn't worsen the image (bottom) significantly.
• Figure 7.5: 2.5D example: Red crosses in the true model (top left) show the locations of SAR sources/receivers. LSL solution without data completion (bottom left) , Born solution (top right). One step of LSL data completion (bottom right) yields a dramatic improvement in the image.