This work studies an inverse scattering problem when limited-aperture data are available that are from just one or a few incident fields. This inverse problem is highly ill-posed due to the limited receivers and a few incident fields employed. Solving inverse scattering problems with limited-aperture data is important in applications as collecting full data is often either unrealistic or too expensive. The direct sampling methods (DSMs) with full-aperture data can effectively and stably estimate the locations and geometric shapes of the unknown scatterers with a very limited number of incident waves. However, a direct application of DSMs to the case of limited receivers would face the resolution limit. To break this limitation, we propose a finite space framework with two specific schemes, and an unsupervised deep learning strategy to construct effective probing functions for the DSMs in the case with limited-aperture data. Several representative numerical experiments are carried out to illustrate and compare the performance of different proposed schemes.

翻译：本研究探讨了在仅有一个或少数几个入射场可用时，利用有限孔径数据进行逆散射问题求解。由于所使用的接收器数量有限且入射场较少，该逆问题具有高度不适定性。利用有限孔径数据求解逆散射问题在实际应用中具有重要意义，因为采集完整数据往往不切实际或成本过高。基于全孔径数据的直接采样法能够利用极少量的入射波，有效且稳定地估计未知散射体的位置与几何形状。然而，将直接采样法直接应用于有限接收器情形会面临分辨率限制。为突破此限制，我们提出了包含两种具体方案的有限空间框架，以及一种无监督深度学习策略，用于在有限孔径数据情况下为直接采样法构建有效的探测函数。本文进行了若干代表性数值实验，以展示和比较所提不同方案的性能。